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Most recent proofs    These are the 100 (Unicode, GIF) or 1000 (Unicode, GIF) most recent proofs in the set.mm database for the Metamath Proof Explorer (and the Hilbert Space Explorer). The set.mm database is maintained on GitHub with master (stable) and develop (development) versions. This page was created from develop commit 212ee7d9, also available here: set.mm (43MB) or set.mm.bz2 (compressed, 13MB).

The original proofs of theorems with recently shortened proofs can often be found by appending "OLD" to the theorem name, for example 19.43OLD for 19.43. The "OLD" versions are usually deleted after a year.

Other links    Email: Norm Megill.    Mailing list: Metamath Google Group Updated 7-Dec-2021 .    Contributing: How can I contribute to Metamath?    Syndication: RSS feed (courtesy of Dan Getz)    Related wikis: Ghilbert site; Ghilbert Google Group.

Recent news items    (7-Aug-2021) Version 0.198 of the metamath program fixes a bug in "write source ... /rewrap" that prevented end-of-sentence punctuation from appearing in column 79, causing some rewrapped lines to be shorter than necessary. Because this affects about 2000 lines in set.mm, you should use version 0.198 or later for rewrapping before submitting to GitHub.

(7-May-2021) Mario Carneiro has written a Metamath verifier in Lean.

(5-May-2021) Marnix Klooster has written a Metamath verifier in Zig.

(24-Mar-2021) Metamath was mentioned in a couple of articles about OpenAI: Researchers find that large language models struggle with math and What Is GPT-F?.

(26-Dec-2020) Version 0.194 of the metamath program adds the keyword "htmlexturl" to the $t comment to specify external versions of theorem pages. This keyward has been added to set.mm, and you must update your local copy of set.mm for "verify markup" to pass with the new program version.

(19-Dec-2020) Aleksandr A. Adamov has translated the Wikipedia Metamath page into Russian.

(19-Nov-2020) Eric Schmidt's checkmm.cpp was used as a test case for C'est, "a non-standard version of the C++20 standard library, with enhanced support for compile-time evaluation." See C++20 Compile-time Metamath Proof Verification using C'est.

(10-Nov-2020) Filip Cernatescu has updated the XPuzzle (Android app) to version 1.2. XPuzzle is a puzzle with math formulas derived from the Metamath system. At the bottom of the web page is a link to the Google Play Store, where the app can be found.

(7-Nov-2020) Richard Penner created a cross-reference guide between Frege's logic notation and the notation used by set.mm.

(4-Sep-2020) Version 0.192 of the metamath program adds the qualifier '/extract' to 'write source'. See 'help write source' and also this Google Group post.

(23-Aug-2020) Version 0.188 of the metamath program adds keywords Conclusion, Fact, Introduction, Paragraph, Scolia, Scolion, Subsection, and Table to bibliographic references. See 'help write bibliography' for the complete current list.

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Last updated on 9-Sep-2025 at 5:21 AM ET.
Recent Additions to the Metamath Proof Explorer   Notes (last updated 7-Dec-2020 )
DateLabelDescription
Theorem
 
3-Sep-2025sbco4 2102 Two ways of exchanging two variables. Both sides of the biconditional exchange 𝑥 and 𝑦, either via two temporary variables 𝑢 and 𝑣, or a single temporary 𝑤. (Contributed by Jim Kingdon, 25-Sep-2018.) Avoid ax-11 2158. (Revised by SN, 3-Sep-2025.)
([𝑦 / 𝑢][𝑥 / 𝑣][𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
 
3-Sep-2025sbco4lem 2101 Lemma for sbco4 2102. It replaces the temporary variable 𝑣 with another temporary variable 𝑤. (Contributed by Jim Kingdon, 26-Sep-2018.) (Proof shortened by Wolf Lammen, 12-Oct-2024.) Avoid ax-11 2158. (Revised by SN, 3-Sep-2025.)
([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
 
2-Sep-2025nfa1w 42623 Replace ax-10 2141 in nfa1 2152 with a substitution hypothesis. (Contributed by SN, 2-Sep-2025.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   Ⅎ𝑥∀𝑥𝜑
 
2-Sep-2025tan3rdpi 42331 The tangent of π / 3 is √3. (Contributed by SN, 2-Sep-2025.)
(tan‘(π / 3)) = (√‘3)
 
2-Sep-2025tanhalfpim 42330 The tangent of π / 2 minus a number is the cotangent, here represented by cos𝐴 / sin𝐴. (Contributed by SN, 2-Sep-2025.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → (sin‘𝐴) ≠ 0)    ⇒   (𝜑 → (tan‘((π / 2) − 𝐴)) = ((cos‘𝐴) / (sin‘𝐴)))
 
2-Sep-2025tan4thpi 26566 The tangent of π / 4. (Contributed by Mario Carneiro, 5-Apr-2015.) (Proof shortened by SN, 2-Sep-2025.)
(tan‘(π / 4)) = 1
 
1-Sep-2025iineq12dv 44998 Equality deduction for indexed intersection. (Contributed by Glauco Siliprandi, 26-Jun-2021.) Remove DV conditions. (Revised by GG, 1-Sep-2025.)
(𝜑 → 𝐴 = 𝐵)    &   ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 = 𝐷)    ⇒   (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷)
 
1-Sep-2025iuneq1i 44977 Equality theorem for indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.) Remove DV conditions. (Revised by GG, 1-Sep-2025.)
𝐴 = 𝐵    ⇒   ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶
 
1-Sep-2025cbvditgvw2 36207 Change bound variable and domain in a directed integral, using implicit substitution. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷    &   (𝑥 = 𝑦 → 𝐸 = 𝐹)    ⇒   ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐹 d𝑦
 
1-Sep-2025cbvprodvw2 36205 Change bound variable and the set of integers in a product, using implicit substitution. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   (𝑗 = 𝑘 → 𝐶 = 𝐷)    ⇒   ∏𝑗 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐷
 
1-Sep-2025cbvsumvw2 36204 Change bound variable and the set of integers in a sum, using implicit substitution. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   (𝑗 = 𝑘 → 𝐶 = 𝐷)    ⇒   Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷
 
1-Sep-2025cbvcsbvw2 36189 Change bound variable of a proper substitution into a class using implicit substitution. General version of cbvcsbv 3933. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   (𝑥 = 𝑦 → 𝐶 = 𝐷)    ⇒   ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌𝐷
 
1-Sep-2025cbvsbcvw2 36188 Change bound variable of a class substitution using implicit substitution. General version of cbvsbcvw 3839. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑦]𝜓)
 
1-Sep-2025ditgeq3sdv 36181 Equality theorem for the directed integral. Deduction form. (Contributed by GG, 1-Sep-2025.)
(𝜑 → 𝐶 = 𝐷)    ⇒   (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ⨜[𝐴 → 𝐵]𝐷 d𝑥)
 
1-Sep-2025ditgeq12d 36180 Equality theorem for the directed integral. Deduction form. (Contributed by GG, 1-Sep-2025.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → 𝐶 = 𝐷)    ⇒   (𝜑 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐸 d𝑥)
 
1-Sep-2025ditgeq123dv 36179 Equality theorem for the directed integral. Deduction form. General version of ditgeq3sdv 36181. (Contributed by GG, 1-Sep-2025.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → 𝐶 = 𝐷)    &   (𝜑 → 𝐸 = 𝐹)    ⇒   (𝜑 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐹 d𝑥)
 
1-Sep-2025itgeq2sdv 36178 Equality theorem for an integral. Deduction form. (Contributed by GG, 1-Sep-2025.)
(𝜑 → 𝐵 = 𝐶)    ⇒   (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥)
 
1-Sep-2025itgeq12sdv 36177 Equality theorem for an integral. Deduction form. General version of itgeq1d 45868 and itgeq2sdv 36178. (Contributed by GG, 1-Sep-2025.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → 𝐶 = 𝐷)    ⇒   (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
 
1-Sep-2025prodeq12sdv 36176 Equality deduction for product. General version of prodeq2sdv 15965. (Contributed by GG, 1-Sep-2025.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → 𝐶 = 𝐷)    ⇒   (𝜑 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐷)
 
1-Sep-2025sumeq12sdv 36175 Equality deduction for sum. General version of sumeq2sdv 15745. (Contributed by GG, 1-Sep-2025.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → 𝐶 = 𝐷)    ⇒   (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷)
 
1-Sep-2025ixpeq12dv 36174 Equality theorem for infinite Cartesian product. Deduction version. (Contributed by GG, 1-Sep-2025.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → 𝐶 = 𝐷)    ⇒   (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐷)
 
1-Sep-2025disjeq12dv 36173 Equality theorem for disjoint collection. Deduction version. (Contributed by GG, 1-Sep-2025.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → 𝐶 = 𝐷)    ⇒   (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷))
 
1-Sep-2025sbequbidv 36172 Deduction substituting both sides of a biconditional. (Contributed by GG, 1-Sep-2025.)
(𝜑 → 𝑢 = 𝑣)    &   (𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → ([𝑢 / 𝑥]𝜓 ↔ [𝑣 / 𝑥]𝜒))
 
1-Sep-2025reueqbidv 36171 Formula-building rule for restricted existential uniqueness quantifier. Deduction form. General version of reubidv 3406. (Contributed by GG, 1-Sep-2025.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒))
 
1-Sep-2025reueqdv 36170 Formula-building rule for restricted existential uniqueness quantifier. Deduction form. (Contributed by GG, 1-Sep-2025.)
(𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜓))
 
1-Sep-2025rmoeqbidv 36169 Formula-building rule for restricted at-most-one quantifier. Deduction form. General version of rmobidv 3405. (Contributed by GG, 1-Sep-2025.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜒))
 
1-Sep-2025rmoeqdv 36168 Formula-building rule for restricted at-most-one quantifier. Deduction form. (Contributed by GG, 1-Sep-2025.)
(𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓))
 
1-Sep-2025ditgeq3i 36167 Equality inference for the directed integral. (Contributed by GG, 1-Sep-2025.)
𝐶 = 𝐷    ⇒   ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ⨜[𝐴 → 𝐵]𝐷 d𝑥
 
1-Sep-2025ditgeq12i 36166 Equality inference for the directed integral. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷    ⇒   ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐸 d𝑥
 
1-Sep-2025ditgeq123i 36165 Equality inference for the directed integral. General version of ditgeq12i 36166 and ditgeq3i 36167. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷    &   𝐸 = 𝐹    ⇒   ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐹 d𝑥
 
1-Sep-2025itgeq2i 36164 Equality inference for an integral. (Contributed by GG, 1-Sep-2025.)
𝐵 = 𝐶    ⇒   ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥
 
1-Sep-2025itgeq1i 36163 Equality inference for an integral. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    ⇒   ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥
 
1-Sep-2025itgeq12i 36162 Equality inference for an integral. General version of itgeq1i 36163 and itgeq2i 36164. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷    ⇒   ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥
 
1-Sep-2025prodeq12si 36161 Equality inference for product. General version of prodeq2si 36160. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷    ⇒   ∏𝑥 ∈ 𝐴 𝐶 = ∏𝑥 ∈ 𝐵 𝐷
 
1-Sep-2025prodeq2si 36160 Equality inference for product. (Contributed by GG, 1-Sep-2025.)
𝐵 = 𝐶    ⇒   ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶
 
1-Sep-2025sumeq12si 36159 Equality inference for sum. General version of sumeq2si 36158. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷    ⇒   Σ𝑥 ∈ 𝐴 𝐶 = Σ𝑥 ∈ 𝐵 𝐷
 
1-Sep-2025sumeq2si 36158 Equality inference for sum. (Contributed by GG, 1-Sep-2025.)
𝐵 = 𝐶    ⇒   Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶
 
1-Sep-2025ixpeq12i 36157 Equality inference for infinite Cartesian product. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷    ⇒   X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐷
 
1-Sep-2025ixpeq1i 36156 Equality inference for infinite Cartesian product. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    ⇒   X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶
 
1-Sep-2025riotaeqi 36155 Equal domains yield equal restricted iotas. Inference version. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    ⇒   (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜑)
 
1-Sep-2025riotaeqbii 36154 Equivalent wff's and equal domains yield equal restricted iotas. Inference version. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   (𝜑 ↔ 𝜓)    ⇒   (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐵 𝜓)
 
1-Sep-2025iineq12i 36153 Equality theorem for indexed intersection. Inference version. General version of iineq1i 36152. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷    ⇒   ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐷
 
1-Sep-2025iineq1i 36152 Equality theorem for indexed intersection. Inference version. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    ⇒   ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑥 ∈ 𝐵 𝐶
 
1-Sep-2025iuneq12i 36151 Equality theorem for indexed union. Inference version. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷    ⇒   ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷
 
1-Sep-2025rabeqbii 36150 Equality theorem for restricted class abstractions. Inference version. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   (𝜑 ↔ 𝜓)    ⇒   {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∈ 𝐵 ∣ 𝜓}
 
1-Sep-2025disjeq12i 36149 Equality theorem for disjoint collection. Inference version. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷    ⇒   (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷)
 
1-Sep-2025disjeq1i 36148 Equality theorem for disjoint collection. Inference version. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    ⇒   (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶)
 
1-Sep-2025sbceqbii 36147 Formula-building inference for class substitution. General version of sbcbii 3865. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   (𝜑 ↔ 𝜓)    ⇒   ([𝐴 / 𝑥]𝜑 ↔ [𝐵 / 𝑥]𝜓)
 
1-Sep-2025reueqbii 36146 Equality inference for restricted existential uniqueness quantifier. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   (𝜓 ↔ 𝜒)    ⇒   (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒)
 
1-Sep-2025reueqi 36145 Equality inference for restricted existential uniqueness quantifier. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    ⇒   (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜓)
 
1-Sep-2025rmoeqbii 36144 Equality inference for restricted at-most-one quantifier. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   (𝜓 ↔ 𝜒)    ⇒   (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜒)
 
1-Sep-2025rmoeqi 36143 Equality inference for restricted at-most-one quantifier. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    ⇒   (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥 ∈ 𝐵 𝜓)
 
1-Sep-2025itgeq1f 25818 Equality theorem for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.) Avoid axioms. (Revised by GG, 1-Sep-2025.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐵    ⇒   (𝐴 = 𝐵 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥)
 
1-Sep-2025prodeq2sdv 15965 Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.) Avoid axioms. (Revised by GG, 1-Sep-2025.)
(𝜑 → 𝐵 = 𝐶)    ⇒   (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶)
 
1-Sep-2025prodeq1i 15958 Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.) Remove DV conditions. (Revised by GG, 1-Sep-2025.)
𝐴 = 𝐵    ⇒   ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶
 
1-Sep-2025iuneq12d 5044 Equality deduction for indexed union, deduction version. (Contributed by Drahflow, 22-Oct-2015.) Remove DV conditions (Revised by GG, 1-Sep-2025.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → 𝐶 = 𝐷)    ⇒   (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐷)
 
1-Sep-2025rabeqbidva 3460 Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.) Remove DV conditions. (Revised by GG, 1-Sep-2025.)
(𝜑 → 𝐴 = 𝐵)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
 
31-Aug-2025asin1half 42332 The arcsine of 1 / 2 is π / 6. (Contributed by SN, 31-Aug-2025.)
(arcsin‘(1 / 2)) = (π / 6)
 
30-Aug-2025ss-ax8 36183 A proof of ax-8 2110 that does not rely on ax-8 2110. It employs df-ss 3993 to perform alpha-renaming and eliminates disjoint variable conditions using ax-9 2118. Contrary to in-ax8 36182, this proof does not rely on df-cleq 2732, therefore using fewer axioms . This method should not be applied to eliminate axiom dependencies. (Contributed by GG, 30-Aug-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
 
26-Aug-20259rp 42285 9 is a positive real. (Contributed by SN, 26-Aug-2025.)
9 ∈ ℝ+
 
26-Aug-20258rp 42284 8 is a positive real. (Contributed by SN, 26-Aug-2025.)
8 ∈ ℝ+
 
26-Aug-20257rp 42283 7 is a positive real. (Contributed by SN, 26-Aug-2025.)
7 ∈ ℝ+
 
26-Aug-20256rp 42282 6 is a positive real. (Contributed by SN, 26-Aug-2025.)
6 ∈ ℝ+
 
26-Aug-20255rp 42281 5 is a positive real. (Contributed by SN, 26-Aug-2025.)
5 ∈ ℝ+
 
26-Aug-20254rp 42280 4 is a positive real. (Contributed by SN, 26-Aug-2025.)
4 ∈ ℝ+
 
26-Aug-2025sq8 42278 The square of 8 is 64. (Contributed by SN, 26-Aug-2025.)
(8↑2) = 64
 
26-Aug-2025sq7 42277 The square of 7 is 49. (Contributed by SN, 26-Aug-2025.)
(7↑2) = 49
 
26-Aug-2025sq6 42276 The square of 6 is 36. (Contributed by SN, 26-Aug-2025.)
(6↑2) = 36
 
26-Aug-2025sq5 42275 The square of 5 is 25. (Contributed by SN, 26-Aug-2025.)
(5↑2) = 25
 
26-Aug-2025sq4 42274 The square of 4 is 16. (Contributed by SN, 26-Aug-2025.)
(4↑2) = 16
 
26-Aug-2025lttrii 42244 'Less than' is transitive. (Contributed by SN, 26-Aug-2025.)
𝐴 ∈ ℝ    &   𝐵 ∈ ℝ    &   𝐶 ∈ ℝ    &   𝐴 < 𝐵    &   𝐵 < 𝐶    ⇒   𝐴 < 𝐶
 
26-Aug-2025sbcov 2257 A composition law for substitution. Version of sbco 2515 with a disjoint variable condition using fewer axioms. (Contributed by NM, 14-May-1993.) (Revised by GG, 7-Aug-2023.) (Proof shortened by SN, 26-Aug-2025.)
([𝑦 / 𝑥][𝑥 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
 
24-Aug-2025usgrexmpl12ngrlic 47844 The graphs 𝐻 and 𝐺 are not locally isomorphic (𝐻 contains a triangle, see usgrexmpl1tri 47830, whereas 𝐺 does not, see usgrexmpl2trifr 47842. (Contributed by AV, 24-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   𝐺 = ⟨𝑉, 𝐸⟩    &   𝐾 = ⟨“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”⟩    &   𝐻 = ⟨𝑉, 𝐾⟩    ⇒    ¬ 𝐺 ≃𝑙𝑔𝑟 𝐻
 
24-Aug-2025grlimgrtri 47810 Local isomorphisms between simple pseudographs map triangles onto triangles. (Contributed by AV, 24-Aug-2025.)
(𝜑 → 𝐺 ∈ USPGraph)    &   (𝜑 → 𝐻 ∈ USPGraph)    &   (𝜑 → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))    &   (𝜑 → 𝑇 ∈ (GrTriangles‘𝐺))    ⇒   (𝜑 → ∃𝑡 𝑡 ∈ (GrTriangles‘𝐻))
 
24-Aug-2025grlimgrtrilem1 47808 Lemma 3 for grlimgrtri 47810. (Contributed by AV, 24-Aug-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑁 = (𝐺 ClNeighbVtx 𝑎)    &   𝐼 = (Edg‘𝐺)    &   𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    ⇒   ((𝐺 ∈ UHGraph ∧ ({𝑎, 𝑏} ∈ 𝐼 ∧ {𝑎, 𝑐} ∈ 𝐼 ∧ {𝑏, 𝑐} ∈ 𝐼)) → ({𝑎, 𝑏} ∈ 𝐾 ∧ {𝑎, 𝑐} ∈ 𝐾 ∧ {𝑏, 𝑐} ∈ 𝐾))
 
24-Aug-2025grimgrtri 47788 Graph isomorphisms map triangles onto triangles. (Contributed by AV, 27-Jul-2025.) (Proof shortened by AV, 24-Aug-2025.)
(𝜑 → 𝐺 ∈ UHGraph)    &   (𝜑 → 𝐻 ∈ UHGraph)    &   (𝜑 → 𝐹 ∈ (𝐺 GraphIso 𝐻))    &   (𝜑 → 𝑇 ∈ (GrTriangles‘𝐺))    ⇒   (𝜑 → (𝐹 “ 𝑇) ∈ (GrTriangles‘𝐻))
 
24-Aug-2025clnbgrssedg 47703 The vertices connected by an edge are a subset of the neigborhood of each of these vertices. (Contributed by AV, 26-May-2025.) (Proof shortened by AV, 24-Aug-2025.)
𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 ClNeighbVtx 𝑋)    ⇒   ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾) → 𝐾 ⊆ 𝑁)
 
24-Aug-2025clnbgredg 47702 A vertices connected by an edge with another vertex is a neigborhood of those vertex. (Contributed by AV, 24-Aug-2025.)
𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 ClNeighbVtx 𝑋)    ⇒   ((𝐺 ∈ UHGraph ∧ (𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾)) → 𝑌 ∈ 𝑁)
 
24-Aug-2025sbievw 2093 Conversion of implicit substitution to explicit substitution. Version of sbie 2510 and sbiev 2318 with more disjoint variable conditions, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by BJ, 18-Jul-2023.) (Proof shortened by SN, 24-Aug-2025.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
 
24-Aug-2025sbbiiev 2092 An equivalence of substitutions (as in sbbii 2076) allowing the additional information that 𝑥 = 𝑡. Version of sbiev 2318 and sbievw 2093 without a disjoint variable condition on 𝜓, useful for substituting only part of 𝜑. (Contributed by SN, 24-Aug-2025.)
(𝑥 = 𝑡 → (𝜑 ↔ 𝜓))    ⇒   ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓)
 
23-Aug-2025grlimgrtrilem2 47809 Lemma 3 for grlimgrtri 47810. (Contributed by AV, 23-Aug-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑁 = (𝐺 ClNeighbVtx 𝑎)    &   𝐼 = (Edg‘𝐺)    &   𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑎))    &   𝐽 = (Edg‘𝐻)    &   𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   (((𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿) ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ 𝑖) = (𝑔‘𝑖) ∧ {𝑏, 𝑐} ∈ 𝐾) → {(𝑓‘𝑏), (𝑓‘𝑐)} ∈ 𝐽)
 
23-Aug-2025grtrimap 47787 Conditions for mapping triangles onto triangles. Lemma for grimgrtri 47788 and grlimgrtri 47810. (Contributed by AV, 23-Aug-2025.)
(𝐹:𝑉–1-1→𝑊 → (((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) ∧ (𝑇 = {𝑎, 𝑏, 𝑐} ∧ (♯‘𝑇) = 3)) → (((𝐹‘𝑎) ∈ 𝑊 ∧ (𝐹‘𝑏) ∈ 𝑊 ∧ (𝐹‘𝑐) ∈ 𝑊) ∧ (𝐹 “ 𝑇) = {(𝐹‘𝑎), (𝐹‘𝑏), (𝐹‘𝑐)} ∧ (♯‘(𝐹 “ 𝑇)) = 3)))
 
23-Aug-2025predgclnbgrel 47701 If a (not necessarily proper) unordered pair containing a vertex is an edge, the other vertex is in the closed neighborhood of the first vertex. (Contributed by AV, 23-Aug-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → 𝑁 ∈ (𝐺 ClNeighbVtx 𝑋))
 
23-Aug-2025numiunnum 36429 An indexed union of sets is numerable if its index set is numerable and there exists a collection of well-orderings on its members. (Contributed by Matthew House, 23-Aug-2025.)
((𝐴 ∈ dom card ∧ ∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝑉 ∧ 𝑆 We 𝐵)) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ dom card)
 
23-Aug-2025weiunwe 36428 A well-ordering on an indexed union can be constructed from a well-ordering on its index set and a collection of well-orderings on its members. (Contributed by Matthew House, 23-Aug-2025.)
𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    ⇒   ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 We 𝐵) → 𝑇 We ∪ 𝑥 ∈ 𝐴 𝐵)
 
23-Aug-2025weiunse 36427 The relation constructed in weiunpo 36422, weiunso 36423, weiunfr 36426, and weiunwe 36428 is set-like if all members of the indexed union are sets. (Contributed by Matthew House, 23-Aug-2025.)
𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    ⇒   ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) → 𝑇 Se ∪ 𝑥 ∈ 𝐴 𝐵)
 
23-Aug-2025weiunfr 36426 A well-founded relation on an indexed union can be constructed from a well-ordering on its index set and a collection of well-founded relations on its members. (Contributed by Matthew House, 23-Aug-2025.)
𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    ⇒   ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) → 𝑇 Fr ∪ 𝑥 ∈ 𝐴 𝐵)
 
23-Aug-2025weiunfrlem2 36425 Lemma for weiunfr 36426. (Contributed by Matthew House, 23-Aug-2025.)
𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    &   𝐶 = (℩𝑠 ∈ (𝐹 “ 𝑟)∀𝑡 ∈ (𝐹 “ 𝑟) ¬ 𝑡𝑅𝑠)    ⇒   ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Fr 𝐵) → 𝑇 Fr ∪ 𝑥 ∈ 𝐴 𝐵)
 
23-Aug-2025weiunfrlem1 36424 Lemma for weiunfr 36426. (Contributed by Matthew House, 23-Aug-2025.)
𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   𝐶 = (℩𝑠 ∈ (𝐹 “ 𝑟)∀𝑡 ∈ (𝐹 “ 𝑟) ¬ 𝑡𝑅𝑠)    ⇒   (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝑟 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑟 ≠ ∅)) → (𝐶 ∈ (𝐹 “ 𝑟) ∧ ∀𝑤 ∈ 𝑟 ¬ (𝐹‘𝑤)𝑅𝐶))
 
23-Aug-2025weiunso 36423 A strict ordering on an indexed union can be constructed from a well-ordering on its index set and a collection of strict orderings on its members. (Contributed by Matthew House, 23-Aug-2025.)
𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    ⇒   ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Or 𝐵) → 𝑇 Or ∪ 𝑥 ∈ 𝐴 𝐵)
 
23-Aug-2025weiunpo 36422 A partial ordering on an indexed union can be constructed from a well-ordering on its index set and a collection of partial orderings on its members. (Contributed by Matthew House, 23-Aug-2025.)
𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    &   𝑇 = {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ∧ ((𝐹‘𝑦)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑦) = (𝐹‘𝑧) ∧ 𝑦⦋(𝐹‘𝑦) / 𝑥⦌𝑆𝑧)))}    ⇒   ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑆 Po 𝐵) → 𝑇 Po ∪ 𝑥 ∈ 𝐴 𝐵)
 
23-Aug-2025weiunlem2 36421 Lemma for weiunpo 36422, weiunso 36423, and weiunse 36427. (Contributed by Matthew House, 23-Aug-2025.)
𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    ⇒   ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → (𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑡 ∈ ⦋(𝐹‘𝑡) / 𝑥⦌𝐵 ∧ ∀𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵∀𝑠 ∈ 𝐴 (𝑡 ∈ ⦋𝑠 / 𝑥⦌𝐵 → ¬ 𝑠𝑅(𝐹‘𝑡))))
 
23-Aug-2025weiunlem1 36420 Lemma for weiunpo 36422, weiunso 36423, weiunfr 36426, and weiunse 36427. (Contributed by Matthew House, 23-Aug-2025.)
𝐹 = (𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ (℩𝑢 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵}∀𝑣 ∈ {𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵} ¬ 𝑣𝑅𝑢))    ⇒   ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → (𝐹:∪ 𝑥 ∈ 𝐴 𝐵⟶𝐴 ∧ ∀𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑤 ∈ ⦋(𝐹‘𝑤) / 𝑥⦌𝐵 ∧ ∀𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵∀𝑣 ∈ 𝐴 (𝑤 ∈ ⦋𝑣 / 𝑥⦌𝐵 → ¬ 𝑣𝑅(𝐹‘𝑤))))
 
23-Aug-2025rexlimdvvva 3220 Inference from Theorem 19.23 of [Margaris] p. 90, for three restricted quantifiers. (Contributed by AV, 23-Aug-2025.)
((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)) → (𝜓 → 𝜒))    ⇒   (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜓 → 𝜒))
 
22-Aug-2025peano2ons 28298 The successor of a surreal ordinal is a surreal ordinal. (Contributed by Scott Fenton, 22-Aug-2025.)
(𝐴 ∈ Ons → (𝐴 +s 1s ) ∈ Ons)
 
22-Aug-2025onmulscl 28297 The surreal ordinals are closed under multiplication. (Contributed by Scott Fenton, 22-Aug-2025.)
((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 ·s 𝐵) ∈ Ons)
 
22-Aug-2025onaddscl 28296 The surreal ordinals are closed under addition. (Contributed by Scott Fenton, 22-Aug-2025.)
((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (𝐴 +s 𝐵) ∈ Ons)
 
21-Aug-2025cbvralsvw 3323 Change bound variable by using a substitution. Version of cbvralsv 3374 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by NM, 20-Nov-2005.) Avoid ax-13 2380. (Revised by GG, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 8-Mar-2025.) Avoid ax-10 2141, ax-12 2178. (Revised by SN, 21-Aug-2025.)
(∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)
 
20-Aug-2025zs12bday 28434 A dyadic fraction has a finite birthday. (Contributed by Scott Fenton, 20-Aug-2025.)
(𝐴 ∈ ℤs[1/2] → ( bday ‘𝐴) ∈ ω)
 
20-Aug-2025zscut 28403 A cut expression for surreal integers. (Contributed by Scott Fenton, 20-Aug-2025.)
(𝐴 ∈ ℤs → 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
 
20-Aug-2025zmulscld 28393 The surreal integers are closed under multiplication. (Contributed by Scott Fenton, 20-Aug-2025.)
(𝜑 → 𝐴 ∈ ℤs)    &   (𝜑 → 𝐵 ∈ ℤs)    ⇒   (𝜑 → (𝐴 ·s 𝐵) ∈ ℤs)
 
20-Aug-2025sltm1d 28141 A surreal is greater than itself minus one. (Contributed by Scott Fenton, 20-Aug-2025.)
(𝜑 → 𝐴 ∈ No )    ⇒   (𝜑 → (𝐴 -s 1s ) <s 𝐴)
 
20-Aug-2025cutmin 27979 If 𝐵 has a minimum, then the minimum may be used alone in the cut. (Contributed by Scott Fenton, 20-Aug-2025.)
(𝜑 → 𝐴 <<s 𝐵)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → ∀𝑦 ∈ 𝐵 𝑋 ≤s 𝑦)    ⇒   (𝜑 → (𝐴 |s 𝐵) = (𝐴 |s {𝑋}))
 
20-Aug-2025cutmax 27978 If 𝐴 has a maximum, then the maximum may be used alone in the cut. (Contributed by Scott Fenton, 20-Aug-2025.)
(𝜑 → 𝐴 <<s 𝐵)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑋)    ⇒   (𝜑 → (𝐴 |s 𝐵) = ({𝑋} |s 𝐵))
 
20-Aug-2025oldfi 27961 The old set of an ordinal natural is finite. (Contributed by Scott Fenton, 20-Aug-2025.)
(𝐴 ∈ ω → ( O ‘𝐴) ∈ Fin)
 
20-Aug-2025madefi 27960 The made set of an ordinal natural is finite. (Contributed by Scott Fenton, 20-Aug-2025.)
(𝐴 ∈ ω → ( M ‘𝐴) ∈ Fin)
 
20-Aug-2025omnaddcl 8753 The naturals are closed under natural addition. (Contributed by Scott Fenton, 20-Aug-2025.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +no 𝐵) ∈ ω)
 
20-Aug-2025naddoa 8752 Natural addition of a natural is the same as regular addition. (Contributed by Scott Fenton, 20-Aug-2025.)
((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 +no 𝐵) = (𝐴 +o 𝐵))
 
20-Aug-2025sbccomlem 3891 Lemma for sbccom 3893. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.) Avoid ax-10 2141, ax-12 2178. (Revised by SN, 20-Aug-2025.)
([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
 
19-Aug-2025zseo 28416 A surreal integer is either even or odd. (Contributed by Scott Fenton, 19-Aug-2025.)
(𝑁 ∈ ℤs → (∃𝑥 ∈ ℤs 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℤs 𝑁 = ((2s ·s 𝑥) +s 1s )))
 
19-Aug-2025n0seo 28415 A non-negative surreal integer is either even or odd. (Contributed by Scott Fenton, 19-Aug-2025.)
(𝑁 ∈ ℕ0s → (∃𝑥 ∈ ℕ0s 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑁 = ((2s ·s 𝑥) +s 1s )))
 
18-Aug-2025pw2cut 28430 Extend halfcut 28426 to arbitrary powers of two. Part of theorem 4.2 of [Gonshor] p. 28. (Contributed by Scott Fenton, 18-Aug-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝑁 ∈ ℕ0s)    &   (𝜑 → 𝐴 <s 𝐵)    &   (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s 𝐵)}) = (𝐴 +s 𝐵))    ⇒   (𝜑 → ({(𝐴 /su (2s↑s𝑁))} |s {(𝐵 /su (2s↑s𝑁))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑁 +s 1s ))))
 
17-Aug-2025usgrlimprop 47807 Properties of a local isomorphism of simple pseudographs. (Contributed by AV, 17-Aug-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))    &   𝐼 = (Edg‘𝐺)    &   𝐽 = (Edg‘𝐻)    &   𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒)))))
 
17-Aug-2025iseqsetvlem 2808 Lemma for iseqsetv-cleq 2809. (Contributed by Wolf Lammen, 17-Aug-2025.) (Proof modification is discouraged.)
(∃𝑥 𝑥 = 𝐴 ↔ ∃𝑧 𝑧 = 𝐴)
 
16-Aug-2025uspgrlimlem4 47805 Lemma 4 for uspgrlim 47806. (Contributed by AV, 16-Aug-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))    &   𝐼 = (Edg‘𝐺)    &   𝐽 = (Edg‘𝐻)    &   𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))) → ((𝑖 ∈ dom (iEdg‘𝐺) ∧ ((iEdg‘𝐺)‘𝑖) ⊆ 𝑁) → (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(((◡(iEdg‘𝐻) ∘ 𝑔) ∘ (iEdg‘𝐺))‘𝑖))))
 
16-Aug-2025uspgrlimlem3 47804 Lemma 3 for uspgrlim 47806. (Contributed by AV, 16-Aug-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))    &   𝐼 = (Edg‘𝐺)    &   𝐽 = (Edg‘𝐻)    &   𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ((𝐺 ∈ USPGraph ∧ ℎ:{𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁}–1-1-onto→𝑅 ∧ ∀𝑖 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑁} (𝑓 “ ((iEdg‘𝐺)‘𝑖)) = ((iEdg‘𝐻)‘(ℎ‘𝑖))) → (𝑒 ∈ 𝐾 → (𝑓 “ 𝑒) = ((((iEdg‘𝐻) ∘ ℎ) ∘ ◡(iEdg‘𝐺))‘𝑒)))
 
16-Aug-2025uspgrlimlem2 47803 Lemma 2 for uspgrlim 47806. (Contributed by AV, 16-Aug-2025.)
𝑀 = (𝐻 ClNeighbVtx 𝑋)    &   𝐽 = (Edg‘𝐻)    &   𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   (𝐻 ∈ USPGraph → (◡(iEdg‘𝐻) “ 𝐿) = {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀})
 
16-Aug-2025uspgrlimlem1 47802 Lemma 1 for uspgrlim 47806. (Contributed by AV, 16-Aug-2025.)
𝑀 = (𝐻 ClNeighbVtx 𝑋)    &   𝐽 = (Edg‘𝐻)    &   𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   (𝐻 ∈ USPGraph → 𝐿 = ((iEdg‘𝐻) “ {𝑥 ∈ dom (iEdg‘𝐻) ∣ ((iEdg‘𝐻)‘𝑥) ⊆ 𝑀}))
 
16-Aug-2025aks5 42154 The AKS Primality test, given an integer 𝑁 greater than or equal to 3, find a coprime 𝑅 such that 𝑅 is big enough. Then, if a bunch of polynomial equalities in the residue ring hold then 𝑁 is a prime power. Currently depends on the axiom ax-exfinfld 42152, since we currently do not have the existence of finite fields in the database. (Contributed by metakunt, 16-Aug-2025.)
𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   𝑋 = (var1‘(ℤ/nℤ‘𝑁))    &   𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))    &   𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))𝑋)(-g‘𝑆)(1r‘𝑆))})    &   (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(.g‘(mulGrp‘𝑆))(𝑋(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))𝑋)(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)(𝑎 gcd 𝑁) = 1)    ⇒   (𝜑 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ 𝑁 = (𝑝↑𝑛))
 
15-Aug-2025uspgrlim 47806 A local isomorphism of simple pseudographs is a bijection between their vertices that preserves neighborhoods, expressed by properties of their edges (not edge functions as in isgrlim2 47797). (Contributed by AV, 15-Aug-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))    &   𝐼 = (Edg‘𝐺)    &   𝐽 = (Edg‘𝐻)    &   𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}    &   𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}    ⇒   ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑒 ∈ 𝐾 (𝑓 “ 𝑒) = (𝑔‘𝑒))))))
 
15-Aug-20253f1oss2 46981 The composition of three bijections as bijection from the image of the converse of the domain onto the image of the converse of the range of the middle bijection. (Contributed by AV, 15-Aug-2025.)
(((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐵 ∧ 𝐷 ⊆ 𝐼)) → ((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷))
 
15-Aug-20253f1oss1 46980 The composition of three bijections as bijection from the image of the domain onto the image of the range of the middle bijection. (Contributed by AV, 15-Aug-2025.)
(((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐸)) → ((𝐻 ∘ 𝐺) ∘ ◡𝐹):(𝐹 “ 𝐶)–1-1-onto→(𝐻 “ 𝐷))
 
15-Aug-2025f1ocoima 7334 The composition of two bijections as bijection onto the image of the range of the first bijection. (Contributed by AV, 15-Aug-2025.)
((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐵 ⊆ 𝐶) → (𝐺 ∘ 𝐹):𝐴–1-1-onto→(𝐺 “ 𝐵))
 
15-Aug-2025dfdif3 4140 Alternate definition of class difference. (Contributed by BJ and Jim Kingdon, 16-Jun-2022.) (Proof shortened by SN, 15-Aug-2025.)
(𝐴 ∖ 𝐵) = {𝑥 ∈ 𝐴 ∣ ∀𝑦 ∈ 𝐵 𝑥 ≠ 𝑦}
 
14-Aug-2025cbvrabv2w 45020 A more general version of cbvrabv 3454. Version of cbvrabv2 45019 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Revised by GG, 14-Aug-2025.)
(𝑥 = 𝑦 → 𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}
 
14-Aug-2025cbvditgdavw2 36256 Change bound variable and limits in a directed integral. Deduction form. (Contributed by GG, 14-Aug-2025.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → 𝐶 = 𝐷)    &   ((𝜑 ∧ 𝑥 = 𝑦) → 𝐸 = 𝐹)    ⇒   (𝜑 → ⨜[𝐴 → 𝐶]𝐸 d𝑥 = ⨜[𝐵 → 𝐷]𝐹 d𝑦)
 
14-Aug-2025cbvitgdavw2 36255 Change bound variable and domain in an integral. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)    &   ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)    ⇒   (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑦)
 
14-Aug-2025cbvproddavw2 36254 Change bound variable and the set of integers in a product. Deduction form. (Contributed by GG, 14-Aug-2025.)
(𝜑 → 𝐴 = 𝐵)    &   ((𝜑 ∧ 𝑗 = 𝑘) → 𝐶 = 𝐷)    ⇒   (𝜑 → ∏𝑗 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐷)
 
14-Aug-2025cbvsumdavw2 36253 Change bound variable and the set of integers in a sum. Deduction form. (Contributed by GG, 14-Aug-2025.)
(𝜑 → 𝐴 = 𝐵)    &   ((𝜑 ∧ 𝑗 = 𝑘) → 𝐶 = 𝐷)    ⇒   (𝜑 → Σ𝑗 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐷)
 
14-Aug-2025cbvixpdavw2 36252 Change bound variable and domain in an indexed Cartesian product. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)    &   ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)    ⇒   (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑦 ∈ 𝐵 𝐷)
 
14-Aug-2025cbvmpo2davw2 36251 Change second bound variable and domains in a maps-to function. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑦 = 𝑧) → 𝐸 = 𝐹)    &   ((𝜑 ∧ 𝑦 = 𝑧) → 𝐶 = 𝐷)    &   ((𝜑 ∧ 𝑦 = 𝑧) → 𝐴 = 𝐵)    ⇒   (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑥 ∈ 𝐵, 𝑧 ∈ 𝐷 ↦ 𝐹))
 
14-Aug-2025cbvmpo1davw2 36250 Change first bound variable and domains in a maps-to function. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑧) → 𝐸 = 𝐹)    &   ((𝜑 ∧ 𝑥 = 𝑧) → 𝐶 = 𝐷)    &   ((𝜑 ∧ 𝑥 = 𝑧) → 𝐴 = 𝐵)    ⇒   (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ 𝐹))
 
14-Aug-2025cbvmpodavw2 36249 Change bound variable and domains in a maps-to function. Deduction form. (Contributed by GG, 14-Aug-2025.)
(((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝐸 = 𝐹)    &   (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)    &   (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝐴 = 𝐵)    ⇒   (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑤 ∈ 𝐷 ↦ 𝐹))
 
14-Aug-2025cbvriotadavw2 36248 Change bound variable and domain in a restricted description binder. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    &   ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)    ⇒   (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑦 ∈ 𝐵 𝜒))
 
14-Aug-2025cbvdisjdavw2 36247 Change bound variable and domain in a disjoint collection. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)    &   ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)    ⇒   (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑦 ∈ 𝐵 𝐷))
 
14-Aug-2025cbvmptdavw2 36246 Change bound variable and domain in a maps-to function. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)    &   ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐷))
 
14-Aug-2025cbviindavw2 36245 Change bound variable and domain in indexed intersections. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)    &   ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)    ⇒   (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑦 ∈ 𝐵 𝐷)
 
14-Aug-2025cbviundavw2 36244 Change bound variable and domain in indexed unions. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)    &   ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)    ⇒   (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑦 ∈ 𝐵 𝐷)
 
14-Aug-2025cbvrabdavw2 36243 Change bound variable and domain in restricted class abstractions. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    &   ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑦 ∈ 𝐵 ∣ 𝜒})
 
14-Aug-2025cbvreudavw2 36242 Change bound variable and quantifier domain in the restricted existential uniqueness quantifier. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    &   ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)    ⇒   (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑦 ∈ 𝐵 𝜒))
 
14-Aug-2025cbvrmodavw2 36241 Change bound variable and quantifier domain in the restricted at-most-one quantifier. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    &   ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵)    ⇒   (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑦 ∈ 𝐵 𝜒))
 
14-Aug-2025cbvditgdavw 36240 Change bound variable in a directed integral. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)    ⇒   (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ⨜[𝐴 → 𝐵]𝐷 d𝑦)
 
14-Aug-2025cbvitgdavw 36239 Change bound variable in an integral. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)    ⇒   (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑦)
 
14-Aug-2025cbvproddavw 36238 Change bound variable in a product. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑗 = 𝑘) → 𝐵 = 𝐶)    ⇒   (𝜑 → ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶)
 
14-Aug-2025cbvsumdavw 36237 Change bound variable in a sum. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑘 = 𝑗) → 𝐵 = 𝐶)    ⇒   (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑗 ∈ 𝐴 𝐶)
 
14-Aug-2025cbvixpdavw 36236 Change bound variable in an indexed Cartesian product. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)    ⇒   (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶)
 
14-Aug-2025cbvoprab13davw 36235 Change the first and third bound variables in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
(((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∣ 𝜒})
 
14-Aug-2025cbvoprab23davw 36234 Change the second and third bound variables in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
(((𝜑 ∧ 𝑦 = 𝑤) ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∣ 𝜒})
 
14-Aug-2025cbvoprab12davw 36233 Change the first and second bound variables in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
(((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑦 = 𝑣) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜒})
 
14-Aug-2025cbvoprab123davw 36232 Change all bound variables in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
((((𝜑 ∧ 𝑥 = 𝑤) ∧ 𝑦 = 𝑢) ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑢⟩, 𝑣⟩ ∣ 𝜒})
 
14-Aug-2025cbvoprab3davw 36231 Change the third bound variable in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑧 = 𝑤) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜒})
 
14-Aug-2025cbvoprab2davw 36230 Change the second bound variable in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑦 = 𝑤) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜒})
 
14-Aug-2025cbvoprab1davw 36229 Change the first bound variable in an operation abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑤) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜒})
 
14-Aug-2025cbvriotadavw 36228 Change bound variable in a restricted description binder. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑦 ∈ 𝐴 𝜒))
 
14-Aug-2025cbviotadavw 36227 Change bound variable in a description binder. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (℩𝑥𝜓) = (℩𝑦𝜒))
 
14-Aug-2025cbvdisjdavw 36226 Change bound variable in a disjoint collection. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)    ⇒   (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶))
 
14-Aug-2025cbvmptdavw 36225 Change bound variable in a maps-to function. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)    ⇒   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶))
 
14-Aug-2025cbvopabdavw 36224 Change bound variables in an ordered-pair class abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
(((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑧, 𝑤⟩ ∣ 𝜒})
 
14-Aug-2025cbvopab2davw 36223 Change the second bound variable in an ordered-pair class abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑦 = 𝑧) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑧⟩ ∣ 𝜒})
 
14-Aug-2025cbvopab1davw 36222 Change the first bound variable in an ordered-pair class abstraction. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑧) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑧, 𝑦⟩ ∣ 𝜒})
 
14-Aug-2025cbviindavw 36221 Change bound variable in indexed intersections. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)    ⇒   (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶)
 
14-Aug-2025cbviundavw 36220 Change bound variable in indexed unions. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)    ⇒   (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶)
 
14-Aug-2025cbvrabdavw 36219 Change bound variable in restricted class abstractions. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑦 ∈ 𝐴 ∣ 𝜒})
 
14-Aug-2025cbvcsbdavw2 36218 Change bound variable of a proper substitution into a class. General version of cbvcsbdavw 36217. Deduction form. (Contributed by GG, 14-Aug-2025.)
(𝜑 → 𝐴 = 𝐵)    &   ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷)    ⇒   (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑦⦌𝐷)
 
14-Aug-2025cbvcsbdavw 36217 Change bound variable of a proper substitution into a class. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶)    ⇒   (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶)
 
14-Aug-2025cbvsbcdavw2 36216 Change bound variable of a class substitution. General version of cbvsbcdavw 36215. Deduction form. (Contributed by GG, 14-Aug-2025.)
(𝜑 → 𝐴 = 𝐵)    &   ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑦]𝜒))
 
14-Aug-2025cbvsbcdavw 36215 Change bound variable of a class substitution. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐴 / 𝑦]𝜒))
 
14-Aug-2025cbvabdavw 36214 Change bound variable in class abstractions. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → {𝑥 ∣ 𝜓} = {𝑦 ∣ 𝜒})
 
14-Aug-2025cbvsbdavw2 36213 Change bound variable in proper substitution. General version of cbvsbdavw 36212. Deduction form. (Contributed by GG, 14-Aug-2025.)
(𝜑 → 𝑧 = 𝑤)    &   ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → ([𝑧 / 𝑥]𝜓 ↔ [𝑤 / 𝑦]𝜒))
 
14-Aug-2025cbvsbdavw 36212 Change bound variable in proper substitution. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → ([𝑧 / 𝑥]𝜓 ↔ [𝑧 / 𝑦]𝜒))
 
14-Aug-2025cbvreudavw 36211 Change bound variable in the restricted existential uniqueness quantifier. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑦 ∈ 𝐴 𝜒))
 
14-Aug-2025cbvrmodavw 36210 Change bound variable in the restricted at-most-one quantifier. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑦 ∈ 𝐴 𝜒))
 
14-Aug-2025cbveudavw 36209 Change bound variable in the existential uniqueness quantifier. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑦𝜒))
 
14-Aug-2025cbvmodavw 36208 Change bound variable in the at-most-one quantifier. Deduction form. (Contributed by GG, 14-Aug-2025.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑦𝜒))
 
14-Aug-2025cbvitgvw2 36206 Change bound variable and domain in an integral, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑦 → 𝐶 = 𝐷)    &   (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑦
 
14-Aug-2025cbvixpvw2 36203 Change bound variable and domain in an indexed Cartesian product, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑦 → 𝐶 = 𝐷)    &   (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   X𝑥 ∈ 𝐴 𝐶 = X𝑦 ∈ 𝐵 𝐷
 
14-Aug-2025cbvmpo2vw2 36202 Change domains and the second bound variable in a maps-to function, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑦 = 𝑧 → 𝐸 = 𝐹)    &   (𝑦 = 𝑧 → 𝐶 = 𝐷)    &   (𝑦 = 𝑧 → 𝐴 = 𝐵)    ⇒   (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑥 ∈ 𝐵, 𝑧 ∈ 𝐷 ↦ 𝐹)
 
14-Aug-2025cbvmpo1vw2 36201 Change domains and the first bound variable in a maps-to function, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑧 → 𝐸 = 𝐹)    &   (𝑥 = 𝑧 → 𝐶 = 𝐷)    &   (𝑥 = 𝑧 → 𝐴 = 𝐵)    ⇒   (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ 𝐹)
 
14-Aug-2025cbvmpovw2 36200 Change bound variables and domains in a maps-to function, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐸 = 𝐹)    &   ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)    &   ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐴 = 𝐵)    ⇒   (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑤 ∈ 𝐷 ↦ 𝐹)
 
14-Aug-2025cbvoprab13vw 36199 Change the first and third bound variables in an operation abstraction, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
((𝑥 = 𝑤 ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))    ⇒   {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 𝑣⟩ ∣ 𝜒}
 
14-Aug-2025cbvoprab23vw 36198 Change the second and third bound variables in an operation abstraction, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
((𝑦 = 𝑤 ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))    ⇒   {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑣⟩ ∣ 𝜒}
 
14-Aug-2025cbvoprab123vw 36197 Change all bound variables in an operation abstraction, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(((𝑥 = 𝑤 ∧ 𝑦 = 𝑢) ∧ 𝑧 = 𝑣) → (𝜓 ↔ 𝜒))    ⇒   {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑢⟩, 𝑣⟩ ∣ 𝜒}
 
14-Aug-2025cbvoprab2vw 36196 Change the second bound variable in an operation abstraction, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑦 = 𝑤 → (𝜓 ↔ 𝜒))    ⇒   {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑤⟩, 𝑧⟩ ∣ 𝜒}
 
14-Aug-2025cbvoprab1vw 36195 Change the first bound variable in an operation abstraction, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑤 → (𝜓 ↔ 𝜒))    ⇒   {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑤, 𝑦⟩, 𝑧⟩ ∣ 𝜒}
 
14-Aug-2025cbvriotavw2 36194 Change bound variable and domain in a restricted description binder, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑦 → 𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐵 𝜓)
 
14-Aug-2025cbvdisjvw2 36193 Change bound variable and domain in a disjoint collection, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑦 → 𝐶 = 𝐷)    &   (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   (Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑦 ∈ 𝐵 𝐷)
 
14-Aug-2025cbvmptvw2 36192 Change bound variable and domain in a maps-to function, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑦 → 𝐶 = 𝐷)    &   (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐷)
 
14-Aug-2025cbviinvw2 36191 Change bound variable and domain in an indexed intersection, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑦 → 𝐶 = 𝐷)    &   (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ∩ 𝑥 ∈ 𝐴 𝐶 = ∩ 𝑦 ∈ 𝐵 𝐷
 
14-Aug-2025cbviunvw2 36190 Change bound variable and domain in indexed unions, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑦 → 𝐶 = 𝐷)    &   (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑦 ∈ 𝐵 𝐷
 
14-Aug-2025cbvreuvw2 36187 Change bound variable and domain in the restricted existential uniqueness quantifier, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑦 → 𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐵 𝜓)
 
14-Aug-2025cbvrmovw2 36186 Change bound variable and domain in the restricted at-most-one quantifier, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑦 → 𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐵 𝜓)
 
14-Aug-2025cbvrexvw2 36185 Change bound variable and domain in the restricted existential quantifier, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑦 → 𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓)
 
14-Aug-2025cbvralvw2 36184 Change bound variable and domain in the restricted universal quantifier, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑦 → 𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)
 
14-Aug-2025sumeq2sdv 15745 Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Proof shortened by Glauco Siliprandi, 5-Apr-2020.) Avoid axioms. (Revised by GG, 14-Aug-2025.)
(𝜑 → 𝐵 = 𝐶)    ⇒   (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶)
 
14-Aug-2025rexima 7270 Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) Reduce DV conditions. (Revised by Matthew House, 14-Aug-2025.)
(𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))    ⇒   ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓))
 
14-Aug-2025ralima 7269 Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) Reduce DV conditions. (Revised by Matthew House, 14-Aug-2025.)
(𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))    ⇒   ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))
 
14-Aug-2025cbviinv 5064 Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) Add disjoint variable condition to avoid ax-13 2380. See cbviinvg 5066 for a less restrictive version requiring more axioms. (Revised by GG, 14-Aug-2025.)
(𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑦 ∈ 𝐴 𝐶
 
14-Aug-2025cbviunv 5063 Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 15-Sep-2003.) Add disjoint variable condition to avoid ax-13 2380. See cbviunvg 5065 for a less restrictive version requiring more axioms. (Revised by GG, 14-Aug-2025.)
(𝑥 = 𝑦 → 𝐵 = 𝐶)    ⇒   ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶
 
14-Aug-2025sbalex 2243 Equivalence of two ways to express proper substitution of a setvar for another setvar disjoint from it in a formula. This proof of their equivalence does not use df-sb 2065.

That both sides of the biconditional express proper substitution is proved by sb5 2277 and sb6 2085. The implication "to the left" is equs4v 1999 and does not require ax-10 2141 nor ax-12 2178. It also holds without disjoint variable condition if we allow more axioms (see equs4 2424). Theorem 6.2 of [Quine] p. 40. Theorem equs5 2468 replaces the disjoint variable condition with a distinctor antecedent. Theorem equs45f 2467 replaces the disjoint variable condition on 𝑥, 𝑡 with the nonfreeness hypothesis of 𝑡 in 𝜑. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2269 in place of equsex 2426 in order to remove dependency on ax-13 2380. (Revised by BJ, 20-Dec-2020.) Revise to remove dependency on df-sb 2065. (Revised by BJ, 21-Sep-2024.) (Proof shortened by SN, 14-Aug-2025.)

(∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
 
14-Aug-2025ax12ev2 2181 Version of ax12v2 2180 rewritten to use an existential quantifier. One direction of sbalex 2243 without the universal quantifier, avoiding ax-10 2141. (Contributed by SN, 14-Aug-2025.)
(∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜑))
 
13-Aug-2025addhalfcut 28429 The cut of a surreal non-negative integer and its successor is the original number plus one half. Part of theorem 4.2 of [Gonshor] p. 30. (Contributed by Scott Fenton, 13-Aug-2025.)
(𝜑 → 𝐴 ∈ ℕ0s)    ⇒   (𝜑 → ({𝐴} |s {(𝐴 +s 1s )}) = (𝐴 +s ( 1s /su 2s)))
 
13-Aug-2025divscan3d 28270 A cancellation law for surreal division. (Contributed by Scott Fenton, 13-Aug-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐵 ≠ 0s )    ⇒   (𝜑 → ((𝐵 ·s 𝐴) /su 𝐵) = 𝐴)
 
13-Aug-2025divsdird 28269 Distribution of surreal division over addition. (Contributed by Scott Fenton, 13-Aug-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐶 ≠ 0s )    ⇒   (𝜑 → ((𝐴 +s 𝐵) /su 𝐶) = ((𝐴 /su 𝐶) +s (𝐵 /su 𝐶)))
 
13-Aug-2025divsrecd 28268 Relationship between surreal division and reciprocal. (Contributed by Scott Fenton, 13-Aug-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐵 ≠ 0s )    ⇒   (𝜑 → (𝐴 /su 𝐵) = (𝐴 ·s ( 1s /su 𝐵)))
 
13-Aug-2025addsbdaylem 28059 Lemma for addsbday 28060. (Contributed by Scott Fenton, 13-Aug-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → ∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))( bday ‘(𝐴 +s 𝑦𝑂)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝑦𝑂)))    &   𝑆 ⊆ (( L ‘𝐵) ∪ ( R ‘𝐵))    ⇒   (𝜑 → ( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)}) ⊆ (( bday ‘𝐴) +no ( bday ‘𝐵)))
 
13-Aug-2025sltp1d 28058 A surreal is less than itself plus one. (Contributed by Scott Fenton, 13-Aug-2025.)
(𝜑 → 𝐴 ∈ No )    ⇒   (𝜑 → 𝐴 <s (𝐴 +s 1s ))
 
12-Aug-2025alan 42614 Alias for 19.26 1869 for easier lookup. (Contributed by SN, 12-Aug-2025.) (New usage is discouraged.)
(∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
 
12-Aug-2025sbalexi 42199 Inference form of sbalex 2243, avoiding ax-10 2141 by using ax-gen 1793. (Contributed by SN, 12-Aug-2025.)
∃𝑥(𝑥 = 𝑦 ∧ 𝜑)    ⇒   ∀𝑥(𝑥 = 𝑦 → 𝜑)
 
12-Aug-2025addsbday 28060 The birthday of the sum of two surreals is less than or equal to the natural ordinal sum of their individual birthdays. Theorem 6.1 of [Gonshor] p. 95. (Contributed by Scott Fenton, 12-Aug-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( bday ‘(𝐴 +s 𝐵)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝐵)))
 
12-Aug-2025dmxp 5948 The domain of a Cartesian product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-10 2141, ax-11 2158, ax-12 2178. (Revised by SN, 12-Aug-2025.)
(𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
 
10-Aug-2025usgrexmpl12ngric 47843 The graphs 𝐻 and 𝐺 are not isomorphic (𝐻 contains a triangle, see usgrexmpl1tri 47830, whereas 𝐺 does not, see usgrexmpl2trifr 47842. (Contributed by AV, 10-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   𝐺 = ⟨𝑉, 𝐸⟩    &   𝐾 = ⟨“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”⟩    &   𝐻 = ⟨𝑉, 𝐾⟩    ⇒    ¬ 𝐺 ≃𝑔𝑟 𝐻
 
10-Aug-2025usgrexmpl2trifr 47842 𝐺 is triangle-free. (Contributed by AV, 10-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   𝐺 = ⟨𝑉, 𝐸⟩    ⇒    ¬ ∃𝑡 𝑡 ∈ (GrTriangles‘𝐺)
 
10-Aug-2025usgrexmpl2nb5 47841 The neighborhood of the sixth vertex of graph 𝐺. (Contributed by AV, 10-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   𝐺 = ⟨𝑉, 𝐸⟩    ⇒   (𝐺 NeighbVtx 5) = {0, 4}
 
10-Aug-2025biorfri 938 A wff is equivalent to its disjunction with falsehood. (Contributed by NM, 23-Mar-1995.) (Proof shortened by Wolf Lammen, 16-Jul-2021.) (Proof shortened by AV, 10-Aug-2025.)
¬ 𝜑    ⇒   (𝜓 ↔ (𝜓 ∨ 𝜑))
 
9-Aug-2025usgrexmpl2nb4 47840 The neighborhood of the fifth vertex of graph 𝐺. (Contributed by AV, 9-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   𝐺 = ⟨𝑉, 𝐸⟩    ⇒   (𝐺 NeighbVtx 4) = {3, 5}
 
9-Aug-2025usgrexmpl2nb3 47839 The neighborhood of the forth vertex of graph 𝐺. (Contributed by AV, 9-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   𝐺 = ⟨𝑉, 𝐸⟩    ⇒   (𝐺 NeighbVtx 3) = {0, 2, 4}
 
9-Aug-2025usgrexmpl2nb2 47838 The neighborhood of the third vertex of graph 𝐺. (Contributed by AV, 9-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   𝐺 = ⟨𝑉, 𝐸⟩    ⇒   (𝐺 NeighbVtx 2) = {1, 3}
 
9-Aug-2025usgrexmpl2nb1 47837 The neighborhood of the second vertex of graph 𝐺. (Contributed by AV, 9-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   𝐺 = ⟨𝑉, 𝐸⟩    ⇒   (𝐺 NeighbVtx 1) = {0, 2}
 
9-Aug-2025usgrexmpl2nb0 47836 The neighborhood of the first vertex of graph 𝐺. (Contributed by AV, 9-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   𝐺 = ⟨𝑉, 𝐸⟩    ⇒   (𝐺 NeighbVtx 0) = {1, 3, 5}
 
9-Aug-2025usgrexmpl2nblem 47835 Lemma for usgrexmpl2nb0 47836 etc. (Contributed by AV, 9-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   𝐺 = ⟨𝑉, 𝐸⟩    ⇒   (𝐾 ∈ ({0, 1, 2} ∪ {3, 4, 5}) → (𝐺 NeighbVtx 𝐾) = {𝑛 ∈ ({0, 1, 2} ∪ {3, 4, 5}) ∣ {𝐾, 𝑛} ∈ ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))})
 
9-Aug-2025aks5lem8 42151 Lemma for aks5. Clean up the conclusion. (Contributed by metakunt, 9-Aug-2025.)
(𝜑 → (♯‘(Base‘𝐾)) ∈ ℕ)    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    &   (𝜑 → 𝑅 ∥ ((♯‘(Base‘𝐾)) − 1))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(.g‘(mulGrp‘𝑆))(𝑋(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))𝑋)(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿))    &   (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))    &   𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))𝑋)(-g‘𝑆)(1r‘𝑆))})    &   𝑋 = (var1‘(ℤ/nℤ‘𝑁))    ⇒   (𝜑 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ 𝑁 = (𝑝↑𝑛))
 
9-Aug-2025aks5lem7 42150 Lemma for aks5. We clean up the hypotheses compared to aks5lem6 42142. (Contributed by metakunt, 9-Aug-2025.)
(𝜑 → (♯‘(Base‘𝐾)) ∈ ℕ)    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    &   (𝜑 → 𝑅 ∥ ((♯‘(Base‘𝐾)) − 1))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(.g‘(mulGrp‘𝑆))(𝑋(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))𝑋)(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿))    &   (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))    &   𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))𝑋)(-g‘𝑆)(1r‘𝑆))})    &   𝑋 = (var1‘(ℤ/nℤ‘𝑁))    ⇒   (𝜑 → 𝑁 = (𝑃↑(𝑃 pCnt 𝑁)))
 
9-Aug-2025unitscyglem5 42149 Lemma for unitscyg (Contributed by metakunt, 9-Aug-2025.)
𝐺 = ((mulGrp‘𝑅) ↾s (Unit‘𝑅))    &   (𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → (Base‘𝑅) ∈ Fin)    &   (𝜑 → 𝐷 ∈ ℕ)    &   (𝜑 → 𝐷 ∥ (♯‘(Base‘𝐺)))    ⇒   (𝜑 → ((mulGrp‘𝑅) PrimRoots 𝐷) ≠ ∅)
 
9-Aug-2025el7g 4713 Members of a set with seven elements. Lemma for usgrexmpl2nb0 47836 etc. (Contributed by AV, 9-Aug-2025.)
(𝑋 ∈ 𝑉 → (𝑋 ∈ ({𝐴} ∪ ({𝐵, 𝐶, 𝐷} ∪ {𝐸, 𝐹, 𝐺})) ↔ (𝑋 = 𝐴 ∨ ((𝑋 = 𝐵 ∨ 𝑋 = 𝐶 ∨ 𝑋 = 𝐷) ∨ (𝑋 = 𝐸 ∨ 𝑋 = 𝐹 ∨ 𝑋 = 𝐺)))))
 
7-Aug-2025usgrgrtrirex 47789 Conditions for a simple graph to contain a triangle. (Contributed by AV, 7-Aug-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑎)    ⇒   (𝐺 ∈ USGraph → (∃𝑡 𝑡 ∈ (GrTriangles‘𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑁 ∃𝑐 ∈ 𝑁 (𝑏 ≠ 𝑐 ∧ {𝑏, 𝑐} ∈ 𝐸)))
 
7-Aug-2025usgrexmpl 29290 𝐺 is a simple graph of five vertices 0, 1, 2, 3, 4, with edges {0, 1}, {1, 2}, {2, 0}, {0, 3}. (Contributed by Alexander van der Vekens, 15-Aug-2017.) (Revised by AV, 21-Oct-2020.) (Proof shortened by AV, 7-Aug-2025.)
𝑉 = (0...4)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 0} {0, 3}”⟩    &   𝐺 = ⟨𝑉, 𝐸⟩    ⇒   𝐺 ∈ USGraph
 
7-Aug-2025zzs12 28433 A surreal integer is a dyadic fraction. (Contributed by Scott Fenton, 7-Aug-2025.)
(𝐴 ∈ ℤs → 𝐴 ∈ ℤs[1/2])
 
7-Aug-2025zs12ex 28432 The class of dyadic fractions is a set. (Contributed by Scott Fenton, 7-Aug-2025.)
ℤs[1/2] ∈ V
 
7-Aug-2025elzs12 28431 Membership in the dyadic fractions. (Contributed by Scott Fenton, 7-Aug-2025.)
(𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))
 
7-Aug-2025pw2bday 28428 The inverses of powers of two have finite birthdays. (Contributed by Scott Fenton, 7-Aug-2025.)
(𝑁 ∈ ℕ0s → ( bday ‘( 1s /su (2s↑s𝑁))) ∈ ω)
 
7-Aug-2025cutpw2 28427 A cut expression for inverses of powers of two. (Contributed by Scott Fenton, 7-Aug-2025.)
(𝑁 ∈ ℕ0s → ( 1s /su (2s↑s(𝑁 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑁))}))
 
7-Aug-2025halfcut 28426 Relate the cut of twice of two numbers to the cut of the numbers. Lemma 4.2 of [Gonshor] p. 28. (Contributed by Scott Fenton, 7-Aug-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐴 <s 𝐵)    &   (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s 𝐵)}) = (𝐴 +s 𝐵))    &   𝐶 = ({𝐴} |s {𝐵})    ⇒   (𝜑 → 𝐶 = ((𝐴 +s 𝐵) /su 2s))
 
7-Aug-2025expsgt0 28425 A non-negative surreal integer power is positive if its base is positive. (Contributed by Scott Fenton, 7-Aug-2025.)
((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑁))
 
7-Aug-2025expsne0 28424 A non-negative surreal integer power is non-zero if its base is non-zero. (Contributed by Scott Fenton, 7-Aug-2025.)
((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ≠ 0s )
 
7-Aug-2025expscl 28423 Closure law for surreal exponentiation. (Contributed by Scott Fenton, 7-Aug-2025.)
((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ No )
 
7-Aug-2025divdivs1d 28267 Surreal division into a fraction. (Contributed by Scott Fenton, 7-Aug-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐵 ≠ 0s )    &   (𝜑 → 𝐶 ≠ 0s )    ⇒   (𝜑 → ((𝐴 /su 𝐵) /su 𝐶) = (𝐴 /su (𝐵 ·s 𝐶)))
 
7-Aug-2025fz0to4untppr 13681 An integer range from 0 to 4 is the union of a triple and a pair. (Contributed by Alexander van der Vekens, 13-Aug-2017.) (Proof shortened by AV, 7-Aug-2025.)
(0...4) = ({0, 1, 2} ∪ {3, 4})
 
6-Aug-2025expsp1 28422 Value of a surreal number raised to a non-negative integer power plus one. (Contributed by Scott Fenton, 6-Aug-2025.)
((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))
 
6-Aug-2025nnsind 28373 Principle of Mathematical Induction (inference schema). (Contributed by Scott Fenton, 6-Aug-2025.)
(𝑥 = 1s → (𝜑 ↔ 𝜓))    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   (𝑥 = (𝑦 +s 1s ) → (𝜑 ↔ 𝜃))    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   𝜓    &   (𝑦 ∈ ℕs → (𝜒 → 𝜃))    ⇒   (𝐴 ∈ ℕs → 𝜏)
 
6-Aug-2025dfnns2 28372 Alternate definition of the positive surreal integers. Compare df-nn 12288. (Contributed by Scott Fenton, 6-Aug-2025.)
ℕs = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)
 
6-Aug-2025iseqsetv-cleq 2809 Alternate proof of iseqsetv-clel 2823. The expression ∃𝑥𝑥 = 𝐴 does not depend on a particular choice of the set variable. The proof here avoids df-clab 2718, df-clel 2819 and ax-8 2110, but instead is based on ax-9 2118, ax-ext 2711 and df-cleq 2732. In particular it still accepts 𝑥 ∈ 𝐴 being a primitive syntax term, not assuming any specific semantics (like elementhood in some form).

Use it in contexts where you want to avoid df-clab 2718, or you need df-cleq 2732 anyway. See the alternative version , not using df-cleq 2732 or ax-ext 2711 or ax-9 2118. (Contributed by Wolf Lammen, 6-Aug-2025.) (Proof modification is discouraged.)

(∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴)
 
3-Aug-2025usgrexmpl2edg 47834 The edges {0, 1}, {1, 2}, {2, 3}, {0, 3}, {3, 4}, {4, 5}, {0, 5} of the graph 𝐺 = ⟨𝑉, 𝐸⟩. (Contributed by AV, 3-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   𝐺 = ⟨𝑉, 𝐸⟩    ⇒   (Edg‘𝐺) = ({{0, 3}} ∪ ({{0, 1}, {1, 2}, {2, 3}} ∪ {{3, 4}, {4, 5}, {0, 5}}))
 
3-Aug-2025usgrexmpl2vtx 47833 The vertices 0, 1, 2, 3, 4, 5 of the graph 𝐺 = ⟨𝑉, 𝐸⟩. (Contributed by AV, 3-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   𝐺 = ⟨𝑉, 𝐸⟩    ⇒   (Vtx‘𝐺) = ({0, 1, 2} ∪ {3, 4, 5})
 
3-Aug-2025usgrexmpl2 47832 𝐺 is a simple graph of six vertices 0, 1, 2, 3, 4, 5, with edges {0, 1}, {1, 2}, {2, 3}, {0, 3}, {3, 4}, {4, 5}, {0, 5}. (Contributed by AV, 3-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    &   𝐺 = ⟨𝑉, 𝐸⟩    ⇒   𝐺 ∈ USGraph
 
3-Aug-2025usgrexmpl2lem 47831 Lemma for usgrexmpl2 47832. (Contributed by AV, 3-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {1, 2} {2, 3} {3, 4} {4, 5} {0, 3} {0, 5}”⟩    ⇒   𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}
 
3-Aug-2025usgrexmpl1tri 47830 𝐺 contains a triangle 0, 1, 2, with corresponding edges {0, 1}, {1, 2}, {0, 2}. (Contributed by AV, 3-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”⟩    &   𝐺 = ⟨𝑉, 𝐸⟩    ⇒   {0, 1, 2} ∈ (GrTriangles‘𝐺)
 
3-Aug-2025usgrexmpl1edg 47829 The edges {0, 1}, {1, 2}, {0, 2}, {0, 3}, {3, 4}, {3, 5}, {4, 5} of the graph 𝐺 = ⟨𝑉, 𝐸⟩. (Contributed by AV, 3-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”⟩    &   𝐺 = ⟨𝑉, 𝐸⟩    ⇒   (Edg‘𝐺) = ({{0, 3}} ∪ ({{0, 1}, {0, 2}, {1, 2}} ∪ {{3, 4}, {3, 5}, {4, 5}}))
 
3-Aug-2025usgrexmpl1vtx 47828 The vertices 0, 1, 2, 3, 4, 5 of the graph 𝐺 = ⟨𝑉, 𝐸⟩. (Contributed by AV, 3-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”⟩    &   𝐺 = ⟨𝑉, 𝐸⟩    ⇒   (Vtx‘𝐺) = ({0, 1, 2} ∪ {3, 4, 5})
 
3-Aug-2025usgrexmpl1 47827 𝐺 is a simple graph of six vertices 0, 1, 2, 3, 4, 5, with edges {0, 1}, {1, 2}, {0, 2}, {0, 3}, {3, 4}, {3, 5}, {4, 5}. (Contributed by AV, 3-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”⟩    &   𝐺 = ⟨𝑉, 𝐸⟩    ⇒   𝐺 ∈ USGraph
 
3-Aug-2025axnulg 35060 A generalization of ax-nul 5324 in which 𝑥 and 𝑦 need not be distinct. Note that it is possible to use axc7e 2322 to derive elirrv 9659 from this theorem, which justifies the dependency on ax-reg 9655. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by BTernaryTau, 3-Aug-2025.) (New usage is discouraged.)
∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
 
3-Aug-2025axsepg2ALT 35051 Alternate proof of axsepg2 35050, derived directly from ax-sep 5317 with no additional set theory axioms. (Contributed by BTernaryTau, 3-Aug-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
 
3-Aug-2025axsepg2 35050 A generalization of ax-sep 5317 in which 𝑦 and 𝑧 need not be distinct. See also axsepg 5318 which instead allows 𝑧 to occur in 𝜑. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by BTernaryTau, 3-Aug-2025.) (New usage is discouraged.)
∃𝑦∀𝑥(𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
 
3-Aug-2025assafld 33642 If an algebra 𝐴 of finite degree over a division ring 𝐾 is an integral domain, then it is a field. Corollary of Proposition 2. in Chapter 5. of [BourbakiAlg2] p. 113. (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐾 = (Scalar‘𝐴)    &   (𝜑 → 𝐴 ∈ AssAlg)    &   (𝜑 → 𝐴 ∈ IDomn)    &   (𝜑 → 𝐾 ∈ DivRing)    &   (𝜑 → (dim‘𝐴) ∈ ℕ0)    ⇒   (𝜑 → 𝐴 ∈ Field)
 
3-Aug-2025assarrginv 33641 If an element 𝑋 of an associative algebra 𝐴 over a division ring 𝐾 is regular, then it is a unit. Proposition 2. in Chapter 5. of [BourbakiAlg2] p. 113. (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐸 = (RLReg‘𝐴)    &   𝑈 = (Unit‘𝐴)    &   𝐾 = (Scalar‘𝐴)    &   (𝜑 → 𝐴 ∈ AssAlg)    &   (𝜑 → 𝐾 ∈ DivRing)    &   (𝜑 → (dim‘𝐴) ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ 𝐸)    ⇒   (𝜑 → 𝑋 ∈ 𝑈)
 
3-Aug-2025assalactf1o 33640 In an associative algebra 𝐴, left-multiplication by a fixed element of the algebra is bijective. See also lactlmhm 33639. (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝐴)    &    · = (.r‘𝐴)    &   𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐶 · 𝑥))    &   (𝜑 → 𝐴 ∈ AssAlg)    &   𝐸 = (RLReg‘𝐴)    &   𝐾 = (Scalar‘𝐴)    &   (𝜑 → 𝐾 ∈ DivRing)    &   (𝜑 → (dim‘𝐴) ∈ ℕ0)    &   (𝜑 → 𝐶 ∈ 𝐸)    ⇒   (𝜑 → 𝐹:𝐵–1-1-onto→𝐵)
 
3-Aug-2025lactlmhm 33639 In an associative algebra 𝐴, left-multiplication by a fixed element of the algebra is a module homomorphism, analogous to ringlghm 20329. (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝐴)    &    · = (.r‘𝐴)    &   𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐶 · 𝑥))    &   (𝜑 → 𝐴 ∈ AssAlg)    &   (𝜑 → 𝐶 ∈ 𝐵)    ⇒   (𝜑 → 𝐹 ∈ (𝐴 LMHom 𝐴))
 
3-Aug-2025lvecendof1f1o 33638 If an endomorphism 𝑈 of a vector space 𝐸 of finite dimension is injective, then it is bijective. Item (b) of Corollary of Proposition 9 in [BourbakiAlg1] p. 298 . (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝐸)    &   (𝜑 → 𝐸 ∈ LVec)    &   (𝜑 → (dim‘𝐸) ∈ ℕ0)    &   (𝜑 → 𝑈 ∈ (𝐸 LMHom 𝐸))    &   (𝜑 → 𝑈:𝐵–1-1→𝐵)    ⇒   (𝜑 → 𝑈:𝐵–1-1-onto→𝐵)
 
3-Aug-2025dimlssid 33637 If the dimension of a linear subspace 𝐿 is the dimension of the whole vector space 𝐸, then 𝐿 is the whole space. (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝐸)    &   (𝜑 → 𝐸 ∈ LVec)    &   (𝜑 → (dim‘𝐸) ∈ ℕ0)    &   (𝜑 → 𝐿 ∈ (LSubSp‘𝐸))    &   (𝜑 → (dim‘(𝐸 ↾s 𝐿)) = (dim‘𝐸))    ⇒   (𝜑 → 𝐿 = 𝐵)
 
3-Aug-2025isunit3 33213 Alternate definition of being a unit. (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝑅)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → (𝑋 ∈ 𝑈 ↔ ∃𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 1 ∧ (𝑦 · 𝑋) = 1 )))
 
3-Aug-2025isunit2 33212 Alternate definition of being a unit. (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝑅)    ⇒   (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ (∃𝑢 ∈ 𝐵 (𝑋 · 𝑢) = 1 ∧ ∃𝑣 ∈ 𝐵 (𝑣 · 𝑋) = 1 )))
 
3-Aug-2025grpsubcld 33018 Closure of group subtraction. (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝐺)    &    − = (-g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 − 𝑌) ∈ 𝐵)
 
3-Aug-2025mndractf1o 33009 An element 𝑋 of a monoid 𝐸 is invertible iff its right-translation 𝐺 is bijective. See also mndlactf1o 33008. Remark in chapter I. of [BourbakiAlg1] p. 17 . (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝐸)    &    0 = (0g‘𝐸)    &    + = (+g‘𝐸)    &   𝐺 = (𝑎 ∈ 𝐵 ↦ (𝑎 + 𝑋))    &   (𝜑 → 𝐸 ∈ Mnd)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝐺:𝐵–1-1-onto→𝐵 ↔ ∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 )))
 
3-Aug-2025mndlactf1o 33008 An element 𝑋 of a monoid 𝐸 is invertible iff its left-translation 𝐹 is bijective. See also grplactf1o 19078. Remark in chapter I. of [BourbakiAlg1] p. 17. (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝐸)    &    0 = (0g‘𝐸)    &    + = (+g‘𝐸)    &   𝐹 = (𝑎 ∈ 𝐵 ↦ (𝑋 + 𝑎))    &   (𝜑 → 𝐸 ∈ Mnd)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝐹:𝐵–1-1-onto→𝐵 ↔ ∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 )))
 
3-Aug-2025mndractfo 33007 An element 𝑋 of a monoid 𝐸 is right-invertible iff its right-translation 𝐺 is surjective. (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝐸)    &    0 = (0g‘𝐸)    &    + = (+g‘𝐸)    &   𝐺 = (𝑎 ∈ 𝐵 ↦ (𝑎 + 𝑋))    &   (𝜑 → 𝐸 ∈ Mnd)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝐺:𝐵–onto→𝐵 ↔ ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))
 
3-Aug-2025mndractf1 33006 If an element 𝑋 of a monoid 𝐸 is right-invertible, with inverse 𝑌, then its left-translation 𝐺 is injective. See also grplactf1o 19078. Remark in chapter I. of [BourbakiAlg1] p. 17 . (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝐸)    &    0 = (0g‘𝐸)    &    + = (+g‘𝐸)    &   𝐺 = (𝑎 ∈ 𝐵 ↦ (𝑎 + 𝑋))    &   (𝜑 → 𝐸 ∈ Mnd)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → (𝑋 + 𝑌) = 0 )    ⇒   (𝜑 → 𝐺:𝐵–1-1→𝐵)
 
3-Aug-2025mndlactfo 33005 An element 𝑋 of a monoid 𝐸 is left-invertible iff its left-translation 𝐹 is surjective. See also grplactf1o 19078. (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝐸)    &    0 = (0g‘𝐸)    &    + = (+g‘𝐸)    &   𝐹 = (𝑎 ∈ 𝐵 ↦ (𝑋 + 𝑎))    &   (𝜑 → 𝐸 ∈ Mnd)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝐹:𝐵–onto→𝐵 ↔ ∃𝑦 ∈ 𝐵 (𝑋 + 𝑦) = 0 ))
 
3-Aug-2025mndlactf1 33004 If an element 𝑋 of a monoid 𝐸 is right-invertible, with inverse 𝑌, then its left-translation 𝐹 is injective. See also grplactf1o 19078. Remark in chapter I. of [BourbakiAlg1] p. 17 . (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝐸)    &    0 = (0g‘𝐸)    &    + = (+g‘𝐸)    &   𝐹 = (𝑎 ∈ 𝐵 ↦ (𝑋 + 𝑎))    &   (𝜑 → 𝐸 ∈ Mnd)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → (𝑌 + 𝑋) = 0 )    ⇒   (𝜑 → 𝐹:𝐵–1-1→𝐵)
 
3-Aug-2025mndlrinvb 33003 In a monoid, if an element has both a left-inverse and a right-inverse, they are equal. (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝐸)    &    0 = (0g‘𝐸)    &    + = (+g‘𝐸)    &   (𝜑 → 𝐸 ∈ Mnd)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → ((∃𝑢 ∈ 𝐵 (𝑋 + 𝑢) = 0 ∧ ∃𝑣 ∈ 𝐵 (𝑣 + 𝑋) = 0 ) ↔ ∃𝑦 ∈ 𝐵 ((𝑋 + 𝑦) = 0 ∧ (𝑦 + 𝑋) = 0 )))
 
3-Aug-2025mndlrinv 33002 In a monoid, if an element 𝑋 has both a left inverse 𝑀 and a right inverse 𝑁, they are equal. (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝐸)    &    0 = (0g‘𝐸)    &    + = (+g‘𝐸)    &   (𝜑 → 𝐸 ∈ Mnd)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑀 ∈ 𝐵)    &   (𝜑 → 𝑁 ∈ 𝐵)    &   (𝜑 → (𝑀 + 𝑋) = 0 )    &   (𝜑 → (𝑋 + 𝑁) = 0 )    ⇒   (𝜑 → 𝑀 = 𝑁)
 
3-Aug-2025mndassd 33001 A monoid operation is associative. (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ Mnd)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
 
3-Aug-2025mndcld 33000 Closure of the operation of a monoid. (Contributed by Thierry Arnoux, 3-Aug-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ Mnd)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
 
3-Aug-2025n0limd 32493 Deduction rule for nonempty classes. (Contributed by Thierry Arnoux, 3-Aug-2025.)
(𝜑 → 𝐴 ≠ ∅)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)    ⇒   (𝜑 → 𝜓)
 
2-Aug-2025usgrexmpl1lem 47826 Lemma for usgrexmpl1 47827. (Contributed by AV, 2-Aug-2025.)
𝑉 = (0...5)    &   𝐸 = ⟨“{0, 1} {0, 2} {1, 2} {0, 3} {3, 4} {3, 5} {4, 5}”⟩    ⇒   𝐸:dom 𝐸–1-1→{𝑒 ∈ 𝒫 𝑉 ∣ (♯‘𝑒) = 2}
 
2-Aug-2025s7f1o 15009 A length 7 word with mutually different symbols is a one-to-one function onto the set of the symbols. (Contributed by AV, 2-Aug-2025.)
((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐷 ∈ 𝑉 ∧ (𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉)) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ (𝐴 ≠ 𝐸 ∧ 𝐴 ≠ 𝐹 ∧ 𝐴 ≠ 𝐺)) ∧ ((𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ (𝐵 ≠ 𝐸 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐺)) ∧ (𝐶 ≠ 𝐷 ∧ (𝐶 ≠ 𝐸 ∧ 𝐶 ≠ 𝐹 ∧ 𝐶 ≠ 𝐺))) ∧ ((𝐷 ≠ 𝐸 ∧ 𝐷 ≠ 𝐹 ∧ 𝐷 ≠ 𝐺) ∧ (𝐸 ≠ 𝐹 ∧ 𝐸 ≠ 𝐺 ∧ 𝐹 ≠ 𝐺)))) → (𝐾 = ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ → 𝐾:(0..^7)–1-1-onto→(({𝐴, 𝐵, 𝐶} ∪ {𝐷}) ∪ {𝐸, 𝐹, 𝐺})))
 
2-Aug-2025hash7g 14529 The size of an unordered set of seven different elements. (Contributed by AV, 2-Aug-2025.)
((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐷 ∈ 𝑉 ∧ (𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉)) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ (𝐴 ≠ 𝐸 ∧ 𝐴 ≠ 𝐹 ∧ 𝐴 ≠ 𝐺)) ∧ ((𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ (𝐵 ≠ 𝐸 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐺)) ∧ (𝐶 ≠ 𝐷 ∧ (𝐶 ≠ 𝐸 ∧ 𝐶 ≠ 𝐹 ∧ 𝐶 ≠ 𝐺))) ∧ ((𝐷 ≠ 𝐸 ∧ 𝐷 ≠ 𝐹 ∧ 𝐷 ≠ 𝐺) ∧ (𝐸 ≠ 𝐹 ∧ 𝐸 ≠ 𝐺 ∧ 𝐹 ≠ 𝐺)))) → (♯‘(({𝐴, 𝐵, 𝐶} ∪ {𝐷}) ∪ {𝐸, 𝐹, 𝐺})) = 7)
 
1-Aug-2025in-ax8 36182 A proof of ax-8 2110 that does not rely on ax-8 2110. It employs df-in 3983 to perform alpha-renaming and eliminates disjoint variable conditions using ax-9 2118. Since the nature of this result is unclear, usage of this theorem is discouraged, and this method should not be applied to eliminate axiom dependencies. (Contributed by GG, 1-Aug-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝑥 ∈ 𝑧 → 𝑦 ∈ 𝑧))
 
1-Aug-2025s3rn 15007 Range of a length 3 string. (Contributed by Thierry Arnoux, 19-Sep-2023.) (Proof shortened by AV, 1-Aug-2025.)
(𝜑 → 𝐼 ∈ 𝐷)    &   (𝜑 → 𝐽 ∈ 𝐷)    &   (𝜑 → 𝐾 ∈ 𝐷)    ⇒   (𝜑 → ran ⟨“𝐼𝐽𝐾”⟩ = {𝐼, 𝐽, 𝐾})
 
1-Aug-2025s2rn 15006 Range of a length 2 string. (Contributed by Thierry Arnoux, 19-Sep-2023.) (Proof shortened by AV, 1-Aug-2025.)
(𝜑 → 𝐼 ∈ 𝐷)    &   (𝜑 → 𝐽 ∈ 𝐷)    ⇒   (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽})
 
31-Jul-2025dvelimexcasei 35046 Eliminate a disjoint variable condition from an existentially quantified statement using cases. Inference form of dvelimexcased 35045. See axnulg 35060 for an example of its use. (Contributed by BTernaryTau, 31-Jul-2025.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)    &   (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜒)    &   (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝜑 → 𝜒)))    &   (∀𝑥 𝑥 = 𝑦 → (𝜓 → 𝜒))    &   ∃𝑧𝜑    &   ∃𝑥𝜓    ⇒   ∃𝑥𝜒
 
31-Jul-2025dvelimexcased 35045 Eliminate a disjoint variable condition from an existentially quantified statement using cases. (Contributed by BTernaryTau, 31-Jul-2025.)
Ⅎ𝑥𝜑    &   (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜃)    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑧 = 𝑥 → (𝜓 → 𝜃)))    &   ((𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (𝜒 → 𝜃))    &   (𝜑 → ∃𝑧𝜓)    &   (𝜑 → ∃𝑥𝜒)    ⇒   (𝜑 → ∃𝑥𝜃)
 
31-Jul-2025dvelimalcasei 35044 Eliminate a disjoint variable condition from a universally quantified statement using cases. Inference form of dvelimalcased 35043. (Contributed by BTernaryTau, 31-Jul-2025.)
(¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑)    &   (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜒)    &   (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝜑 → 𝜒)))    &   (∀𝑥 𝑥 = 𝑦 → (𝜓 → 𝜒))    &   ∀𝑧𝜑    &   ∀𝑥𝜓    ⇒   ∀𝑥𝜒
 
31-Jul-2025dvelimalcased 35043 Eliminate a disjoint variable condition from a universally quantified statement using cases. (Contributed by BTernaryTau, 31-Jul-2025.)
Ⅎ𝑥𝜑    &   (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑)    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜃)    &   ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑧 = 𝑥 → (𝜓 → 𝜃)))    &   ((𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (𝜒 → 𝜃))    &   (𝜑 → ∀𝑧𝜓)    &   (𝜑 → ∀𝑥𝜒)    ⇒   (𝜑 → ∀𝑥𝜃)
 
31-Jul-2025cbvex1v 35042 Rule used to change bound variables, using implicit substitution. (Contributed by BTernaryTau, 31-Jul-2025.)
Ⅎ𝑥𝜑    &   Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   (𝜑 → (∃𝑥𝜓 → ∃𝑦𝜒))
 
31-Jul-2025nfan1c 35041 Variant of nfan 1898 and commuted form of nfan1 2201. (Contributed by BTernaryTau, 31-Jul-2025.)
Ⅎ𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   Ⅎ𝑥(𝜓 ∧ 𝜑)
 
30-Jul-2025bj-sylggt 36576 Stronger form of sylgt 1820, closer to ax-2 7. (Contributed by BJ, 30-Jul-2025.)
((𝜑 → ∀𝑥(𝜓 → 𝜒)) → ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜒)))
 
30-Jul-2025s7rn 15008 Range of a length 7 string. (Contributed by AV, 30-Jul-2025.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝐸 ∈ 𝑉)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝐺 ∈ 𝑉)    ⇒   (𝜑 → ran ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (({𝐴, 𝐵, 𝐶} ∪ {𝐷}) ∪ {𝐸, 𝐹, 𝐺}))
 
30-Jul-2025fz0to5un2tp 13682 An integer range from 0 to 5 is the union of two triples. (Contributed by AV, 30-Jul-2025.)
(0...5) = ({0, 1, 2} ∪ {3, 4, 5})
 
30-Jul-2025cbvexeqsetf 3503 The expression ∃𝑥𝑥 = 𝐴 means "𝐴 is a set" even if 𝐴 contains 𝑥 as a bound variable. This lemma helps minimizing axiom or df-clab 2718 usage in some cases. Extracted from the proof of issetft 3504. (Contributed by Wolf Lammen, 30-Jul-2025.)
(Ⅎ𝑥𝐴 → (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑦 𝑦 = 𝐴))
 
28-Jul-2025spcimgft 3558 Closed theorem form of spcimgf 3562. (Contributed by Wolf Lammen, 28-Jul-2025.)
(((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓))) → (𝐴 ∈ 𝑉 → (∀𝑥𝜑 → 𝜓)))
 
27-Jul-2025spcimgfi1 3559 A closed version of spcimgf 3562. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by Wolf Lammen, 27-Jul-2025.)
Ⅎ𝑥𝜓    &   Ⅎ𝑥𝐴    ⇒   (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → 𝜓)))
 
26-Jul-2025grtriclwlk3 47786 A triangle induces a closed walk of length 3 . (Contributed by AV, 26-Jul-2025.)
(𝜑 → 𝑇 ∈ (GrTriangles‘𝐺))    &   (𝜑 → 𝑃:(0..^3)–1-1-onto→𝑇)    ⇒   (𝜑 → 𝑃 ∈ (3 ClWWalksN 𝐺))
 
26-Jul-2025grtrissvtx 47785 A triangle is a subset of the vertices (of a graph). (Contributed by AV, 26-Jul-2025.)
𝑉 = (Vtx‘𝐺)    ⇒   (𝑇 ∈ (GrTriangles‘𝐺) → 𝑇 ⊆ 𝑉)
 
26-Jul-2025nnsge1 28356 A positive surreal integer is greater than or equal to one. (Contributed by Scott Fenton, 26-Jul-2025.)
(𝑁 ∈ ℕs → 1s ≤s 𝑁)
 
26-Jul-2025n0s0suc 28355 A non-negative surreal integer is either zero or a successor. (Contributed by Scott Fenton, 26-Jul-2025.)
(𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ ∃𝑥 ∈ ℕ0s 𝐴 = (𝑥 +s 1s )))
 
25-Jul-2025grtrif1o 47783 Any bijection onto a triangle preserves the edges of the triangle. (Contributed by AV, 25-Jul-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   ((𝑇 ∈ (GrTriangles‘𝐺) ∧ 𝐹:(0..^3)–1-1-onto→𝑇) → ({(𝐹‘0), (𝐹‘1)} ∈ 𝐸 ∧ {(𝐹‘0), (𝐹‘2)} ∈ 𝐸 ∧ {(𝐹‘1), (𝐹‘2)} ∈ 𝐸))
 
25-Jul-2025grtriprop 47782 The properties of a triangle. (Contributed by AV, 25-Jul-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   (𝑇 ∈ (GrTriangles‘𝐺) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
 
25-Jul-2025expsnnval 28419 Value of surreal exponentiation at a natural number. (Contributed by Scott Fenton, 25-Jul-2025.)
((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (𝐴↑s𝑁) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁))
 
25-Jul-2025uzsind 28401 Induction on the upper surreal integers that start at 𝑀. (Contributed by Scott Fenton, 25-Jul-2025.)
(𝑗 = 𝑀 → (𝜑 ↔ 𝜓))    &   (𝑗 = 𝑘 → (𝜑 ↔ 𝜒))    &   (𝑗 = (𝑘 +s 1s ) → (𝜑 ↔ 𝜃))    &   (𝑗 = 𝑁 → (𝜑 ↔ 𝜏))    &   (𝑀 ∈ ℤs → 𝜓)    &   ((𝑀 ∈ ℤs ∧ 𝑘 ∈ ℤs ∧ 𝑀 ≤s 𝑘) → (𝜒 → 𝜃))    ⇒   ((𝑀 ∈ ℤs ∧ 𝑁 ∈ ℤs ∧ 𝑀 ≤s 𝑁) → 𝜏)
 
25-Jul-2025peano5uzs 28400 Peano's inductive postulate for upper surreal integers. (Contributed by Scott Fenton, 25-Jul-2025.)
(𝜑 → 𝑁 ∈ ℤs)    &   (𝜑 → 𝑁 ∈ 𝐴)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥 +s 1s ) ∈ 𝐴)    ⇒   (𝜑 → {𝑘 ∈ ℤs ∣ 𝑁 ≤s 𝑘} ⊆ 𝐴)
 
25-Jul-2025zn0subs 28399 The non-negative difference of surreal integers is a non-negative integer. (Contributed by Scott Fenton, 25-Jul-2025.)
((𝑀 ∈ ℤs ∧ 𝑁 ∈ ℤs) → (𝑀 ≤s 𝑁 ↔ (𝑁 -s 𝑀) ∈ ℕ0s))
 
25-Jul-2025elznns 28398 Surreal integer property expressed in terms of positive integers and non-negative integers. (Contributed by Scott Fenton, 25-Jul-2025.)
(𝑁 ∈ ℤs ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ ( -us ‘𝑁) ∈ ℕ0s)))
 
25-Jul-2025elnnzs 28397 Positive surreal integer property expressed in terms of integers. (Contributed by Scott Fenton, 25-Jul-2025.)
(𝑁 ∈ ℕs ↔ (𝑁 ∈ ℤs ∧ 0s <s 𝑁))
 
25-Jul-2025eln0zs 28396 Non-negative surreal integer property expressed in terms of integers. (Contributed by Scott Fenton, 25-Jul-2025.)
(𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℤs ∧ 0s ≤s 𝑁))
 
25-Jul-2025elzs2 28395 A surreal integer is either a positive integer, zero, or the negative of a positive integer. (Contributed by Scott Fenton, 25-Jul-2025.)
(𝑁 ∈ ℤs ↔ (𝑁 ∈ No ∧ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ∨ ( -us ‘𝑁) ∈ ℕs)))
 
25-Jul-2025zsubscld 28392 The surreal integers are closed under subtraction. (Contributed by Scott Fenton, 25-Jul-2025.)
(𝜑 → 𝐴 ∈ ℤs)    &   (𝜑 → 𝐵 ∈ ℤs)    ⇒   (𝜑 → (𝐴 -s 𝐵) ∈ ℤs)
 
25-Jul-2025zaddscld 28391 The surreal integers are closed under addition. (Contributed by Scott Fenton, 25-Jul-2025.)
(𝜑 → 𝐴 ∈ ℤs)    &   (𝜑 → 𝐵 ∈ ℤs)    ⇒   (𝜑 → (𝐴 +s 𝐵) ∈ ℤs)
 
25-Jul-2025zaddscl 28390 The surreal integers are closed under addition. (Contributed by Scott Fenton, 25-Jul-2025.)
((𝐴 ∈ ℤs ∧ 𝐵 ∈ ℤs) → (𝐴 +s 𝐵) ∈ ℤs)
 
25-Jul-2025nnzsubs 28381 The difference of two surreal positive integers is an integer. (Contributed by Scott Fenton, 25-Jul-2025.)
((𝐴 ∈ ℕs ∧ 𝐵 ∈ ℕs) → (𝐴 -s 𝐵) ∈ ℤs)
 
25-Jul-2025addsubs4d 28140 Rearrangement of four terms in mixed addition and subtraction. Surreal version. (Contributed by Scott Fenton, 25-Jul-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐷 ∈ No )    ⇒   (𝜑 → ((𝐴 +s 𝐵) -s (𝐶 +s 𝐷)) = ((𝐴 -s 𝐶) +s (𝐵 -s 𝐷)))
 
25-Jul-2025cflem 10308 A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set 𝐴. (Contributed by NM, 24-Apr-2004.) Avoid ax-11 2158. (Revised by BTernaryTau, 25-Jul-2025.)
(𝐴 ∈ 𝑉 → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
 
24-Jul-2025exps1 28421 Surreal exponentiation to one. (Contributed by Scott Fenton, 24-Jul-2025.)
(𝐴 ∈ No → (𝐴↑s 1s ) = 𝐴)
 
24-Jul-2025exps0 28420 Surreal exponentiation to zero. (Contributed by Scott Fenton, 24-Jul-2025.)
(𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
 
24-Jul-2025expsval 28418 The value of surreal exponentiation. (Contributed by Scott Fenton, 24-Jul-2025.)
((𝐴 ∈ No ∧ 𝐵 ∈ ℤs) → (𝐴↑s𝐵) = if(𝐵 = 0s , 1s , if( 0s <s 𝐵, (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵))))))
 
24-Jul-20251zs 28387 One is a surreal integer. (Contributed by Scott Fenton, 24-Jul-2025.)
1s ∈ ℤs
 
24-Jul-2025negs1s 28069 An expression for negative surreal one. (Contributed by Scott Fenton, 24-Jul-2025.)
( -us ‘ 1s ) = (∅ |s { 0s })
 
24-Jul-2025pwnss 5370 The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Proof shortened by BJ, 24-Jul-2025.)
(𝐴 ∈ 𝑉 → ¬ 𝒫 𝐴 ⊆ 𝐴)
 
24-Jul-2025rabexg 5355 Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.) (Proof shortened by BJ, 24-Jul-2025.)
(𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
 
24-Jul-2025sbiev 2318 Conversion of implicit substitution to explicit substitution. Version of sbie 2510 with a disjoint variable condition, not requiring ax-13 2380. See sbievw 2093 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by Wolf Lammen, 18-Jan-2023.) Remove dependence on ax-10 2141 and shorten proof. (Revised by BJ, 18-Jul-2023.) (Proof shortened by SN, 24-Jul-2025.)
Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
 
23-Jul-2025grtriproplem 47780 Lemma for grtriprop 47782. (Contributed by AV, 23-Jul-2025.)
((𝑓:(0..^3)–1-1-onto→{𝑥, 𝑦, 𝑧} ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸)) → ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸))
 
23-Jul-2025nohalf 28417 An explicit expression for one half. This theorem avoids the axiom of infinity. (Contributed by Scott Fenton, 23-Jul-2025.)
( 1s /su 2s) = ({ 0s } |s { 1s })
 
23-Jul-20252ne0s 28414 Surreal two is non-zero. (Contributed by Scott Fenton, 23-Jul-2025.)
2s ≠ 0s
 
23-Jul-20252sno 28413 Surreal two is a surreal number. (Contributed by Scott Fenton, 23-Jul-2025.)
2s ∈ No
 
23-Jul-20252nns 28412 Surreal two is a surreal natural. (Contributed by Scott Fenton, 23-Jul-2025.)
2s ∈ ℕs
 
23-Jul-2025no2times 28411 Version of 2times 12423 for surreal numbers. (Contributed by Scott Fenton, 23-Jul-2025.)
(𝐴 ∈ No → (2s ·s 𝐴) = (𝐴 +s 𝐴))
 
23-Jul-2025fvf1tp 13834 Values of a one-to-one function between two sets with three elements. Actually, such a function is a bijection. (Contributed by AV, 23-Jul-2025.)
(𝐹:(0..^3)–1-1→{𝑋, 𝑌, 𝑍} → ((((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑋 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑌)) ∨ (((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑍) ∨ ((𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑍 ∧ (𝐹‘2) = 𝑋)) ∨ (((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑋 ∧ (𝐹‘2) = 𝑌) ∨ ((𝐹‘0) = 𝑍 ∧ (𝐹‘1) = 𝑌 ∧ (𝐹‘2) = 𝑋))))
 
22-Jul-2025fvf1pr 7338 Values of a one-to-one function between two sets with two elements. Actually, such a function is a bijection. (Contributed by AV, 22-Jul-2025.)
(((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ 𝐹:{𝐴, 𝐵}–1-1→{𝑋, 𝑌}) → (((𝐹‘𝐴) = 𝑋 ∧ (𝐹‘𝐵) = 𝑌) ∨ ((𝐹‘𝐴) = 𝑌 ∧ (𝐹‘𝐵) = 𝑋)))
 
21-Jul-2025tpf1o 14544 A bijection onto a (proper) triple. (Contributed by AV, 21-Jul-2025.)
𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))    &   𝑇 = {𝐴, 𝐵, 𝐶}    ⇒   (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (♯‘𝑇) = 3) → 𝐹:(0..^3)–1-1-onto→𝑇)
 
21-Jul-2025hash3tpb 14538 A set of size three is a proper unordered triple. (Contributed by AV, 21-Jul-2025.)
(𝑉 ∈ 𝑊 → ((♯‘𝑉) = 3 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ∧ 𝑉 = {𝑎, 𝑏, 𝑐})))
 
21-Jul-2025hash3tpexb 14537 A set of size three is an unordered triple if and only if it contains three different elements. (Contributed by AV, 21-Jul-2025.)
(𝑉 ∈ 𝑊 → ((♯‘𝑉) = 3 ↔ ∃𝑎∃𝑏∃𝑐((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ∧ 𝑉 = {𝑎, 𝑏, 𝑐})))
 
21-Jul-2025hash3tpde 14536 A set of size three is an unordered triple of three different elements. (Contributed by AV, 21-Jul-2025.)
((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → ∃𝑎∃𝑏∃𝑐((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ∧ 𝑉 = {𝑎, 𝑏, 𝑐}))
 
21-Jul-2025unexg 7772 The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.) Prove unexg 7772 first and then unex 7773 and unexb 7776 from it. (Revised by BJ, 21-Jul-2025.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
 
21-Jul-2025rexeqtrrdv 3339 Substitution of equal classes into a restricted existential quantifier. (Contributed by Matthew House, 21-Jul-2025.)
(𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    &   (𝜑 → 𝐵 = 𝐴)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
 
21-Jul-2025raleqtrrdv 3338 Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025.)
(𝜑 → ∀𝑥 ∈ 𝐴 𝜓)    &   (𝜑 → 𝐵 = 𝐴)    ⇒   (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
 
21-Jul-2025rexeqtrdv 3337 Substitution of equal classes into a restricted existential quantifier. (Contributed by Matthew House, 21-Jul-2025.)
(𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    &   (𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
 
21-Jul-2025raleqtrdv 3336 Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025.)
(𝜑 → ∀𝑥 ∈ 𝐴 𝜓)    &   (𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
 
21-Jul-2025r3ex 3204 Triple existential quantification. (Contributed by AV, 21-Jul-2025.)
(∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝜑 ↔ ∃𝑥∃𝑦∃𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝜑))
 
20-Jul-2025isgrtri 47784 A triangle in a graph. (Contributed by AV, 20-Jul-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   (𝑇 ∈ (GrTriangles‘𝐺) ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 ∃𝑧 ∈ 𝑉 (𝑇 = {𝑥, 𝑦, 𝑧} ∧ (♯‘𝑇) = 3 ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑥, 𝑧} ∈ 𝐸 ∧ {𝑦, 𝑧} ∈ 𝐸)))
 
20-Jul-2025grtri 47781 The triangles in a graph. (Contributed by AV, 20-Jul-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   (𝐺 ∈ 𝑊 → (GrTriangles‘𝐺) = {𝑡 ∈ 𝒫 𝑉 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝐸 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝐸 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝐸))})
 
20-Jul-2025df-grtri 47779 Definition of a triangles in a graph. A triangle in a graph is a set of three (different) vertices completely connected with each other. Such vertices induce a closed walk of length 3, see grtriclwlk3 47786. (TODO: and a cycle of length 3 ,see grtricycl ). (Contributed by AV, 20-Jul-2025.)
GrTriangles = (𝑔 ∈ V ↦ ⦋(Vtx‘𝑔) / 𝑣⦌⦋(Edg‘𝑔) / 𝑒⦌{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))})
 
20-Jul-2025tpfo 14543 A function onto a (proper) triple. (Contributed by AV, 20-Jul-2025.)
𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))    &   𝑇 = {𝐴, 𝐵, 𝐶}    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐹:(0..^3)–onto→𝑇)
 
20-Jul-2025tpf 14542 A function into a (proper) triple. (Contributed by AV, 20-Jul-2025.)
𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))    &   𝑇 = {𝐴, 𝐵, 𝐶}    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐹:(0..^3)⟶𝑇)
 
20-Jul-2025tpf1ofv2 14541 The value of a one-to-one function onto a triple at 2. (Contributed by AV, 20-Jul-2025.)
𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))    ⇒   (𝐶 ∈ 𝑉 → (𝐹‘2) = 𝐶)
 
20-Jul-2025tpf1ofv1 14540 The value of a one-to-one function onto a triple at 1. (Contributed by AV, 20-Jul-2025.)
𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))    ⇒   (𝐵 ∈ 𝑉 → (𝐹‘1) = 𝐵)
 
20-Jul-2025tpf1ofv0 14539 The value of a one-to-one function onto a triple at 0. (Contributed by AV, 20-Jul-2025.)
𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))    ⇒   (𝐴 ∈ 𝑉 → (𝐹‘0) = 𝐴)
 
19-Jul-2025cbvrabw 3481 Rule to change the bound variable in a restricted class abstraction, using implicit substitution. Version of cbvrab 3487 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by Andrew Salmon, 11-Jul-2011.) Avoid ax-13 2380. (Revised by GG, 10-Jan-2024.) Avoid ax-10 2141. (Revised by Wolf Lammen, 19-Jul-2025.)
Ⅎ𝑥𝐴    &   Ⅎ𝑦𝐴    &   Ⅎ𝑦𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐴 ∣ 𝜓}
 
18-Jul-2025raleleq 3350 All elements of a class are elements of a class equal to this class. (Contributed by AV, 30-Oct-2020.) (Proof shortened by Wolf Lammen, 18-Jul-2025.)
(𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵)
 
14-Jul-2025unitscyglem4 42148 Lemma for unitscyg (Contributed by metakunt, 14-Jul-2025.)
𝐵 = (Base‘𝐺)    &    ↑ = (.g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝐵 ∈ Fin)    &   (𝜑 → ∀𝑛 ∈ ℕ (♯‘{𝑥 ∈ 𝐵 ∣ (𝑛 ↑ 𝑥) = (0g‘𝐺)}) ≤ 𝑛)    &   (𝜑 → 𝐷 ∈ ℕ)    &   (𝜑 → 𝐷 ∥ (♯‘𝐵))    ⇒   (𝜑 → (♯‘{𝑦 ∈ 𝐵 ∣ ((od‘𝐺)‘𝑦) = 𝐷}) = (ϕ‘𝐷))
 
14-Jul-2025unitscyglem3 42147 Lemma for unitscyg. (Contributed by metakunt, 14-Jul-2025.)
𝐵 = (Base‘𝐺)    &    ↑ = (.g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝐵 ∈ Fin)    &   (𝜑 → ∀𝑛 ∈ ℕ (♯‘{𝑥 ∈ 𝐵 ∣ (𝑛 ↑ 𝑥) = (0g‘𝐺)}) ≤ 𝑛)    ⇒   (𝜑 → ∀𝑑 ∈ ℕ ((𝑑 ∥ (♯‘𝐵) ∧ {𝑥 ∈ 𝐵 ∣ ((od‘𝐺)‘𝑥) = 𝑑} ≠ ∅) → (♯‘{𝑥 ∈ 𝐵 ∣ ((od‘𝐺)‘𝑥) = 𝑑}) = (ϕ‘𝑑)))
 
14-Jul-2025grpods 42144 Relate sums of elements of orders and roots of unity. (Contributed by metakunt, 14-Jul-2025.)
𝐵 = (Base‘𝐺)    &    ↑ = (.g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝐵 ∈ Fin)    &   (𝜑 → 𝑁 ∈ ℕ)    ⇒   (𝜑 → Σ𝑘 ∈ {𝑚 ∈ (1...𝑁) ∣ 𝑚 ∥ 𝑁} (♯‘{𝑥 ∈ 𝐵 ∣ ((od‘𝐺)‘𝑥) = 𝑘}) = (♯‘{𝑥 ∈ 𝐵 ∣ (𝑁 ↑ 𝑥) = (0g‘𝐺)}))
 
14-Jul-2025isprimroot2 42044 Alternative way of creating primitive roots. (Contributed by metakunt, 14-Jul-2025.)
(𝜑 → 𝑅 ∈ CMnd)    &   (𝜑 → 𝐾 ∈ ℕ)    &   (𝜑 → 𝑀 ∈ (Base‘𝑅))    &   (𝜑 → ((od‘𝑅)‘𝑀) = 𝐾)    ⇒   (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))
 
13-Jul-2025exfinfldd 42153 For any prime 𝑃 and any positive integer 𝑁 there exists a field 𝑘 such that 𝑘 contains 𝑃↑𝑁 elements. (Contributed by metakunt, 13-Jul-2025.)
(𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑁 ∈ ℕ)    ⇒   (𝜑 → ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑃↑𝑁) ∧ (chr‘𝑘) = 𝑃))
 
13-Jul-2025ax-exfinfld 42152 Existence axiom for finite fields, eventually we want to construct them. (Contributed by metakunt, 13-Jul-2025.)
∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ∃𝑘 ∈ Field ((♯‘(Base‘𝑘)) = (𝑝↑𝑛) ∧ (chr‘𝑘) = 𝑝)
 
13-Jul-2025unitscyglem2 42146 Lemma for unitscyg. (Contributed by metakunt, 13-Jul-2025.)
𝐵 = (Base‘𝐺)    &    ↑ = (.g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝐵 ∈ Fin)    &   (𝜑 → ∀𝑛 ∈ ℕ (♯‘{𝑥 ∈ 𝐵 ∣ (𝑛 ↑ 𝑥) = (0g‘𝐺)}) ≤ 𝑛)    &   (𝜑 → 𝐷 ∈ ℕ)    &   (𝜑 → 𝐷 ∥ (♯‘𝐵))    &   (𝜑 → 𝐴 ∈ 𝐵)    &   (𝜑 → ((od‘𝐺)‘𝐴) = 𝐷)    &   (𝜑 → ∀𝑐 ∈ ℕ (𝑐 < 𝐷 → ((𝑐 ∥ (♯‘𝐵) ∧ {𝑥 ∈ 𝐵 ∣ ((od‘𝐺)‘𝑥) = 𝑐} ≠ ∅) → (♯‘{𝑥 ∈ 𝐵 ∣ ((od‘𝐺)‘𝑥) = 𝑐}) = (ϕ‘𝑐))))    ⇒   (𝜑 → (♯‘{𝑥 ∈ 𝐵 ∣ ((od‘𝐺)‘𝑥) = 𝐷}) = (ϕ‘𝐷))
 
13-Jul-2025unitscyglem1 42145 Lemma for unitscyg. (Contributed by metakunt, 13-Jul-2025.)
𝐵 = (Base‘𝐺)    &    ↑ = (.g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝐵 ∈ Fin)    &   (𝜑 → ∀𝑛 ∈ ℕ (♯‘{𝑥 ∈ 𝐵 ∣ (𝑛 ↑ 𝑥) = (0g‘𝐺)}) ≤ 𝑛)    &   (𝜑 → 𝐴 ∈ 𝐵)    ⇒   (𝜑 → (♯‘{𝑥 ∈ 𝐵 ∣ (((od‘𝐺)‘𝐴) ↑ 𝑥) = (0g‘𝐺)}) = ((od‘𝐺)‘𝐴))
 
13-Jul-2025indstrd 42143 Strong induction, deduction version. (Contributed by Steven Nguyen, 13-Jul-2025.)
(𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    &   (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    &   ((𝜑 ∧ 𝑥 ∈ ℕ ∧ ∀𝑦 ∈ ℕ (𝑦 < 𝑥 → 𝜒)) → 𝜓)    &   (𝜑 → 𝐴 ∈ ℕ)    ⇒   (𝜑 → 𝜃)
 
11-Jul-2025gricgrlic 47825 Isomorphic hypergraphs are locally isomorphic. (Contributed by AV, 12-Jun-2025.) (Proof shortened by AV, 11-Jul-2025.)
((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) → (𝐺 ≃𝑔𝑟 𝐻 → 𝐺 ≃𝑙𝑔𝑟 𝐻))
 
11-Jul-2025gricer 47767 Isomorphism is an equivalence relation on hypergraphs. (Contributed by AV, 3-May-2025.) (Proof shortened by AV, 11-Jul-2025.)
( ≃𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph
 
9-Jul-2025div0 11976 Division into zero is zero. (Contributed by NM, 14-Mar-2005.) (Proof shortened by SN, 9-Jul-2025.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0 / 𝐴) = 0)
 
9-Jul-2025divid 11974 A number divided by itself is one. (Contributed by NM, 1-Aug-2004.) (Proof shortened by SN, 9-Jul-2025.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴 / 𝐴) = 1)
 
9-Jul-2025div11 11971 One-to-one relationship for division. (Contributed by NM, 20-Apr-2006.) (Proof shortened by Mario Carneiro, 27-May-2016.) (Proof shortened by SN, 9-Jul-2025.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) = (𝐵 / 𝐶) ↔ 𝐴 = 𝐵))
 
8-Jul-2025grpinv11 19041 The group inverse is one-to-one. (Contributed by NM, 22-Mar-2015.) (Proof shortened by SN, 8-Jul-2025.)
𝐵 = (Base‘𝐺)    &   𝑁 = (invg‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → ((𝑁‘𝑋) = (𝑁‘𝑌) ↔ 𝑋 = 𝑌))
 
7-Jul-2025fimgmcyclem 42481 Lemma for fimgmcyc 42482. (Contributed by SN, 7-Jul-2025.)
(𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))    ⇒   (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
 
6-Jul-2025fisdomnn 42232 A finite set is dominated by the set of natural numbers. (Contributed by SN, 6-Jul-2025.)
(𝐴 ∈ Fin → 𝐴 ≺ ℕ)
 
6-Jul-2025constrelextdg2 33729 If the 𝑁-th step (𝐶‘𝑁) of the construction of constuctible numbers is included in a subfield 𝐹 of the complex numbers, then any element 𝑋 of the next step (𝐶‘suc 𝑁) is either in 𝐹 or in a quadratic extension of 𝐹. (Contributed by Thierry Arnoux, 6-Jul-2025.)
𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   𝐾 = (ℂfld ↾s 𝐹)    &   𝐿 = (ℂfld ↾s (ℂfld fldGen (𝐹 ∪ {𝑋})))    &   (𝜑 → 𝐹 ∈ (SubDRing‘ℂfld))    &   (𝜑 → 𝑁 ∈ On)    &   (𝜑 → (𝐶‘𝑁) ⊆ 𝐹)    &   (𝜑 → 𝑋 ∈ (𝐶‘suc 𝑁))    ⇒   (𝜑 → (𝑋 ∈ 𝐹 ∨ (𝐿[:]𝐾) = 2))
 
6-Jul-2025constrfin 33728 Each step of the construction of constructible numbers is finite. (Contributed by Thierry Arnoux, 6-Jul-2025.)
𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   (𝜑 → 𝑁 ∈ ω)    ⇒   (𝜑 → (𝐶‘𝑁) ∈ Fin)
 
6-Jul-2025constrrtcc 33718 In the construction of constructible numbers, circle-circle intersections are roots of a quadratic equation. (Contributed by Thierry Arnoux, 6-Jul-2025.)
(𝜑 → 𝑆 ⊆ ℂ)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   (𝜑 → 𝐵 ∈ 𝑆)    &   (𝜑 → 𝐶 ∈ 𝑆)    &   (𝜑 → 𝐷 ∈ 𝑆)    &   (𝜑 → 𝐸 ∈ 𝑆)    &   (𝜑 → 𝐹 ∈ 𝑆)    &   (𝜑 → 𝑋 ∈ ℂ)    &   (𝜑 → 𝐴 ≠ 𝐷)    &   (𝜑 → (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐶)))    &   (𝜑 → (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝐹)))    &   𝑃 = ((𝐵 − 𝐶) · (∗‘(𝐵 − 𝐶)))    &   𝑄 = ((𝐸 − 𝐹) · (∗‘(𝐸 − 𝐹)))    &   𝑀 = (((𝑄 − ((∗‘𝐷) · (𝐷 + 𝐴))) − (𝑃 − ((∗‘𝐴) · (𝐷 + 𝐴)))) / ((∗‘𝐷) − (∗‘𝐴)))    &   𝑁 = -(((((∗‘𝐴) · (𝐷 · 𝐴)) − (𝑃 · 𝐷)) − (((∗‘𝐷) · (𝐷 · 𝐴)) − (𝑄 · 𝐴))) / ((∗‘𝐷) − (∗‘𝐴)))    ⇒   (𝜑 → ((𝑋↑2) + ((𝑀 · 𝑋) + 𝑁)) = 0)
 
6-Jul-2025constrrtcclem 33717 In the construction of constructible numbers, circle-circle intersections are roots of a quadratic equation. Case of non-degenerate circles. (Contributed by Thierry Arnoux, 6-Jul-2025.)
(𝜑 → 𝑆 ⊆ ℂ)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   (𝜑 → 𝐵 ∈ 𝑆)    &   (𝜑 → 𝐶 ∈ 𝑆)    &   (𝜑 → 𝐷 ∈ 𝑆)    &   (𝜑 → 𝐸 ∈ 𝑆)    &   (𝜑 → 𝐹 ∈ 𝑆)    &   (𝜑 → 𝑋 ∈ ℂ)    &   (𝜑 → 𝐴 ≠ 𝐷)    &   (𝜑 → (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐶)))    &   (𝜑 → (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝐹)))    &   𝑃 = ((𝐵 − 𝐶) · (∗‘(𝐵 − 𝐶)))    &   𝑄 = ((𝐸 − 𝐹) · (∗‘(𝐸 − 𝐹)))    &   𝑀 = (((𝑄 − ((∗‘𝐷) · (𝐷 + 𝐴))) − (𝑃 − ((∗‘𝐴) · (𝐷 + 𝐴)))) / ((∗‘𝐷) − (∗‘𝐴)))    &   𝑁 = -(((((∗‘𝐴) · (𝐷 · 𝐴)) − (𝑃 · 𝐷)) − (((∗‘𝐷) · (𝐷 · 𝐴)) − (𝑄 · 𝐴))) / ((∗‘𝐷) − (∗‘𝐴)))    &   (𝜑 → 𝐵 ≠ 𝐶)    &   (𝜑 → 𝐸 ≠ 𝐹)    ⇒   (𝜑 → ((𝑋↑2) + ((𝑀 · 𝑋) + 𝑁)) = 0)
 
6-Jul-2025constrrtlc2 33716 In the construction of constructible numbers, line-circle intersections are one of the original points, in a degenerate case. (Contributed by Thierry Arnoux, 6-Jul-2025.)
(𝜑 → 𝑆 ⊆ ℂ)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   (𝜑 → 𝐵 ∈ 𝑆)    &   (𝜑 → 𝐶 ∈ 𝑆)    &   (𝜑 → 𝐸 ∈ 𝑆)    &   (𝜑 → 𝐹 ∈ 𝑆)    &   (𝜑 → 𝑇 ∈ ℝ)    &   (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))    &   (𝜑 → (abs‘(𝑋 − 𝐶)) = (abs‘(𝐸 − 𝐹)))    &   (𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → 𝑋 = 𝐴)
 
6-Jul-2025constrrtlc1 33715 In the construction of constructible numbers, line-circle intersections are roots of a quadratic equation, non-degenerate case. (Contributed by Thierry Arnoux, 6-Jul-2025.)
(𝜑 → 𝑆 ⊆ ℂ)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   (𝜑 → 𝐵 ∈ 𝑆)    &   (𝜑 → 𝐶 ∈ 𝑆)    &   (𝜑 → 𝐸 ∈ 𝑆)    &   (𝜑 → 𝐹 ∈ 𝑆)    &   (𝜑 → 𝑇 ∈ ℝ)    &   (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))    &   (𝜑 → (abs‘(𝑋 − 𝐶)) = (abs‘(𝐸 − 𝐹)))    &   𝑄 = (((∗‘𝐵) − (∗‘𝐴)) / (𝐵 − 𝐴))    &   𝑀 = (((((∗‘𝐴) − (𝐴 · 𝑄)) − (∗‘𝐶)) − (𝐶 · 𝑄)) / 𝑄)    &   𝑁 = (-((𝐶 · (((∗‘𝐴) − (𝐴 · 𝑄)) − (∗‘𝐶))) + ((𝐸 − 𝐹) · ((∗‘𝐸) − (∗‘𝐹)))) / 𝑄)    &   (𝜑 → 𝐴 ≠ 𝐵)    ⇒   (𝜑 → (((𝑋↑2) + ((𝑀 · 𝑋) + 𝑁)) = 0 ∧ 𝑄 ≠ 0))
 
6-Jul-2025constrrtll 33714 In the construction of constructible numbers, line-line intersections are solutions of linear equations, and can therefore be completely constructed. (Contributed by Thierry Arnoux, 6-Jul-2025.)
(𝜑 → 𝑆 ⊆ ℂ)    &   (𝜑 → 𝐴 ∈ 𝑆)    &   (𝜑 → 𝐵 ∈ 𝑆)    &   (𝜑 → 𝐶 ∈ 𝑆)    &   (𝜑 → 𝐷 ∈ 𝑆)    &   (𝜑 → 𝑇 ∈ ℝ)    &   (𝜑 → 𝑅 ∈ ℝ)    &   (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))    &   (𝜑 → 𝑋 = (𝐶 + (𝑅 · (𝐷 − 𝐶))))    &   (𝜑 → (ℑ‘((∗‘(𝐵 − 𝐴)) · (𝐷 − 𝐶))) ≠ 0)    &   𝑁 = (𝐴 + (((((𝐴 − 𝐶) · ((∗‘𝐷) − (∗‘𝐶))) − (((∗‘𝐴) − (∗‘𝐶)) · (𝐷 − 𝐶))) / ((((∗‘𝐵) − (∗‘𝐴)) · (𝐷 − 𝐶)) − ((𝐵 − 𝐴) · ((∗‘𝐷) − (∗‘𝐶))))) · (𝐵 − 𝐴)))    ⇒   (𝜑 → 𝑋 = 𝑁)
 
6-Jul-2025cnfldfld 33328 The complex numbers form a field. (Contributed by Thierry Arnoux, 6-Jul-2025.)
ℂfld ∈ Field
 
6-Jul-2025subrgmcld 33205 A subring is closed under multiplication. (Contributed by Thierry Arnoux, 6-Jul-2025.)
· = (.r‘𝑅)    &   (𝜑 → 𝐴 ∈ (SubRing‘𝑅))    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐴)    ⇒   (𝜑 → (𝑋 · 𝑌) ∈ 𝐴)
 
6-Jul-2025subgsubcld 33020 A subgroup is closed under group subtraction. (Contributed by Thierry Arnoux, 6-Jul-2025.)
− = (-g‘𝐺)    &   (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑 → 𝑋 ∈ 𝑆)    &   (𝜑 → 𝑌 ∈ 𝑆)    ⇒   (𝜑 → (𝑋 − 𝑌) ∈ 𝑆)
 
6-Jul-2025quad3d 32749 Variant of quadratic equation with discriminant expanded. (Contributed by Filip Cernatescu, 19-Oct-2019.) Deduction version. (Revised by Thierry Arnoux, 6-Jul-2025.)
(𝜑 → 𝑋 ∈ ℂ)    &   (𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 ≠ 0)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → ((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0)    ⇒   (𝜑 → (𝑋 = ((-𝐵 + (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴)) ∨ 𝑋 = ((-𝐵 − (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴))))
 
6-Jul-2025muldivdid 32746 Distribution of division over addition with a multiplication. (Contributed by Thierry Arnoux, 6-Jul-2025.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐵 ≠ 0)    ⇒   (𝜑 → (((𝐴 · 𝐵) + 𝐶) / 𝐵) = (𝐴 + (𝐶 / 𝐵)))
 
6-Jul-2025submuladdd 32745 The product of a difference and a sum. Cf. addmulsub 11746. (Contributed by Thierry Arnoux, 6-Jul-2025.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐷 ∈ ℂ)    ⇒   (𝜑 → ((𝐴 − 𝐵) · (𝐶 + 𝐷)) = (((𝐴 · 𝐶) + (𝐴 · 𝐷)) − ((𝐵 · 𝐶) + (𝐵 · 𝐷))))
 
6-Jul-2025rabrexfi 32526 Conditions for a class abstraction with a restricted existential quantification to be finite. (Contributed by Thierry Arnoux, 6-Jul-2025.)
(𝜑 → 𝐵 ∈ Fin)    &   ((𝜑 ∧ 𝑦 ∈ 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ Fin)    ⇒   (𝜑 → {𝑥 ∈ 𝐴 ∣ ∃𝑦 ∈ 𝐵 𝜓} ∈ Fin)
 
6-Jul-20253unrab 32523 Union of three restricted class abstractions. (Contributed by Thierry Arnoux, 6-Jul-2025.)
(({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ 𝜓}) ∪ {𝑥 ∈ 𝐴 ∣ 𝜒}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ 𝜓 ∨ 𝜒)}
 
6-Jul-2025rabsstp 32522 Conditions for a restricted class abstraction to be a subset of an unordered triplet. (Contributed by Thierry Arnoux, 6-Jul-2025.)
({𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ {𝑋, 𝑌, 𝑍} ↔ ∀𝑥 ∈ 𝑉 (𝜑 → (𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍)))
 
6-Jul-2025rabsspr 32521 Conditions for a restricted class abstraction to be a subset of an unordered pair. (Contributed by Thierry Arnoux, 6-Jul-2025.)
({𝑥 ∈ 𝑉 ∣ 𝜑} ⊆ {𝑋, 𝑌} ↔ ∀𝑥 ∈ 𝑉 (𝜑 → (𝑥 = 𝑋 ∨ 𝑥 = 𝑌)))
 
5-Jul-2025rexor 42616 Alias for r19.43 3128 for easier lookup. (Contributed by SN, 5-Jul-2025.) (New usage is discouraged.)
(∃𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
 
5-Jul-2025exor 42615 Alias for 19.43 1881 for easier lookup. (Contributed by SN, 5-Jul-2025.) (New usage is discouraged.)
(∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
 
5-Jul-2025sn-eluzp1l 42289 Shorter proof of eluzp1l 12924. (Contributed by NM, 12-Sep-2005.) (Revised by SN, 5-Jul-2025.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 < 𝑁)
 
5-Jul-2025eluzp1 42288 Membership in a successor upper set of integers. (Contributed by SN, 5-Jul-2025.)
(𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) ↔ (𝑁 ∈ ℤ ∧ 𝑀 < 𝑁)))
 
4-Jul-2025sn-negex12 42385 A combination of cnegex 11465 and cnegex2 11466, this proof takes cnre 11281 𝐴 = 𝑟 + i · 𝑠 and shows that i · -𝑠 + -𝑟 is both a left and right inverse. (Contributed by SN, 5-May-2024.) (Proof shortened by SN, 4-Jul-2025.)
(𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ ((𝐴 + 𝑏) = 0 ∧ (𝑏 + 𝐴) = 0))
 
4-Jul-2025rsubrotld 42260 Rotate the variables left in an equation with subtraction on the right, converting it into an addition.

EDITORIAL: The label for this theorem is questionable. Do not move until it would have 7 uses: current additional uses: (none). (Contributed by SN, 4-Jul-2025.)

(𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐴 = (𝐵 − 𝐶))    ⇒   (𝜑 → 𝐵 = (𝐶 + 𝐴))
 
3-Jul-2025fiabv 42484 In a finite domain (a finite field), the only absolute value is the trivial one (abvtrivg 20850). (Contributed by SN, 3-Jul-2025.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝑇 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, 1))    &   (𝜑 → 𝑅 ∈ Domn)    &   (𝜑 → 𝐵 ∈ Fin)    ⇒   (𝜑 → 𝐴 = {𝑇})
 
3-Jul-2025fidomncyc 42483 Version of odcl2 19601 for multiplicative groups of finite domains (that is, a finite monoid where nonzero elements are cancellable): one (1) is a multiple of any nonzero element. (Contributed by SN, 3-Jul-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    &    ↑ = (.g‘(mulGrp‘𝑅))    &   (𝜑 → 𝑅 ∈ Domn)    &   (𝜑 → 𝐵 ∈ Fin)    &   (𝜑 → 𝐴 ∈ (𝐵 ∖ { 0 }))    ⇒   (𝜑 → ∃𝑛 ∈ ℕ (𝑛 ↑ 𝐴) = 1 )
 
3-Jul-2025fimgmcyc 42482 Version of odcl2 19601 for finite magmas: the multiples of an element 𝐴 ∈ 𝐵 are eventually periodic. (Contributed by SN, 3-Jul-2025.)
𝐵 = (Base‘𝑀)    &    · = (.g‘𝑀)    &   (𝜑 → 𝑀 ∈ Mgm)    &   (𝜑 → 𝐵 ∈ Fin)    &   (𝜑 → 𝐴 ∈ 𝐵)    ⇒   (𝜑 → ∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴))
 
3-Jul-2025abvexp 42480 Move exponentiation in and out of absolute value. (Contributed by SN, 3-Jul-2025.)
𝐴 = (AbsVal‘𝑅)    &    ↑ = (.g‘(mulGrp‘𝑅))    &   𝐵 = (Base‘𝑅)    &   (𝜑 → 𝑅 ∈ NzRing)    &   (𝜑 → 𝐹 ∈ 𝐴)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑁 ∈ ℕ0)    ⇒   (𝜑 → (𝐹‘(𝑁 ↑ 𝑋)) = ((𝐹‘𝑋)↑𝑁))
 
3-Jul-2025asclf1 42479 Two ways of saying the scalar injection is one-to-one. (Contributed by SN, 3-Jul-2025.)
𝐴 = (algSc‘𝑊)    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)    &    0 = (0g‘𝑊)    &   𝑁 = (0g‘𝑆)    &   (𝜑 → 𝑊 ∈ Ring)    &   (𝜑 → 𝑊 ∈ LMod)    ⇒   (𝜑 → (𝐴:𝐾–1-1→𝐵 ↔ ∀𝑠 ∈ 𝐾 ((𝐴‘𝑠) = 0 → 𝑠 = 𝑁)))
 
3-Jul-2025expeqidd 42305 A nonnegative real number is zero or one if and only if it is itself when raised to an integer greater than one. (Contributed by SN, 3-Jul-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘2))    &   (𝜑 → 0 ≤ 𝐴)    ⇒   (𝜑 → ((𝐴↑𝑁) = 𝐴 ↔ (𝐴 = 0 ∨ 𝐴 = 1)))
 
3-Jul-2025expeq1d 42304 A nonnegative real number is one if and only if it is one when raised to a positive integer. (Contributed by SN, 3-Jul-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 0 ≤ 𝐴)    ⇒   (𝜑 → ((𝐴↑𝑁) = 1 ↔ 𝐴 = 1))
 
3-Jul-2025explt1d 42303 A nonnegative real number less than one raised to a positive integer is less than one. (Contributed by SN, 3-Jul-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 𝐴 < 1)    ⇒   (𝜑 → (𝐴↑𝑁) < 1)
 
3-Jul-2025ply1idvr1 22311 The identity of a polynomial ring expressed as power of the polynomial variable. (Contributed by AV, 14-Aug-2019.) (Proof shortened by SN, 3-Jul-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑋 = (var1‘𝑅)    &   𝑁 = (mulGrp‘𝑃)    &    ↑ = (.g‘𝑁)    ⇒   (𝑅 ∈ Ring → (0 ↑ 𝑋) = (1r‘𝑃))
 
3-Jul-2025eqsnd 4855 Deduce that a set is a singleton. (Contributed by Thierry Arnoux, 10-May-2023.) (Proof shortened by SN, 3-Jul-2025.)
((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = 𝐵)    &   (𝜑 → 𝐵 ∈ 𝐴)    ⇒   (𝜑 → 𝐴 = {𝐵})
 
1-Jul-2025constrconj 33727 If a point 𝑋 of the complex plane is constructible, so is its conjugate (∗‘𝑋). (Proposed by Saveliy Skresanov, 25-Jun-2025.) (Contributed by Thierry Arnoux, 1-Jul-2025.)
𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   (𝜑 → 𝑁 ∈ On)    &   (𝜑 → 𝑋 ∈ (𝐶‘𝑁))    ⇒   (𝜑 → (∗‘𝑋) ∈ (𝐶‘𝑁))
 
1-Jul-2025constrmon 33726 The construction of constructible numbers is monotonous, i.e. if the ordinal 𝑀 is less than the ordinal 𝑁, which is denoted by 𝑀 ∈ 𝑁, then the 𝑀-th step of the constructible numbers is included in the 𝑁-th step. (Contributed by Thierry Arnoux, 1-Jul-2025.)
𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   (𝜑 → 𝑁 ∈ On)    &   (𝜑 → 𝑀 ∈ 𝑁)    ⇒   (𝜑 → (𝐶‘𝑀) ⊆ (𝐶‘𝑁))
 
1-Jul-2025cjsubd 32747 Complex conjugate distributes over subtraction. (Contributed by Thierry Arnoux, 1-Jul-2025.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    ⇒   (𝜑 → (∗‘(𝐴 − 𝐵)) = ((∗‘𝐴) − (∗‘𝐵)))
 
29-Jun-2025sn-suprubd 42443 suprubd 12251 without ax-mulcom 11242, proven trivially from sn-suprcld 42442. (Contributed by SN, 29-Jun-2025.)
(𝜑 → 𝐴 ⊆ ℝ)    &   (𝜑 → 𝐴 ≠ ∅)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    &   (𝜑 → 𝐵 ∈ 𝐴)    ⇒   (𝜑 → 𝐵 ≤ sup(𝐴, ℝ, < ))
 
29-Jun-2025sn-suprcld 42442 suprcld 12252 without ax-mulcom 11242, proven trivially from sn-sup3d 42441. (Contributed by SN, 29-Jun-2025.)
(𝜑 → 𝐴 ⊆ ℝ)    &   (𝜑 → 𝐴 ≠ ∅)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    ⇒   (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
 
29-Jun-2025sn-sup3d 42441 sup3 12246 without ax-mulcom 11242, proven trivially from sn-sup2 42440. (Contributed by SN, 29-Jun-2025.)
(𝜑 → 𝐴 ⊆ ℝ)    &   (𝜑 → 𝐴 ≠ ∅)    &   (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    ⇒   (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
 
29-Jun-2025supinf 42230 The supremum is the infimum of the upper bounds. (Contributed by SN, 29-Jun-2025.)
(𝜑 → < Or 𝐴)    &   (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐵 𝑦 < 𝑧)))    ⇒   (𝜑 → sup(𝐵, 𝐴, < ) = inf({𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐵 ¬ 𝑥 < 𝑤}, 𝐴, < ))
 
29-Jun-20253jcadALT 35647 Alternate proof of 3jcad 1129. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) Use 3jcad instead. (New usage is discouraged.)
(𝜑 → (𝜓 → 𝜒))    &   (𝜑 → (𝜓 → 𝜃))    &   (𝜑 → (𝜓 → 𝜏))    ⇒   (𝜑 → (𝜓 → (𝜒 ∧ 𝜃 ∧ 𝜏)))
 
29-Jun-2025pm3.48ALT 35646 Alternate proof of pm3.48 964. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜃)))
 
29-Jun-2025orbi2iALT 35645 Alternate proof of orbi2i 911. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ↔ 𝜓)    ⇒   ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓))
 
29-Jun-20252thALT 35644 Alternate proof of 2th 264. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    ⇒   (𝜑 ↔ 𝜓)
 
29-Jun-2025wevgblacfn 35068 If 𝑅 is a well-ordering of the universe, then 𝐹 is a global choice function. Here 𝐹 maps each set 𝑧 to its minimal element with respect to 𝑅 (except when 𝑧 is the empty set, in which case it is mapped to the empty set, though this is only done for convenience). (Contributed by BTernaryTau, 29-Jun-2025.)
𝐹 = (𝑧 ∈ V ↦ ∪ {𝑦 ∈ 𝑧 ∣ ∀𝑥 ∈ 𝑧 ¬ 𝑥𝑅𝑦})    ⇒   (𝑅 We V → (𝐹 Fn V ∧ ∀𝑧(𝑧 ≠ ∅ → (𝐹‘𝑧) ∈ 𝑧)))
 
29-Jun-2025bian1d 32476 Adding a superfluous conjunct in a biconditional. (Contributed by Thierry Arnoux, 26-Feb-2017.) (Proof shortened by Hongxiu Chen, 29-Jun-2025.)
(𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    ⇒   (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜒 ∧ 𝜃)))
 
29-Jun-2025mulgt1 12150 The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.) (Proof shortened by SN, 29-Jun-2025.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (1 < 𝐴 ∧ 1 < 𝐵)) → 1 < (𝐴 · 𝐵))
 
29-Jun-20253jaob 1426 Disjunction of three antecedents. (Contributed by NM, 13-Sep-2011.) (Proof shortened by Hongxiu Chen, 29-Jun-2025.)
(((𝜑 ∨ 𝜒 ∨ 𝜃) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓) ∧ (𝜃 → 𝜓)))
 
28-Jun-2025bj-ru1 36902 A version of Russell's paradox ru 3802 not mentioning the universal class. (see also bj-ru 36903). (Contributed by BJ, 12-Oct-2019.) Remove usage of ax-10 2141, ax-11 2158, ax-12 2178 by using eqabbw 2818 following BTernaryTau's similar revision of ru 3802. (Revised by BJ, 28-Jun-2025.) (Proof modification is discouraged.)
¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥}
 
25-Jun-2025drngmulrcan 42474 Cancellation of a nonzero factor on the right for multiplication. (mulcan2ad 11920 analog). (Contributed by SN, 14-Aug-2024.) (Proof shortened by SN, 25-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ DivRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝑍 ≠ 0 )    &   (𝜑 → (𝑋 · 𝑍) = (𝑌 · 𝑍))    ⇒   (𝜑 → 𝑋 = 𝑌)
 
25-Jun-2025drngmullcan 42473 Cancellation of a nonzero factor on the left for multiplication. (mulcanad 11919 analog). (Contributed by SN, 14-Aug-2024.) (Proof shortened by SN, 25-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ DivRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝑍 ≠ 0 )    &   (𝜑 → (𝑍 · 𝑋) = (𝑍 · 𝑌))    ⇒   (𝜑 → 𝑋 = 𝑌)
 
25-Jun-2025aks5lem6 42142 Connect results of section 5 and Theorem 6.1 AKS. (Contributed by metakunt, 25-Jun-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    &   (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))    &   𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(-g‘𝑆)(1r‘𝑆))})    &   𝑋 = (var1‘(ℤ/nℤ‘𝑁))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(.g‘(mulGrp‘𝑆))(𝑋(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))𝑋)(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿))    ⇒   (𝜑 → 𝑁 = (𝑃↑(𝑃 pCnt 𝑁)))
 
25-Jun-2025constrss 33725 Constructed points are in the next generation constructed points. (Contributed by Thierry Arnoux, 25-Jun-2025.)
𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   (𝜑 → 𝑁 ∈ On)    ⇒   (𝜑 → (𝐶‘𝑁) ⊆ (𝐶‘suc 𝑁))
 
25-Jun-2025constr01 33724 0 and 1 are in all steps of the construction of constructible points. (Contributed by Thierry Arnoux, 25-Jun-2025.)
𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   (𝜑 → 𝑁 ∈ On)    ⇒   (𝜑 → {0, 1} ⊆ (𝐶‘𝑁))
 
25-Jun-2025constrsslem 33723 Lemma for constrss 33725. This lemma requires the additional condition that 0 is the constructible number; that condition is removed in constrss 33725. (Proposed by Saveliy Skresanov, 23-JUn-2025.) (Contributed by Thierry Arnoux, 25-Jun-2025.)
𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   (𝜑 → 𝑁 ∈ On)    &   (𝜑 → 0 ∈ (𝐶‘𝑁))    ⇒   (𝜑 → (𝐶‘𝑁) ⊆ (𝐶‘suc 𝑁))
 
25-Jun-2025constrsscn 33722 Closure of the constructible points in the complex numbers. (Contributed by Thierry Arnoux, 25-Jun-2025.)
𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   (𝜑 → 𝑁 ∈ On)    ⇒   (𝜑 → (𝐶‘𝑁) ⊆ ℂ)
 
25-Jun-2025constrlim 33721 Limit step of the construction of constructible numbers. (Contributed by Thierry Arnoux, 25-Jun-2025.)
𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   (𝜑 → 𝑁 ∈ 𝑉)    &   (𝜑 → Lim 𝑁)    ⇒   (𝜑 → (𝐶‘𝑁) = ∪ 𝑛 ∈ 𝑁 (𝐶‘𝑛))
 
25-Jun-2025constrsuc 33720 Membership in the successor step of the construction of constructible numbers. (Contributed by Thierry Arnoux, 25-Jun-2025.)
𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   (𝜑 → 𝑁 ∈ On)    &   𝑆 = (𝐶‘𝑁)    ⇒   (𝜑 → (𝑋 ∈ (𝐶‘suc 𝑁) ↔ (𝑋 ∈ ℂ ∧ (∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ∃𝑐 ∈ 𝑆 ∃𝑑 ∈ 𝑆 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑋 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑋 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ∃𝑐 ∈ 𝑆 ∃𝑒 ∈ 𝑆 ∃𝑓 ∈ 𝑆 ∃𝑡 ∈ ℝ (𝑋 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ∃𝑐 ∈ 𝑆 ∃𝑑 ∈ 𝑆 ∃𝑒 ∈ 𝑆 ∃𝑓 ∈ 𝑆 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓)))))))
 
25-Jun-2025constr0 33719 The first step of the construction of constructible numbers is the pair {0, 1}. In this theorem and the following, we use (𝐶‘𝑁) for the 𝑁-th intermediate iteration of the constructible number. (Contributed by Thierry Arnoux, 25-Jun-2025.)
𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    ⇒   (𝐶‘∅) = {0, 1}
 
25-Jun-2025re0cj 32748 The conjugate of a pure imaginary number is its negative. (Contributed by Thierry Arnoux, 25-Jun-2025.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → (ℜ‘𝐴) = 0)    ⇒   (𝜑 → (∗‘𝐴) = -𝐴)
 
25-Jun-2025abvtrivg 20850 The trivial absolute value. This theorem is not true for rings with zero divisors, which violate the multiplication axiom; abvdom 20847 is the converse of this theorem. (Contributed by SN, 25-Jun-2025.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝐹 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, 1))    ⇒   (𝑅 ∈ Domn → 𝐹 ∈ 𝐴)
 
25-Jun-2025drngmul0or 20776 A product is zero iff one of its factors is zero. (Contributed by NM, 8-Oct-2014.) (Proof shortened by SN, 25-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ DivRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0 ∨ 𝑌 = 0 )))
 
25-Jun-2025drngmcl 20766 The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011.) (Proof shortened by SN, 25-Jun-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    ⇒   ((𝑅 ∈ DivRing ∧ 𝑋 ∈ (𝐵 ∖ { 0 }) ∧ 𝑌 ∈ (𝐵 ∖ { 0 })) → (𝑋 · 𝑌) ∈ (𝐵 ∖ { 0 }))
 
23-Jun-2025dfac5lem4 10189 Lemma for dfac5 10192. (Contributed by NM, 11-Apr-2004.) Avoid ax-11 2158. (Revised by BTernaryTau, 23-Jun-2025.)
𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 ∈ ℎ 𝑢 = ({𝑡} × 𝑡))}    &   (𝜑 ↔ ∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))    ⇒   (𝜑 → ∃𝑦∀𝑧 ∈ 𝐴 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
 
23-Jun-2025fodomfib 9391 Equivalence of an onto mapping and dominance for a nonempty finite set. Unlike fodomb 10589 for arbitrary sets, this theorem does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.) Avoid ax-pow 5383. (Revised by BTernaryTau, 23-Jun-2025.)
(𝐴 ∈ Fin → ((𝐴 ≠ ∅ ∧ ∃𝑓 𝑓:𝐴–onto→𝐵) ↔ (∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴)))
 
23-Jun-2025fodomfir 9390 There exists a mapping from a finite set onto any nonempty set that it dominates, proved without using the Axiom of Power Sets (unlike fodomr 9188). (Contributed by BTernaryTau, 23-Jun-2025.)
((𝐴 ∈ Fin ∧ ∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵)
 
23-Jun-2025dmcosseq 5994 Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-11 2158. (Revised by BTernaryTau, 23-Jun-2025.)
(ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) = dom 𝐵)
 
23-Jun-2025excomimw 2043 Weak version of excomim 2164. Uses only Tarski's FOL axiom schemes. (Contributed by BTernaryTau, 23-Jun-2025.)
(𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)
 
22-Jun-2025prcinf 35062 Any proper class is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. This proof holds regardless of whether the Axiom of Infinity is accepted or negated. (Contributed by BTernaryTau, 22-Jun-2025.)
(¬ 𝐴 ∈ V → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))
 
22-Jun-2025axnulALT2 35061 Alternate proof of axnul 5323, proved from propositional calculus, ax-gen 1793, ax-4 1807, ax-5 1909, and ax-inf2 9704. (Contributed by BTernaryTau, 22-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥
 
22-Jun-2025rtelextdg2 33710 If an element 𝑋 is a solution of a quadratic equation, then it is either in the base field, or the degree of its field extension is exactly 2. (Contributed by Thierry Arnoux, 22-Jun-2025.)
𝐾 = (𝐸 ↾s 𝐹)    &   𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝑋})))    &    0 = (0g‘𝐸)    &   𝑃 = (Poly1‘𝐾)    &   𝑉 = (Base‘𝐸)    &    · = (.r‘𝐸)    &    + = (+g‘𝐸)    &    ↑ = (.g‘(mulGrp‘𝐸))    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝑋 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ 𝐹)    &   (𝜑 → 𝐵 ∈ 𝐹)    &   (𝜑 → ((2 ↑ 𝑋) + ((𝐴 · 𝑋) + 𝐵)) = 0 )    ⇒   (𝜑 → (𝑋 ∈ 𝐹 ∨ (𝐿[:]𝐾) = 2))
 
22-Jun-2025rtelextdg2lem 33709 Lemma for rtelextdg2 33710: If an element 𝑋 is a solution of a quadratic equation, then the degree of its field extension is at most 2. (Contributed by Thierry Arnoux, 22-Jun-2025.)
𝐾 = (𝐸 ↾s 𝐹)    &   𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝑋})))    &    0 = (0g‘𝐸)    &   𝑃 = (Poly1‘𝐾)    &   𝑉 = (Base‘𝐸)    &    · = (.r‘𝐸)    &    + = (+g‘𝐸)    &    ↑ = (.g‘(mulGrp‘𝐸))    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝑋 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ 𝐹)    &   (𝜑 → 𝐵 ∈ 𝐹)    &   (𝜑 → ((2 ↑ 𝑋) + ((𝐴 · 𝑋) + 𝐵)) = 0 )    &   𝑌 = (var1‘𝐾)    &    ⊕ = (+g‘𝑃)    &    ⊗ = (.r‘𝑃)    &    ∧ = (.g‘(mulGrp‘𝑃))    &   𝑈 = (algSc‘𝑃)    &   𝐺 = ((2 ∧ 𝑌) ⊕ (((𝑈‘𝐴) ⊗ 𝑌) ⊕ (𝑈‘𝐵)))    ⇒   (𝜑 → (𝐿[:]𝐾) ≤ 2)
 
22-Jun-2025minplymindeg 33693 The minimal polynomial of 𝐴 is minimal among the nonzero annihilators of 𝐴 with regard to degree. (Contributed by Thierry Arnoux, 22-Jun-2025.)
𝑂 = (𝐸 evalSub1 𝐹)    &   𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   𝐵 = (Base‘𝐸)    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐴 ∈ 𝐵)    &    0 = (0g‘𝐸)    &   𝑀 = (𝐸 minPoly 𝐹)    &   𝐷 = (deg1‘(𝐸 ↾s 𝐹))    &   𝑍 = (0g‘𝑃)    &   𝑈 = (Base‘𝑃)    &   (𝜑 → ((𝑂‘𝐻)‘𝐴) = 0 )    &   (𝜑 → 𝐻 ∈ 𝑈)    &   (𝜑 → 𝐻 ≠ 𝑍)    ⇒   (𝜑 → (𝐷‘(𝑀‘𝐴)) ≤ (𝐷‘𝐻))
 
22-Jun-2025fldgenfldext 33670 A subfield 𝐹 extended with a set 𝐴 forms a field extension. (Contributed by Thierry Arnoux, 22-Jun-2025.)
𝐵 = (Base‘𝐸)    &   𝐾 = (𝐸 ↾s 𝐹)    &   𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ 𝐴)))    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐴 ⊆ 𝐵)    ⇒   (𝜑 → 𝐿/FldExt𝐾)
 
22-Jun-2025deg1vr 33571 The degree of the variable polynomial is 1. (Contributed by Thierry Arnoux, 22-Jun-2025.)
𝐷 = (deg1‘𝑅)    &   𝑃 = (Poly1‘𝑅)    &   𝑋 = (var1‘𝑅)    &   (𝜑 → 𝑅 ∈ NzRing)    ⇒   (𝜑 → (𝐷‘𝑋) = 1)
 
22-Jun-2025coe1vr1 33570 Polynomial coefficient of the variable. (Contributed by Thierry Arnoux, 22-Jun-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑋 = (var1‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &    0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    ⇒   (𝜑 → (coe1‘𝑋) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 1, 1 , 0 )))
 
22-Jun-2025coe1zfv 33569 The coefficients of the zero univariate polynomial. (Contributed by Thierry Arnoux, 22-Jun-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑍 = (0g‘𝑃)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑁 ∈ ℕ0)    ⇒   (𝜑 → ((coe1‘𝑍)‘𝑁) = 0 )
 
22-Jun-2025evl1deg2 33559 Evaluation of a univariate polynomial of degree 2. (Contributed by Thierry Arnoux, 22-Jun-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑂 = (eval1‘𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &    · = (.r‘𝑅)    &    + = (+g‘𝑅)    &    ↑ = (.g‘(mulGrp‘𝑅))    &   𝐹 = (coe1‘𝑀)    &   𝐸 = (deg1‘𝑅)    &   𝐴 = (𝐹‘2)    &   𝐵 = (𝐹‘1)    &   𝐶 = (𝐹‘0)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑀 ∈ 𝑈)    &   (𝜑 → (𝐸‘𝑀) = 2)    &   (𝜑 → 𝑋 ∈ 𝐾)    ⇒   (𝜑 → ((𝑂‘𝑀)‘𝑋) = ((𝐴 · (2 ↑ 𝑋)) + ((𝐵 · 𝑋) + 𝐶)))
 
22-Jun-2025ressasclcl 33553 Closure of the univariate polynomial evaluation for scalars. (Contributed by Thierry Arnoux, 22-Jun-2025.)
𝑊 = (Poly1‘𝑈)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝐴 = (algSc‘𝑊)    &   𝐵 = (Base‘𝑊)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝑋 ∈ 𝑅)    ⇒   (𝜑 → (𝐴‘𝑋) ∈ 𝐵)
 
22-Jun-2025gsumtp 33031 Group sum of an unordered triple. (Contributed by Thierry Arnoux, 22-Jun-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝑘 = 𝑀 → 𝐴 = 𝐶)    &   (𝑘 = 𝑁 → 𝐴 = 𝐷)    &   (𝑘 = 𝑂 → 𝐴 = 𝐸)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝑀 ∈ 𝑉)    &   (𝜑 → 𝑁 ∈ 𝑊)    &   (𝜑 → 𝑂 ∈ 𝑋)    &   (𝜑 → 𝑀 ≠ 𝑁)    &   (𝜑 → 𝑁 ≠ 𝑂)    &   (𝜑 → 𝑀 ≠ 𝑂)    &   (𝜑 → 𝐶 ∈ 𝐵)    &   (𝜑 → 𝐷 ∈ 𝐵)    &   (𝜑 → 𝐸 ∈ 𝐵)    ⇒   (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀, 𝑁, 𝑂} ↦ 𝐴)) = ((𝐶 + 𝐷) + 𝐸))
 
21-Jun-2025r1peuqusdeg1 35603 Uniqueness of polynomial remainder in terms of a quotient structure in the sense of the right hand side of r1pid2 26213. (Contributed by SN, 21-Jun-2025.)
𝑃 = (Poly1‘𝑅)    &   𝐼 = ((RSpan‘𝑃)‘{𝐹})    &   𝑇 = (𝑃 /s (𝑃 ~QG 𝐼))    &   𝑄 = (Base‘𝑇)    &   𝑁 = (Unic1p‘𝑅)    &   𝐷 = (deg1‘𝑅)    &   (𝜑 → 𝑅 ∈ Domn)    &   (𝜑 → 𝐹 ∈ 𝑁)    &   (𝜑 → 𝑍 ∈ 𝑄)    ⇒   (𝜑 → ∃!𝑞 ∈ 𝑍 (𝐷‘𝑞) < (𝐷‘𝐹))
 
21-Jun-2025df-plfl 35597 Define the field extension that augments a field with the root of the given irreducible polynomial, and extends the norm if one exists and the extension is unique. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Thierry Arnoux and Steven Nguyen, 21-Jun-2025.)
polyFld = (𝑟 ∈ V, 𝑝 ∈ V ↦ ⦋(Poly1‘𝑟) / 𝑠⦌⦋((RSpan‘𝑠)‘{𝑝}) / 𝑖⦌⦋(𝑐 ∈ (Base‘𝑟) ↦ [(𝑐( ·𝑠 ‘𝑠)(1r‘𝑠))](𝑠 ~QG 𝑖)) / 𝑓⦌⟨⦋(𝑠 /s (𝑠 ~QG 𝑖)) / 𝑡⦌((𝑡 toNrmGrp (℩𝑛 ∈ (AbsVal‘𝑡)(𝑛 ∘ 𝑓) = (norm‘𝑟))) sSet ⟨(le‘ndx), ⦋(𝑧 ∈ (Base‘𝑡) ↦ (℩𝑞 ∈ 𝑧 (𝑞(rem1p‘𝑟)𝑝) = 𝑞)) / 𝑔⦌(◡𝑔 ∘ ((le‘𝑠) ∘ 𝑔))⟩), 𝑓⟩)
 
21-Jun-2025idomrcan 33240 Right-cancellation law for integral domains. (Contributed by Thierry Arnoux, 22-Mar-2025.) (Proof shortened by SN, 21-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ (𝐵 ∖ { 0 }))    &   (𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → (𝑋 · 𝑍) = (𝑌 · 𝑍))    ⇒   (𝜑 → 𝑋 = 𝑌)
 
21-Jun-2025r1pid2 26213 Identity law for polynomial remainder operation: it leaves a polynomial 𝐴 unchanged iff the degree of 𝐴 is less than the degree of the divisor 𝐵. (Contributed by Thierry Arnoux, 2-Apr-2025.) Generalize to domains. (Revised by SN, 21-Jun-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑈 = (Base‘𝑃)    &   𝑁 = (Unic1p‘𝑅)    &   𝐸 = (rem1p‘𝑅)    &   𝐷 = (deg1‘𝑅)    &   (𝜑 → 𝑅 ∈ Domn)    &   (𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐵 ∈ 𝑁)    ⇒   (𝜑 → ((𝐴𝐸𝐵) = 𝐴 ↔ (𝐷‘𝐴) < (𝐷‘𝐵)))
 
21-Jun-2025domneq0r 20740 Right multiplication by a nonzero element does not change zeroness in a domain. Compare rrgeq0 20716. (Contributed by SN, 21-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 }))    &   (𝜑 → 𝑅 ∈ Domn)    ⇒   (𝜑 → ((𝑋 · 𝑌) = 0 ↔ 𝑋 = 0 ))
 
21-Jun-2025domnrcan 20739 Right-cancellation law for domains. (Contributed by SN, 21-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ (𝐵 ∖ { 0 }))    &   (𝜑 → 𝑅 ∈ Domn)    &   (𝜑 → (𝑋 · 𝑍) = (𝑌 · 𝑍))    ⇒   (𝜑 → 𝑋 = 𝑌)
 
21-Jun-2025domnrcanb 20738 Right-cancellation law for domains, biconditional version of domnrcan 20739. (Contributed by SN, 21-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ (𝐵 ∖ { 0 }))    &   (𝜑 → 𝑅 ∈ Domn)    ⇒   (𝜑 → ((𝑋 · 𝑍) = (𝑌 · 𝑍) ↔ 𝑋 = 𝑌))
 
21-Jun-2025domnlcan 20737 Left-cancellation law for domains. (Contributed by Thierry Arnoux, 22-Mar-2025.) (Proof shortened by SN, 21-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝑅 ∈ Domn)    &   (𝜑 → (𝑋 · 𝑌) = (𝑋 · 𝑍))    ⇒   (𝜑 → 𝑌 = 𝑍)
 
21-Jun-2025domnlcanb 20736 Left-cancellation law for domains, biconditional version of domnlcan 20737. (Contributed by Thierry Arnoux, 8-Jun-2025.) Shorten this theorem and domnlcan 20737 overall. (Revised by SN, 21-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝑅 ∈ Domn)    ⇒   (𝜑 → ((𝑋 · 𝑌) = (𝑋 · 𝑍) ↔ 𝑌 = 𝑍))
 
21-Jun-2025isdomn2 20727 A ring is a domain iff all nonzero elements are regular elements. (Contributed by Mario Carneiro, 28-Mar-2015.) (Proof shortened by SN, 21-Jun-2025.)
𝐵 = (Base‘𝑅)    &   𝐸 = (RLReg‘𝑅)    &    0 = (0g‘𝑅)    ⇒   (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))
 
21-Jun-2025nzrpropd 20540 If two structures have the same components (properties), one is a nonzero ring iff the other one is. (Contributed by SN, 21-Jun-2025.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → 𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    ⇒   (𝜑 → (𝐾 ∈ NzRing ↔ 𝐿 ∈ NzRing))
 
20-Jun-2025opprablb 42461 A class is an Abelian group if and only if its opposite (ring) is an Abelian group. (Contributed by SN, 20-Jun-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ Abel ↔ 𝑂 ∈ Abel)
 
20-Jun-2025opprgrpb 42460 A class is a group if and only if its opposite (ring) is a group. (Contributed by SN, 20-Jun-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ Grp ↔ 𝑂 ∈ Grp)
 
20-Jun-2025opprmndb 42459 A class is a monoid if and only if its opposite (ring) is a monoid. (Contributed by SN, 20-Jun-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ Mnd ↔ 𝑂 ∈ Mnd)
 
20-Jun-2025ply1divalg3 35602 Uniqueness of polynomial remainder: convert the subtraction in ply1divalg2 26190 to addition. (Contributed by SN, 20-Jun-2025.)
𝑃 = (Poly1‘𝑅)    &   𝐷 = (deg1‘𝑅)    &   𝐵 = (Base‘𝑃)    &    + = (+g‘𝑃)    &    ∙ = (.r‘𝑃)    &   𝐶 = (Unic1p‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐺 ∈ 𝐶)    ⇒   (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 + (𝑞 ∙ 𝐺))) < (𝐷‘𝐺))
 
20-Jun-2025rexxfr3d 35598 Transfer existential quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by SN, 20-Jun-2025.)
(𝑥 = 𝑋 → (𝜓 ↔ 𝜒))    &   (𝜑 → (𝑥 ∈ 𝐴 ↔ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋))    &   (𝜑 → 𝑋 ∈ 𝑉)    ⇒   (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))
 
20-Jun-2025isdomn4r 20735 A ring is a domain iff it is nonzero and the right cancellation law for multiplication holds. (Contributed by SN, 20-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ (𝐵 ∖ { 0 })((𝑎 · 𝑐) = (𝑏 · 𝑐) → 𝑎 = 𝑏)))
 
20-Jun-2025opprnzrb 20541 The opposite of a nonzero ring is nonzero, bidirectional form of opprnzr 20542. (Contributed by SN, 20-Jun-2025.)
𝑂 = (oppr‘𝑅)    ⇒   (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing)
 
20-Jun-2025fodomfi 9372 An onto function implies dominance of domain over range, for finite sets. Unlike fodomg 10585 for arbitrary sets, this theorem does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Mario Carneiro, 16-Nov-2014.) Avoid ax-pow 5383. (Revised by BTernaryTau, 20-Jun-2025.)
((𝐴 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴)
 
20-Jun-2025ru 3802 Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14.

In the late 1800s, Frege's Axiom of (unrestricted) Comprehension, expressed in our notation as 𝐴 ∈ V, asserted that any collection of sets 𝐴 is a set i.e. belongs to the universe V of all sets. In particular, by substituting {𝑥 ∣ 𝑥 ∉ 𝑥} (the "Russell class") for 𝐴, it asserted {𝑥 ∣ 𝑥 ∉ 𝑥} ∈ V, meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V. This contradiction was discovered by Russell in 1901 (published in 1903), invalidating the Comprehension Axiom and leading to the collapse of Frege's system, which Frege acknowledged in the second edition of his Grundgesetze der Arithmetik.

In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom ssex 5339 asserting that 𝐴 is a set only when it is smaller than some other set 𝐵. However, Zermelo was then faced with a "chicken and egg" problem of how to show 𝐵 is a set, leading him to introduce the set-building axioms of Null Set 0ex 5325, Pairing prex 5452, Union uniex 7770, Power Set pwex 5398, and Infinity omex 9706 to give him some starting sets to work with (all of which, before Russell's Paradox, were immediate consequences of Frege's Comprehension). In 1922 Fraenkel strengthened the Subset Axiom with our present Replacement Axiom funimaex 6661 (whose modern formalization is due to Skolem, also in 1922). Thus, in a very real sense Russell's Paradox spawned the invention of ZF set theory and completely revised the foundations of mathematics!

Another mainstream formalization of set theory, devised by von Neumann, Bernays, and Goedel, uses class variables rather than setvar variables as its primitives. The axiom system NBG in [Mendelson] p. 225 is suitable for a Metamath encoding. NBG is a conservative extension of ZF in that it proves exactly the same theorems as ZF that are expressible in the language of ZF. An advantage of NBG is that it is finitely axiomatizable - the Axiom of Replacement can be broken down into a finite set of formulas that eliminate its wff metavariable. Finite axiomatizability is required by some proof languages (although not by Metamath). There is a stronger version of NBG called Morse-Kelley (axiom system MK in [Mendelson] p. 287).

Russell himself continued in a different direction, avoiding the paradox with his "theory of types". Quine extended Russell's ideas to formulate his New Foundations set theory (axiom system NF of [Quine] p. 331). In NF, the collection of all sets is a set, contrarily to ZF and NBG set theories. Russell's paradox has other consequences: when classes are too large (beyond the size of those used in standard mathematics), the axiom of choice ac4 10538 and Cantor's theorem canth 7396 are provably false. (See ncanth 7397 for some intuition behind the latter.) Recent results (as of 2014) seem to show that NF is equiconsistent to Z (ZF in which ax-sep 5317 replaces ax-rep 5303) with ax-sep 5317 restricted to only bounded quantifiers. NF is finitely axiomatizable and can be encoded in Metamath using the axioms from T. Hailperin, "A set of axioms for logic", J. Symb. Logic 9:1-19 (1944).

Under our ZF set theory, every set is a member of the Russell class by elirrv 9659 (derived from the Axiom of Regularity), so for us the Russell class equals the universe V (Theorem ruv 9665). See ruALT 9666 for an alternate proof of ru 3802 derived from that fact. (Contributed by NM, 7-Aug-1994.) Remove use of ax-13 2380. (Revised by BJ, 12-Oct-2019.) Remove use of ax-10 2141, ax-11 2158, and ax-12 2178. (Revised by BTernaryTau, 20-Jun-2025.) (Proof modification is discouraged.)

{𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V
 
20-Jun-2025vtocl 3570 Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993.) Remove dependency on ax-10 2141. (Revised by BJ, 29-Nov-2020.) (Proof shortened by SN, 20-Apr-2024.) (Proof shortened by Wolf Lammen, 20-Jun-2025.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   𝜑    ⇒   𝜓
 
19-Jun-2025ellcsrspsn 35601 Membership in a left coset in a quotient of a ring by the span of a singleton (that is, by the ideal generated by an element). This characterization comes from eqglact 19213 and elrspsn 21267. (Contributed by SN, 19-Jun-2025.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    · = (.r‘𝑅)    &    ∼ = (𝑅 ~QG 𝐼)    &   𝑈 = (𝑅 /s ∼ )    &   𝐼 = ((RSpan‘𝑅)‘{𝑀})    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑀 ∈ 𝐵)    &   (𝜑 → 𝑋 ∈ (Base‘𝑈))    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 (𝑋 = [𝑥] ∼ ∧ 𝑋 = {𝑧 ∣ ∃𝑦 ∈ 𝐵 𝑧 = (𝑥 + (𝑦 · 𝑀))}))
 
19-Jun-2025rspssbasd 35600 The span of a set of ring elements is a set of ring elements. (Contributed by SN, 19-Jun-2025.)
𝐾 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐺 ⊆ 𝐵)    ⇒   (𝜑 → (𝐾‘𝐺) ⊆ 𝐵)
 
19-Jun-2025rexxfr3dALT 35599 Longer proof of rexxfr3d 35598 using ax-11 2158 instead of ax-12 2178, without the disjoint variable condition 𝐴𝑥𝑦. (Contributed by SN, 19-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑋 → (𝜓 ↔ 𝜒))    &   (𝜑 → (𝑥 ∈ 𝐴 ↔ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋))    &   (𝜑 → 𝑋 ∈ 𝑉)    ⇒   (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))
 
19-Jun-2025fldext2chn 33711 In a non-empty tower 𝑇 of quadratic field extensions, the degree of the extension of the first member by the last is a power of two. (Contributed by Thierry Arnoux, 19-Jun-2025.)
< = {⟨𝑓, 𝑒⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) = 2)}    &   (𝜑 → 𝑇 ∈ ( < ChainField))    &   (𝜑 → (𝑇‘0) = 𝑄)    &   (𝜑 → (lastS‘𝑇) = 𝐹)    &   (𝜑 → 0 < (♯‘𝑇))    ⇒   (𝜑 → ∃𝑛 ∈ ℕ0 (𝐹[:]𝑄) = (2↑𝑛))
 
19-Jun-2025chnso 32978 A chain induces a total order. (Contributed by Thierry Arnoux, 19-Jun-2025.)
(( < Po 𝐴 ∧ 𝐶 ∈ ( < Chain𝐴)) → < Or ran 𝐶)
 
19-Jun-2025chnlt 32977 Compare any two elements in a chain. (Contributed by Thierry Arnoux, 19-Jun-2025.)
(𝜑 → < Po 𝐴)    &   (𝜑 → 𝐶 ∈ ( < Chain𝐴))    &   (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶)))    &   (𝜑 → 𝐼 ∈ (0..^𝐽))    ⇒   (𝜑 → (𝐶‘𝐼) < (𝐶‘𝐽))
 
19-Jun-2025chnub 32976 In a chain, the last element is an upper bound. (Contributed by Thierry Arnoux, 19-Jun-2025.)
(𝜑 → < Po 𝐴)    &   (𝜑 → 𝐶 ∈ ( < Chain𝐴))    &   (𝜑 → 𝐼 ∈ (0..^((♯‘𝐶) − 1)))    ⇒   (𝜑 → (𝐶‘𝐼) < (lastS‘𝐶))
 
19-Jun-2025chnind 32975 Induction over a chain. See nnind 12305 for an explanation about the hypotheses. (Contributed by Thierry Arnoux, 19-Jun-2025.)
(𝑐 = ∅ → (𝜓 ↔ 𝜒))    &   (𝑐 = 𝑑 → (𝜓 ↔ 𝜃))    &   (𝑐 = (𝑑 ++ ⟨“𝑥”⟩) → (𝜓 ↔ 𝜏))    &   (𝑐 = 𝐶 → (𝜓 ↔ 𝜂))    &   (𝜑 → 𝐶 ∈ ( < Chain𝐴))    &   (𝜑 → 𝜒)    &   (((((𝜑 ∧ 𝑑 ∈ ( < Chain𝐴)) ∧ 𝑥 ∈ 𝐴) ∧ (𝑑 = ∅ ∨ (lastS‘𝑑) < 𝑥)) ∧ 𝜃) → 𝜏)    ⇒   (𝜑 → 𝜂)
 
19-Jun-2025pfxchn 32974 A prefix of a chain is still a chain. (Contributed by Thierry Arnoux, 19-Jun-2025.)
(𝜑 → 𝐶 ∈ ( < Chain𝐴))    &   (𝜑 → 𝐿 ∈ (0...(♯‘𝐶)))    ⇒   (𝜑 → (𝐶 prefix 𝐿) ∈ ( < Chain𝐴))
 
19-Jun-2025chnltm1 32973 Basic property of a chain. (Contributed by Thierry Arnoux, 19-Jun-2025.)
(𝜑 → 𝐶 ∈ ( < Chain𝐴))    &   (𝜑 → 𝑁 ∈ (dom 𝐶 ∖ {0}))    ⇒   (𝜑 → (𝐶‘(𝑁 − 1)) < (𝐶‘𝑁))
 
19-Jun-2025chnwrd 32972 A chain is an ordered sequence, i.e. a word. (Contributed by Thierry Arnoux, 19-Jun-2025.)
(𝜑 → 𝐶 ∈ ( < Chain𝐴))    ⇒   (𝜑 → 𝐶 ∈ Word 𝐴)
 
19-Jun-2025ischn 32971 Property of being a chain. (Contributed by Thierry Arnoux, 19-Jun-2025.)
(𝐶 ∈ ( < Chain𝐴) ↔ (𝐶 ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom 𝐶 ∖ {0})(𝐶‘(𝑛 − 1)) < (𝐶‘𝑛)))
 
19-Jun-2025df-chn 32970 Define the class of (finite) chains. A chain is defined to be a sequence of objects, where each object is less than the next one in the sequence. The term "chain" is usually used in order theory. In the context of algebra, chains are often called "towers", for example for fields, or "series", for example for subgroup or subnormal series. (Contributed by Thierry Arnoux, 19-Jun-2025.)
( < Chain𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)}
 
19-Jun-2025ccatdmss 32908 The domain of a concatenated word is a superset of the domain of the first word. (Contributed by Thierry Arnoux, 19-Jun-2025.)
(𝜑 → 𝐴 ∈ Word 𝑆)    &   (𝜑 → 𝐵 ∈ Word 𝑆)    ⇒   (𝜑 → dom 𝐴 ⊆ dom (𝐴 ++ 𝐵))
 
17-Jun-2025aks5lem5a 42141 Lemma for AKS, section 5, connect to Theorem 6.1. (Contributed by metakunt, 17-Jun-2025.)
(𝜑 → 𝐾 ∈ Field)    &   𝑃 = (chr‘𝐾)    &   (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁))    &   𝐵 = (𝑆 /s (𝑆 ~QG 𝐿))    &   𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(-g‘𝑆)(1r‘𝑆))})    &   (𝜑 → 𝑅 ∈ ℕ)    &    ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)[(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎)))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((ℤRHom‘𝑆)‘𝑎))](𝑆 ~QG 𝐿))    ⇒   (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
 
17-Jun-2025aks5lem4a 42140 Lemma for AKS section 5, reduce hypotheses. (Contributed by metakunt, 17-Jun-2025.)
(𝜑 → 𝐾 ∈ Field)    &   𝑃 = (chr‘𝐾)    &   (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁))    &   𝐵 = (𝑆 /s (𝑆 ~QG 𝐿))    &   𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(-g‘𝑆)(1r‘𝑆))})    &   (𝜑 → 𝑅 ∈ ℕ)    &    ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))    &   (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   (𝜑 → 𝐴 ∈ ℤ)    &   (𝜑 → [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))](𝑆 ~QG 𝐿))    ⇒   (𝜑 → (𝑁(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘𝑀)) = (((eval1‘𝐾)‘((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁(.g‘(mulGrp‘𝐾))𝑀)))
 
17-Jun-2025aks5lem3a 42139 Lemma for AKS section 5. (Contributed by metakunt, 17-Jun-2025.)
(𝜑 → 𝐾 ∈ Field)    &   𝑃 = (chr‘𝐾)    &   (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁))    &   𝐵 = (𝑆 /s (𝑆 ~QG 𝐿))    &   𝐿 = ((RSpan‘𝑆)‘{((𝑅(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(-g‘𝑆)(1r‘𝑆))})    &   (𝜑 → 𝑅 ∈ ℕ)    &    ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   𝑆 = (Poly1‘(ℤ/nℤ‘𝑁))    &   𝐹 = (𝑝 ∈ (Base‘(Poly1‘(ℤ/nℤ‘𝑁))) ↦ (𝐺 ∘ 𝑝))    &   𝐺 = (𝑞 ∈ (Base‘(ℤ/nℤ‘𝑁)) ↦ ∪ ((ℤRHom‘𝐾) “ 𝑞))    &   𝐻 = (𝑟 ∈ (Base‘(Poly1‘𝐾)) ↦ (((eval1‘𝐾)‘𝑟)‘𝑀))    &   (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   𝐼 = (𝑠 ∈ (Base‘𝐵) ↦ ∪ ((𝐻 ∘ 𝐹) “ 𝑠))    &   (𝜑 → 𝐴 ∈ ℤ)    &   (𝜑 → [(𝑁(.g‘(mulGrp‘𝑆))((var1‘(ℤ/nℤ‘𝑁))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴))))](𝑆 ~QG 𝐿) = [((𝑁(.g‘(mulGrp‘𝑆))(var1‘(ℤ/nℤ‘𝑁)))(+g‘𝑆)((algSc‘𝑆)‘((ℤRHom‘(ℤ/nℤ‘𝑁))‘𝐴)))](𝑆 ~QG 𝐿))    ⇒   (𝜑 → (𝑁(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘𝑀)) = (((eval1‘𝐾)‘((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝐴))))‘(𝑁(.g‘(mulGrp‘𝐾))𝑀)))
 
17-Jun-2025ply1asclzrhval 42138 Transfer results from algebraic scalars and ZR ring homomorphisms. (Contributed by metakunt, 17-Jun-2025.)
𝑊 = (Poly1‘𝑅)    &   𝐴 = (algSc‘𝑊)    &   𝐵 = (ℤRHom‘𝑊)    &   𝐶 = (ℤRHom‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑋 ∈ ℤ)    ⇒   (𝜑 → (𝐴‘(𝐶‘𝑋)) = (𝐵‘𝑋))
 
16-Jun-2025mapex 7973 The class of all functions mapping one set to another is a set. Remark after Definition 10.24 of [Kunen] p. 31. (Contributed by Raph Levien, 4-Dec-2003.) (Proof shortened by AV, 16-Jun-2025.)
((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)
 
14-Jun-20252sqr3minply 33730 The polynomial ((𝑋↑3) − 2) is the minimal polynomial for (2↑𝑐(1 / 3)) over ℚ, and its degree is 3. (Contributed by Thierry Arnoux, 14-Jun-2025.)
𝑄 = (ℂfld ↾s ℚ)    &    − = (-g‘𝑃)    &    ↑ = (.g‘(mulGrp‘𝑃))    &   𝑃 = (Poly1‘𝑄)    &   𝐾 = (algSc‘𝑃)    &   𝑋 = (var1‘𝑄)    &   𝐷 = (deg1‘𝑄)    &   𝐹 = ((3 ↑ 𝑋) − (𝐾‘2))    &   𝐴 = (2↑𝑐(1 / 3))    &   𝑀 = (ℂfld minPoly ℚ)    ⇒   (𝐹 = (𝑀‘𝐴) ∧ (𝐷‘𝐹) = 3)
 
14-Jun-2025evl1deg3 33560 Evaluation of a univariate polynomial of degree 3. (Contributed by Thierry Arnoux, 14-Jun-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑂 = (eval1‘𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &    · = (.r‘𝑅)    &    + = (+g‘𝑅)    &    ↑ = (.g‘(mulGrp‘𝑅))    &   𝐹 = (coe1‘𝑀)    &   𝐸 = (deg1‘𝑅)    &   𝐴 = (𝐹‘3)    &   𝐵 = (𝐹‘2)    &   𝐶 = (𝐹‘1)    &   𝐷 = (𝐹‘0)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑀 ∈ 𝑈)    &   (𝜑 → (𝐸‘𝑀) = 3)    &   (𝜑 → 𝑋 ∈ 𝐾)    ⇒   (𝜑 → ((𝑂‘𝑀)‘𝑋) = (((𝐴 · (3 ↑ 𝑋)) + (𝐵 · (2 ↑ 𝑋))) + ((𝐶 · 𝑋) + 𝐷)))
 
14-Jun-2025df-ufd 33524 Define the class of unique factorization domains. A unique factorization domain (UFD for short), is an integral domain such that every nonzero prime ideal contains a prime element (this is a characterization due to Irving Kaplansky). A UFD is sometimes also called a "factorial ring" following the terminology of Bourbaki. (Contributed by Mario Carneiro, 17-Feb-2015.) Exclude the 0 prime ideal. (Revised by Thierry Arnoux, 9-May-2025.) Exclude the 0 ring. (Revised by Thierry Arnoux, 14-Jun-2025.)
UFD = {𝑟 ∈ IDomn ∣ ∀𝑖 ∈ ((PrmIdeal‘𝑟) ∖ {{(0g‘𝑟)}})(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅}
 
14-Jun-2025expgt0b 32812 A real number 𝐴 raised to an odd integer power is positive iff it is positive. (Contributed by SN, 4-Mar-2023.) Use the more standard ¬ 2 ∥ 𝑁 (Revised by Thierry Arnoux, 14-Jun-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝑁)    ⇒   (𝜑 → (0 < 𝐴 ↔ 0 < (𝐴↑𝑁)))
 
12-Jun-2025gricbri 47759 Implications of two graphs being isomorphic. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 5-May-2025.) (Proof shortened by AV, 12-Jun-2025.)
𝑉 = (Vtx‘𝐴)    &   𝑊 = (Vtx‘𝐵)    &   𝐼 = (iEdg‘𝐴)    &   𝐽 = (iEdg‘𝐵)    ⇒   (𝐴 ≃𝑔𝑟 𝐵 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))
 
12-Jun-2025gricrcl 47757 Reverse closure of the "is isomorphic to" relation for graphs. (Contributed by AV, 12-Jun-2025.)
(𝐺 ≃𝑔𝑟 𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V))
 
11-Jun-2025grlicen 47824 Locally isomorphic graphs have equinumerous sets of vertices. (Contributed by AV, 11-Jun-2025.)
𝐵 = (Vtx‘𝑅)    &   𝐶 = (Vtx‘𝑆)    ⇒   (𝑅 ≃𝑙𝑔𝑟 𝑆 → 𝐵 ≈ 𝐶)
 
11-Jun-2025grlicer 47823 Local isomorphism is an equivalence relation on hypergraphs. (Contributed by AV, 11-Jun-2025.)
( ≃𝑙𝑔𝑟 ∩ (UHGraph × UHGraph)) Er UHGraph
 
11-Jun-2025brinxper 8786 Conditions for a reflexive, symmetric and transitive binary relation to be an equivalence relation over a class 𝑉. (Contributed by AV, 11-Jun-2025.)
(𝑥 ∈ 𝑉 → 𝑥 ∼ 𝑥)    &   (𝑥 ∈ 𝑉 → (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥))    &   (𝑥 ∈ 𝑉 → ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧))    ⇒   ( ∼ ∩ (𝑉 × 𝑉)) Er 𝑉
 
10-Jun-2025grlictr 47822 Graph local isomorphism is transitive. (Contributed by AV, 10-Jun-2025.)
((𝑅 ≃𝑙𝑔𝑟 𝑆 ∧ 𝑆 ≃𝑙𝑔𝑟 𝑇) → 𝑅 ≃𝑙𝑔𝑟 𝑇)
 
9-Jun-2025grlicsymb 47821 Graph local isomorphism is symmetric in both directions for hypergraphs. (Contributed by AV, 9-Jun-2025.)
((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴 ≃𝑙𝑔𝑟 𝐵 ↔ 𝐵 ≃𝑙𝑔𝑟 𝐴))
 
9-Jun-2025grlicsym 47820 Graph local isomorphism is symmetric for hypergraphs. (Contributed by AV, 9-Jun-2025.)
(𝐺 ∈ UHGraph → (𝐺 ≃𝑙𝑔𝑟 𝑆 → 𝑆 ≃𝑙𝑔𝑟 𝐺))
 
9-Jun-2025grlicref 47819 Graph local isomorphism is reflexive for hypergraphs. (Contributed by AV, 9-Jun-2025.)
(𝐺 ∈ UHGraph → 𝐺 ≃𝑙𝑔𝑟 𝐺)
 
9-Jun-2025grilcbri2 47818 Implications of two graphs being locally isomorphic. (Contributed by AV, 9-Jun-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   𝑁 = (𝐺 ClNeighbVtx 𝑋)    &   𝑀 = (𝐻 ClNeighbVtx (𝑓‘𝑋))    &   𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}    &   𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}    ⇒   (𝐺 ≃𝑙𝑔𝑟 𝐻 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ (𝑋 ∈ 𝑉 → ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
 
9-Jun-2025dfgrlic3 47817 Alternate, explicit definition of the "is locally isomorphic to" relation for two graphs. (Contributed by AV, 9-Jun-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   𝑀 = (𝐻 ClNeighbVtx (𝑓‘𝑣))    &   𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}    &   𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}    ⇒   ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
 
9-Jun-2025grilcbri 47816 Implications of two graphs being locally isomorphic. (Contributed by AV, 9-Jun-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    ⇒   (𝐺 ≃𝑙𝑔𝑟 𝐻 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑣)))))
 
9-Jun-2025dfgrlic2 47815 Alternate, explicit definition of the "is locally isomorphic to" relation for two graphs. (Contributed by AV, 9-Jun-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    ⇒   ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝑓‘𝑣))))))
 
9-Jun-2025grlicrcl 47814 Reverse closure of the "is locally isomorphic to" relation for graphs. (Contributed by AV, 9-Jun-2025.)
(𝐺 ≃𝑙𝑔𝑟 𝑆 → (𝐺 ∈ V ∧ 𝑆 ∈ V))
 
9-Jun-2025grlicrel 47813 The "is locally isomorphic to" relation for graphs is a relation. (Contributed by AV, 9-Jun-2025.)
Rel ≃𝑙𝑔𝑟
 
9-Jun-2025brgrilci 47812 Prove that two graphs are locally isomorphic by an explicit local isomorphism. (Contributed by AV, 9-Jun-2025.)
(𝐹 ∈ (𝑅 GraphLocIso 𝑆) → 𝑅 ≃𝑙𝑔𝑟 𝑆)
 
9-Jun-2025brgrlic 47811 The relation "is locally isomorphic to" for graphs. (Contributed by AV, 9-Jun-2025.)
(𝑅 ≃𝑙𝑔𝑟 𝑆 ↔ (𝑅 GraphLocIso 𝑆) ≠ ∅)
 
9-Jun-2025f1oabexg 7974 The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.) (Proof shortened by AV, 9-Jun-2025.)
𝐹 = {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)}    ⇒   ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V)
 
9-Jun-2025fabexg 7970 Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008.) (Proof shortened by AV, 9-Jun-2025.)
𝐹 = {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)}    ⇒   ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V)
 
8-Jun-2025ply1dg3rt0irred 33564 If a cubic polynomial over a field has no roots, it is irreducible. (Proposed by Saveliy Skresanov, 5-Jun-2025.) (Contributed by Thierry Arnoux, 8-Jun-2025.)
0 = (0g‘𝐹)    &   𝑂 = (eval1‘𝐹)    &   𝐷 = (deg1‘𝐹)    &   𝑃 = (Poly1‘𝐹)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → 𝐹 ∈ Field)    &   (𝜑 → 𝑄 ∈ 𝐵)    &   (𝜑 → (◡(𝑂‘𝑄) “ { 0 }) = ∅)    &   (𝜑 → (𝐷‘𝑄) = 3)    ⇒   (𝜑 → 𝑄 ∈ (Irred‘𝑃))
 
8-Jun-2025ply1mulrtss 33563 The roots of a factor 𝐹 are also roots of the product of polynomials (𝐹 · 𝐺). (Contributed by Thierry Arnoux, 8-Jun-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑈 = (Base‘𝑃)    &   𝑂 = (eval1‘𝑅)    &   𝐷 = (deg1‘𝑅)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝐹 ∈ 𝑈)    &   (𝜑 → 𝐺 ∈ 𝑈)    &    · = (.r‘𝑃)    ⇒   (𝜑 → (◡(𝑂‘𝐹) “ { 0 }) ⊆ (◡(𝑂‘(𝐹 · 𝐺)) “ { 0 }))
 
8-Jun-2025ply1dg1rtn0 33562 Polynomials of degree 1 over a field always have some roots. (Contributed by Thierry Arnoux, 8-Jun-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑈 = (Base‘𝑃)    &   𝑂 = (eval1‘𝑅)    &   𝐷 = (deg1‘𝑅)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ Field)    &   (𝜑 → 𝐺 ∈ 𝑈)    &   (𝜑 → (𝐷‘𝐺) = 1)    ⇒   (𝜑 → (◡(𝑂‘𝐺) “ { 0 }) ≠ ∅)
 
8-Jun-2025ply1dg1rt 33561 Express the root − 𝐵 / 𝐴 of a polynomial 𝐴 · 𝑋 + 𝐵 of degree 1 over a field. (Contributed by Thierry Arnoux, 8-Jun-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑈 = (Base‘𝑃)    &   𝑂 = (eval1‘𝑅)    &   𝐷 = (deg1‘𝑅)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ Field)    &   (𝜑 → 𝐺 ∈ 𝑈)    &   (𝜑 → (𝐷‘𝐺) = 1)    &   𝑁 = (invg‘𝑅)    &    / = (/r‘𝑅)    &   𝐶 = (coe1‘𝐺)    &   𝐴 = (𝐶‘1)    &   𝐵 = (𝐶‘0)    &   𝑍 = ((𝑁‘𝐵) / 𝐴)    ⇒   (𝜑 → (◡(𝑂‘𝐺) “ { 0 }) = {𝑍})
 
8-Jun-2025evl1deg1 33558 Evaluation of a univariate polynomial of degree 1. (Contributed by Thierry Arnoux, 8-Jun-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑂 = (eval1‘𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &    · = (.r‘𝑅)    &    + = (+g‘𝑅)    &   𝐶 = (coe1‘𝑀)    &   𝐷 = (deg1‘𝑅)    &   𝐴 = (𝐶‘1)    &   𝐵 = (𝐶‘0)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑀 ∈ 𝑈)    &   (𝜑 → (𝐷‘𝑀) = 1)    &   (𝜑 → 𝑋 ∈ 𝐾)    ⇒   (𝜑 → ((𝑂‘𝑀)‘𝑋) = ((𝐴 · 𝑋) + 𝐵))
 
8-Jun-2025evl1fvf 33546 The univariate polynomial evaluation function as a function, with domain and codomain. (Contributed by Thierry Arnoux, 8-Jun-2025.)
𝑂 = (eval1‘𝑅)    &   𝑃 = (Poly1‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑 → 𝑅 ∈ CRing)    &   𝐵 = (Base‘𝑅)    &   (𝜑 → 𝑄 ∈ 𝑈)    ⇒   (𝜑 → (𝑂‘𝑄):𝐵⟶𝐵)
 
8-Jun-2025domnlcanbOLD 33242 Obsolete version of domnlcanb 20736 as of 21-Jun-2025. (Contributed by Thierry Arnoux, 8-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝑅 ∈ Domn)    ⇒   (𝜑 → ((𝑋 · 𝑌) = (𝑋 · 𝑍) ↔ 𝑌 = 𝑍))
 
8-Jun-2025domnmuln0rd 33238 In a domain, factors of a nonzero product are nonzero. (Contributed by Thierry Arnoux, 8-Jun-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ Domn)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → (𝑋 · 𝑌) ≠ 0 )    ⇒   (𝜑 → (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ))
 
8-Jun-2025nn0disj01 32814 The pair {0, 1} does not overlap the rest of the nonnegative integers. (Contributed by Thierry Arnoux, 8-Jun-2025.)
({0, 1} ∩ (ℤ≥‘2)) = ∅
 
8-Jun-2025nn0split01 32813 Split 0 and 1 from the nonnegative integers. (Contributed by Thierry Arnoux, 8-Jun-2025.)
ℕ0 = ({0, 1} ∪ (ℤ≥‘2))
 
7-Jun-2025grimedg 47777 Graph isomorphisms map edges onto the corresponding edges. (Contributed by AV, 7-Jun-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (Edg‘𝐺)    &   𝐸 = (Edg‘𝐻)    ⇒   ((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → (𝐾 ∈ 𝐼 ↔ ((𝐹 “ 𝐾) ∈ 𝐸 ∧ 𝐾 ⊆ 𝑉)))
 
7-Jun-2025sn-isghm 42621 Longer proof of isghm 19249, unsuccessfully attempting to simplify isghm 19249 using elovmpo 7689 according to an editorial note (now removed). (Contributed by SN, 7-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑋 = (Base‘𝑆)    &   𝑌 = (Base‘𝑇)    &    + = (+g‘𝑆)    &    ⨣ = (+g‘𝑇)    ⇒   (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))
 
7-Jun-2025aks5lem2 42137 Lemma for section 5 https://www3.nd.edu/%7eandyp/notes/AKS.pdf. Construct the quotient for the AKS reduction. (Contributed by metakunt, 7-Jun-2025.)
(𝜑 → 𝐾 ∈ Field)    &   𝑃 = (chr‘𝐾)    &   (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁))    &   𝐹 = (𝑝 ∈ (Base‘(Poly1‘(ℤ/nℤ‘𝑁))) ↦ (𝐺 ∘ 𝑝))    &   𝐺 = (𝑞 ∈ (Base‘(ℤ/nℤ‘𝑁)) ↦ ∪ ((ℤRHom‘𝐾) “ 𝑞))    &   𝐻 = (𝑟 ∈ (Base‘(Poly1‘𝐾)) ↦ (((eval1‘𝐾)‘𝑟)‘𝑀))    &   (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   𝐼 = (𝑠 ∈ (Base‘𝐴) ↦ ∪ ((𝐻 ∘ 𝐹) “ 𝑠))    &   𝐴 = ((Poly1‘(ℤ/nℤ‘𝑁)) /s ((Poly1‘(ℤ/nℤ‘𝑁)) ~QG 𝐿))    &   𝐿 = ((RSpan‘(Poly1‘(ℤ/nℤ‘𝑁)))‘{((𝑅(.g‘(mulGrp‘(Poly1‘(ℤ/nℤ‘𝑁))))(var1‘(ℤ/nℤ‘𝑁)))(-g‘(Poly1‘(ℤ/nℤ‘𝑁)))(1r‘(Poly1‘(ℤ/nℤ‘𝑁))))})    &   (𝜑 → 𝑅 ∈ ℕ)    ⇒   (𝜑 → (𝐼 ∈ (𝐴 RingHom 𝐾) ∧ ∀𝑔 ∈ (Base‘(Poly1‘(ℤ/nℤ‘𝑁)))(𝐼‘[𝑔]((Poly1‘(ℤ/nℤ‘𝑁)) ~QG 𝐿)) = ((𝐻 ∘ 𝐹)‘𝑔)))
 
7-Jun-2025aks5lem1 42136 Section 5 of https://www3.nd.edu/%7eandyp/notes/AKS.pdf. Construction of a ring homomorphism out of Zn X to K. (Contributed by metakunt, 7-Jun-2025.)
(𝜑 → 𝐾 ∈ Field)    &   𝑃 = (chr‘𝐾)    &   (𝜑 → (𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ ∧ 𝑃 ∥ 𝑁))    &   𝐹 = (𝑝 ∈ (Base‘(Poly1‘(ℤ/nℤ‘𝑁))) ↦ (𝐺 ∘ 𝑝))    &   𝐺 = (𝑞 ∈ (Base‘(ℤ/nℤ‘𝑁)) ↦ ∪ ((ℤRHom‘𝐾) “ 𝑞))    &   𝐻 = (𝑟 ∈ (Base‘(Poly1‘𝐾)) ↦ (((eval1‘𝐾)‘𝑟)‘𝑀))    &   (𝜑 → 𝑀 ∈ (Base‘𝐾))    ⇒   (𝜑 → (𝐻 ∘ 𝐹) ∈ ((Poly1‘(ℤ/nℤ‘𝑁)) RingHom 𝐾))
 
7-Jun-2025rhmqusspan 42135 Ring homomorphism out of a quotient given an ideal spanned by a singleton. (Contributed by metakunt, 7-Jun-2025.)
0 = (0g‘𝐻)    &   (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))    &   𝐾 = (◡𝐹 “ { 0 })    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   (𝜑 → 𝐺 ∈ CRing)    &   𝑁 = ((RSpan‘𝐺)‘{𝑋})    &   (𝜑 → 𝑋 ∈ (Base‘𝐺))    &   (𝜑 → (𝐹‘𝑋) = 0 )    ⇒   (𝜑 → (𝐽 ∈ (𝑄 RingHom 𝐻) ∧ ∀𝑔 ∈ (Base‘𝐺)(𝐽‘[𝑔](𝐺 ~QG 𝑁)) = (𝐹‘𝑔)))
 
7-Jun-2025cnvopab 6164 The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-10 2141, ax-12 2178. (Revised by SN, 7-Jun-2025.)
◡{⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑦, 𝑥⟩ ∣ 𝜑}
 
6-Jun-2025dfufd2 33535 Alternative definition of unique factorization domain (UFD). This is often the textbook definition. Chapter VII, Paragraph 3, Section 3, Proposition 2 of [BourbakiCAlg2], p. 228. (Contributed by Thierry Arnoux, 6-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    ⇒   (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑥 ∈ ((𝐵 ∖ 𝑈) ∖ { 0 })∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)))
 
6-Jun-2025dfufd2lem 33534 Lemma for dfufd2 33535. (Contributed by Thierry Arnoux, 6-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &   (𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → 𝐼 ∈ (PrmIdeal‘𝑅))    &   (𝜑 → 𝐹 ∈ Word 𝑃)    &   (𝜑 → (𝑀 Σg 𝐹) ∈ 𝐼)    &   (𝜑 → (𝑀 Σg 𝐹) ≠ 0 )    ⇒   (𝜑 → (𝐼 ∩ 𝑃) ≠ ∅)
 
6-Jun-2025domnprodn0 33239 In a domain, a finite product of nonzero terms is nonzero. (Contributed by Thierry Arnoux, 6-Jun-2025.)
𝐵 = (Base‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ Domn)    &   (𝜑 → 𝐹 ∈ Word (𝐵 ∖ { 0 }))    ⇒   (𝜑 → (𝑀 Σg 𝐹) ≠ 0 )
 
5-Jun-2025cjex 42237 The conjugate function is a set. (Contributed by SN, 5-Jun-2025.)
∗ ∈ V
 
5-Jun-2025absex 42236 The absolute value function is a set. (Contributed by SN, 5-Jun-2025.)
abs ∈ V
 
5-Jun-2025subex 42235 The subtraction operation is a set. (Contributed by SN, 5-Jun-2025.)
− ∈ V
 
5-Jun-2025leex 42234 The less-than-or-equal-to relation is a set. (Contributed by SN, 5-Jun-2025.)
≤ ∈ V
 
5-Jun-2025ltex 42233 The less-than relation is a set. (Contributed by SN, 5-Jun-2025.)
< ∈ V
 
5-Jun-2025elno 27700 Membership in the surreals. (Contributed by Scott Fenton, 11-Jun-2011.) (Proof shortened by SF, 14-Apr-2012.) Avoid ax-rep 5303. (Revised by SN, 5-Jun-2025.)
(𝐴 ∈ No ↔ ∃𝑥 ∈ On 𝐴:𝑥⟶{1o, 2o})
 
5-Jun-2025isghm 19249 Property of being a homomorphism of groups. (Contributed by Stefan O'Rear, 31-Dec-2014.) (Proof shortened by SN, 5-Jun-2025.)
𝑋 = (Base‘𝑆)    &   𝑌 = (Base‘𝑇)    &    + = (+g‘𝑆)    &    ⨣ = (+g‘𝑇)    ⇒   (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹‘𝑢) ⨣ (𝐹‘𝑣)))))
 
5-Jun-20251div0 11943 You can't divide by zero, because division explicitly excludes zero from the domain of the function. Thus, by the definition of function value, it evaluates to the empty set. (This theorem is for information only and normally is not referenced by other proofs. To be meaningful, it assumes that ∅ is not a complex number, which depends on the particular complex number construction that is used.) (Contributed by Mario Carneiro, 1-Apr-2014.) (Proof shortened by SN, 5-Jun-2025.) (New usage is discouraged.)
(1 / 0) = ∅
 
5-Jun-2025ceqsexv2d 3545 Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.) Shorten, reduce dv conditions. (Revised by Wolf Lammen, 5-Jun-2025.) (Proof shortened by SN, 5-Jun-2025.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   𝜓    ⇒   ∃𝑥𝜑
 
4-Jun-2025zndvdchrrhm 41920 Construction of a ring homomorphism from ℤ/nℤ to 𝑅 when the characteristic of 𝑅 divides 𝑁. (Contributed by metakunt, 4-Jun-2025.)
(𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → (chr‘𝑅) ∈ ℤ)    &   (𝜑 → (chr‘𝑅) ∥ 𝑁)    &   𝑍 = (ℤ/nℤ‘𝑁)    &   𝐹 = (𝑥 ∈ (Base‘𝑍) ↦ ∪ ((ℤRHom‘𝑅) “ 𝑥))    ⇒   (𝜑 → 𝐹 ∈ (𝑍 RingHom 𝑅))
 
4-Jun-2025rhmzrhval 41919 Evaluation of integers across a ring homomorphism. (Contributed by metakunt, 4-Jun-2025.)
(𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆))    &   (𝜑 → 𝑋 ∈ ℤ)    &   𝑀 = (ℤRHom‘𝑅)    &   𝑁 = (ℤRHom‘𝑆)    ⇒   (𝜑 → (𝐹‘(𝑀‘𝑋)) = (𝑁‘𝑋))
 
3-Jun-20251arithufd 33533 Existence of a factorization into irreducible elements in a unique factorization domain. Any non-zero, non-unit element 𝑋 of a UFD 𝑅 can be written as a product of primes 𝑓. As shown in 1arithidom 33522, that factorization is unique, up to the order of the factors and multiplication by units. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &   (𝜑 → 𝑅 ∈ UFD)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → ¬ 𝑋 ∈ 𝑈)    &   (𝜑 → 𝑋 ≠ 0 )    ⇒   (𝜑 → ∃𝑓 ∈ Word 𝑃𝑋 = (𝑀 Σg 𝑓))
 
3-Jun-20251arithufdlem4 33532 Lemma for 1arithufd 33533. Nonzero ring, non-field case. Those trivial cases are handled in the final proof. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &   (𝜑 → 𝑅 ∈ UFD)    &   (𝜑 → ¬ 𝑅 ∈ DivRing)    &   𝑆 = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)}    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → ¬ 𝑋 ∈ 𝑈)    &   (𝜑 → 𝑋 ≠ 0 )    ⇒   (𝜑 → 𝑋 ∈ 𝑆)
 
3-Jun-20251arithufdlem3 33531 Lemma for 1arithufd 33533. If a product (𝑌 · 𝑋) can be written as a product of primes, with 𝑋 non-unit, nonzero, so can 𝑋. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &   (𝜑 → 𝑅 ∈ UFD)    &   (𝜑 → ¬ 𝑅 ∈ DivRing)    &   𝑆 = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)}    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → ¬ 𝑋 ∈ 𝑈)    &   (𝜑 → 𝑋 ≠ 0 )    &    · = (.r‘𝑅)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → (𝑌 · 𝑋) ∈ 𝑆)    ⇒   (𝜑 → 𝑋 ∈ 𝑆)
 
3-Jun-20251arithufdlem2 33530 Lemma for 1arithufd 33533. The set 𝑆 of elements which can be written as a product of primes is multiplicatively closed. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &   (𝜑 → 𝑅 ∈ UFD)    &   (𝜑 → ¬ 𝑅 ∈ DivRing)    &   𝑆 = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)}    &    · = (.r‘𝑅)    &   (𝜑 → 𝑋 ∈ 𝑆)    &   (𝜑 → 𝑌 ∈ 𝑆)    ⇒   (𝜑 → (𝑋 · 𝑌) ∈ 𝑆)
 
3-Jun-20251arithufdlem1 33529 Lemma for 1arithufd 33533. The set 𝑆 of elements which can be written as a product of primes is not empty. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &   (𝜑 → 𝑅 ∈ UFD)    &   (𝜑 → ¬ 𝑅 ∈ DivRing)    &   𝑆 = {𝑥 ∈ 𝐵 ∣ ∃𝑓 ∈ Word 𝑃𝑥 = (𝑀 Σg 𝑓)}    ⇒   (𝜑 → 𝑆 ≠ ∅)
 
3-Jun-2025pidufd 33528 Every principal ideal domain is a unique factorization domain. (Contributed by Thierry Arnoux, 3-Jun-2025.)
(𝜑 → 𝑅 ∈ PID)    ⇒   (𝜑 → 𝑅 ∈ UFD)
 
3-Jun-2025ufdidom 33527 A nonzero unique factorization domain is an integral domain. (Contributed by Thierry Arnoux, 3-Jun-2025.)
(𝜑 → 𝑅 ∈ UFD)    ⇒   (𝜑 → 𝑅 ∈ IDomn)
 
3-Jun-2025ufdprmidl 33526 In a unique factorization domain 𝑅, a nonzero prime ideal 𝐽 contains a prime element 𝑝. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝐼 = (PrmIdeal‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ UFD)    &   (𝜑 → 𝐽 ∈ 𝐼)    &   (𝜑 → 𝐽 ≠ { 0 })    ⇒   (𝜑 → ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽)
 
3-Jun-2025unitmulrprm 33513 A ring unit multiplied by a ring prime is a ring prime. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝑃 = (RPrime‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → 𝐼 ∈ 𝑈)    &   (𝜑 → 𝑄 ∈ 𝑃)    ⇒   (𝜑 → (𝐼 · 𝑄) ∈ 𝑃)
 
3-Jun-2025krullndrng 33466 Krull's theorem for non-division-rings: Existence of a nonzero maximal ideal. (Contributed by Thierry Arnoux, 3-Jun-2025.)
0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ NzRing)    &   (𝜑 → ¬ 𝑅 ∈ DivRing)    ⇒   (𝜑 → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝑚 ≠ { 0 })
 
3-Jun-2025drngmxidlr 33463 If a ring's only maximal ideal is the zero ideal, it is a division ring. See also drngmxidl 33462. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝑀 = (MaxIdeal‘𝑅)    &   (𝜑 → 𝑅 ∈ NzRing)    &   (𝜑 → 𝑀 = {{ 0 }})    ⇒   (𝜑 → 𝑅 ∈ DivRing)
 
3-Jun-2025ssdifidlprm 33443 If the set 𝑆 of ssdifidl 33442 is multiplicatively closed, then the ideal 𝑖 is prime. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝐵 = (Base‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑 → 𝑆 ∈ (SubMnd‘𝑀))    &   𝑀 = (mulGrp‘𝑅)    &   (𝜑 → (𝑆 ∩ 𝐼) = ∅)    &   𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ 𝐼 ⊆ 𝑝)}    ⇒   (𝜑 → ∃𝑖 ∈ 𝑃 (𝑖 ∈ (PrmIdeal‘𝑅) ∧ ∀𝑗 ∈ 𝑃 ¬ 𝑖 ⊊ 𝑗))
 
3-Jun-2025ssdifidl 33442 Let 𝑅 be a ring, and let 𝐼 be an ideal of 𝑅 disjoint with a set 𝑆. Then there exists an ideal 𝑖, maximal among the set 𝑃 of ideals containing 𝐼 and disjoint with 𝑆. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝐵 = (Base‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑 → 𝑆 ⊆ 𝐵)    &   (𝜑 → (𝑆 ∩ 𝐼) = ∅)    &   𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ 𝐼 ⊆ 𝑝)}    ⇒   (𝜑 → ∃𝑖 ∈ 𝑃 ∀𝑗 ∈ 𝑃 ¬ 𝑖 ⊊ 𝑗)
 
3-Jun-2025ssdifidllem 33441 Lemma for ssdifidl 33442: The set 𝑃 used in the proof of ssdifidl 33442 satisfies the condition of Zorn's Lemma. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝐵 = (Base‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑 → 𝑆 ⊆ 𝐵)    &   (𝜑 → (𝑆 ∩ 𝐼) = ∅)    &   𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ ((𝑆 ∩ 𝑝) = ∅ ∧ 𝐼 ⊆ 𝑝)}    &   (𝜑 → 𝑍 ⊆ 𝑃)    &   (𝜑 → 𝑍 ≠ ∅)    &   (𝜑 → [⊊] Or 𝑍)    ⇒   (𝜑 → ∪ 𝑍 ∈ 𝑃)
 
3-Jun-2025lidlmcld 33404 An ideal is closed under left-multiplication by elements of the full ring. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐼 ∈ 𝑈)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐼)    ⇒   (𝜑 → (𝑋 · 𝑌) ∈ 𝐼)
 
3-Jun-2025lpirlidllpi 33359 In a principal ideal ring, ideals are principal. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝐵 = (Base‘𝑅)    &   𝐼 = (LIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑 → 𝑅 ∈ LPIR)    &   (𝜑 → 𝐽 ∈ 𝐼)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 𝐽 = (𝐾‘{𝑥}))
 
3-Jun-2025ringdi22 33203 Expand the product of two sums in a ring. (Contributed by Thierry Arnoux, 3-Jun-2025.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝑇 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋 + 𝑌) · (𝑍 + 𝑇)) = (((𝑋 · 𝑍) + (𝑌 · 𝑍)) + ((𝑋 · 𝑇) + (𝑌 · 𝑇))))
 
3-Jun-2025subgcld 33019 A subgroup is closed under group operation. (Contributed by Thierry Arnoux, 3-Jun-2025.)
+ = (+g‘𝐺)    &   (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑 → 𝑋 ∈ 𝑆)    &   (𝜑 → 𝑌 ∈ 𝑆)    ⇒   (𝜑 → (𝑋 + 𝑌) ∈ 𝑆)
 
3-Jun-2025an82ds 32475 Inference exchanging the last antecedent with the second one. (Contributed by Thierry Arnoux, 3-Jun-2025.)
((((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) → 𝜌)    ⇒   ((((((((𝜑 ∧ 𝜎) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜓) → 𝜌)
 
3-Jun-2025an72ds 32474 Inference exchanging the last antecedent with the second one. (Contributed by Thierry Arnoux, 3-Jun-2025.)
(((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) → 𝜎)    ⇒   (((((((𝜑 ∧ 𝜁) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜓) → 𝜎)
 
3-Jun-2025an62ds 32473 Inference exchanging the last antecedent with the second one. (Contributed by Thierry Arnoux, 3-Jun-2025.)
((((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) → 𝜁)    ⇒   ((((((𝜑 ∧ 𝜂) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜓) → 𝜁)
 
3-Jun-2025an52ds 32472 Inference exchanging the last antecedent with the second. (Contributed by Thierry Arnoux, 3-Jun-2025.)
(((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂)    ⇒   (((((𝜑 ∧ 𝜏) ∧ 𝜒) ∧ 𝜃) ∧ 𝜓) → 𝜂)
 
3-Jun-2025an42ds 32471 Inference exchanging the last antecedent with the second one. See also an32s 651. (Contributed by Thierry Arnoux, 3-Jun-2025.)
((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ((((𝜑 ∧ 𝜃) ∧ 𝜒) ∧ 𝜓) → 𝜏)
 
3-Jun-2025qusmulcrng 21311 Value of the ring operation in a quotient ring of a commutative ring. (Contributed by Thierry Arnoux, 1-Sep-2024.) (Proof shortened by metakunt, 3-Jun-2025.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    × = (.r‘𝑄)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → ([𝑋](𝑅 ~QG 𝐼) × [𝑌](𝑅 ~QG 𝐼)) = [(𝑋 · 𝑌)](𝑅 ~QG 𝐼))
 
2-Jun-2025uhgrimgrlim 47801 An isomorphism of hypergraphs is a local isomorphism between the two graphs. (Contributed by AV, 2-Jun-2025.)
((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))
 
2-Jun-2025clnbgrgrim 47776 Graph isomorphisms between hypergraphs map closed neighborhoods onto closed neighborhoods. (Contributed by AV, 2-Jun-2025.)
𝑉 = (Vtx‘𝐺)    ⇒   ((((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻)) ∧ 𝑋 ∈ 𝑉) → (𝐻 ClNeighbVtx (𝐹‘𝑋)) = (𝐹 “ (𝐺 ClNeighbVtx 𝑋)))
 
2-Jun-2025clnbgrgrimlem 47775 Lemma for clnbgrgrim 47776: For two isomorphic hypergraphs, if there is an edge connecting the image of a vertex of the first graph with a vertex of the second graph, the vertex of the second graph is the image of a neighbor of the vertex of the first graph. (Contributed by AV, 2-Jun-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐸 = (Edg‘𝐻)    ⇒   (((𝐺 ∈ UHGraph ∧ 𝐻 ∈ UHGraph) ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊)) → ((𝐾 ∈ 𝐸 ∧ {(𝐹‘𝑋), 𝑌} ⊆ 𝐾) → ∃𝑛 ∈ (𝐺 ClNeighbVtx 𝑋)(𝐹‘𝑛) = 𝑌))
 
31-May-2025uhgrimisgrgric 47773 For isomorphic hypergraphs, the induced subgraph of a subset of vertices of one graph is isomorphic to the subgraph induced by the image of the subset. (Contributed by AV, 31-May-2025.)
𝑉 = (Vtx‘𝐺)    ⇒   ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ (𝐺 GraphIso 𝐻) ∧ 𝑁 ⊆ 𝑉) → (𝐺 ISubGr 𝑁) ≃𝑔𝑟 (𝐻 ISubGr (𝐹 “ 𝑁)))
 
31-May-2025uhgrimisgrgriclem 47772 Lemma for uhgrimisgrgric 47773. (Contributed by AV, 31-May-2025.)
(((𝐹:𝑉–1-1-onto→𝑊 ∧ 𝐺:𝐴⟶𝒫 𝑉) ∧ (𝑁 ⊆ 𝑉 ∧ 𝐼:𝐴–1-1-onto→𝐵) ∧ ∀𝑖 ∈ 𝐴 (𝐻‘(𝐼‘𝑖)) = (𝐹 “ (𝐺‘𝑖))) → ((𝐽 ∈ 𝐵 ∧ (𝐻‘𝐽) ⊆ (𝐹 “ 𝑁)) ↔ ∃𝑘 ∈ 𝐴 ((𝐺‘𝑘) ⊆ 𝑁 ∧ (𝐼‘𝑘) = 𝐽)))
 
31-May-2025vtocl4ga 3599 Implicit substitution of 4 classes for 4 setvar variables. (Contributed by AV, 22-Jan-2019.) (Proof shortened by Wolf Lammen, 31-May-2025.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   (𝑧 = 𝐶 → (𝜒 ↔ 𝜌))    &   (𝑤 = 𝐷 → (𝜌 ↔ 𝜃))    &   (((𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝑅) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑇)) → 𝜑)    ⇒   (((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑅) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑇)) → 𝜃)
 
31-May-2025vtocl3ga 3595 Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.) Reduce axiom usage. (Revised by GG, 3-Oct-2024.) (Proof shortened by Wolf Lammen, 31-May-2025.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))    &   ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑)    ⇒   ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃)
 
31-May-2025vtocl3gaf 3593 Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.) (Proof shortened by Wolf Lammen, 31-May-2025.)
Ⅎ𝑥𝐴    &   Ⅎ𝑦𝐴    &   Ⅎ𝑧𝐴    &   Ⅎ𝑦𝐵    &   Ⅎ𝑧𝐵    &   Ⅎ𝑧𝐶    &   Ⅎ𝑥𝜓    &   Ⅎ𝑦𝜒    &   Ⅎ𝑧𝜃    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))    &   ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑇) → 𝜑)    ⇒   ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑇) → 𝜃)
 
31-May-2025vtocl2gaf 3591 Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.) (Proof shortened by Wolf Lammen, 31-May-2025.)
Ⅎ𝑥𝐴    &   Ⅎ𝑦𝐴    &   Ⅎ𝑦𝐵    &   Ⅎ𝑥𝜓    &   Ⅎ𝑦𝜒    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑)    ⇒   ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒)
 
31-May-2025vtocle 3567 Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.) Avoid df-clab 2718. (Revised by Wolf Lammen, 31-May-2025.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → 𝜑)    ⇒   𝜑
 
30-May-20251arithidomlem1 33520 Lemma for 1arithidom 33522. (Contributed by Thierry Arnoux, 30-May-2025.)
𝑈 = (Unit‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &    · = (.r‘𝑅)    &   𝐽 = (0..^(♯‘𝐹))    &   (𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → 𝐹 ∈ Word 𝑃)    &   (𝜑 → 𝐺 ∈ Word 𝑃)    &   (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg 𝐺))    &   (𝜑 → 𝑄 ∈ 𝑃)    &   (𝜑 → ∀𝑔 ∈ Word 𝑃(∃𝑘 ∈ 𝑈 (𝑀 Σg 𝐹) = (𝑘 · (𝑀 Σg 𝑔)) → ∃𝑤∃𝑢 ∈ (𝑈 ↑m (0..^(♯‘𝐹)))(𝑤:(0..^(♯‘𝐹))–1-1-onto→(0..^(♯‘𝐹)) ∧ 𝑔 = (𝑢 ∘f · (𝐹 ∘ 𝑤)))))    &   (𝜑 → 𝐻 ∈ Word 𝑃)    &   (𝜑 → ∃𝑘 ∈ 𝑈 (𝑀 Σg (𝐹 ++ ⟨“𝑄”⟩)) = (𝑘 · (𝑀 Σg 𝐻)))    &   (𝜑 → 𝐾 ∈ (0..^(♯‘𝐻)))    &   (𝜑 → 𝑄(∥r‘𝑅)(𝐻‘𝐾))    &   (𝜑 → 𝑇 ∈ 𝑈)    &   (𝜑 → (𝑇 · 𝑄) = (𝐻‘𝐾))    &   (𝜑 → 𝑆:(0..^(♯‘𝐻))–1-1-onto→(0..^(♯‘𝐻)))    &   (𝜑 → (𝐻 ∘ 𝑆) = (((𝐻 ∘ 𝑆) prefix ((♯‘𝐻) − 1)) ++ ⟨“(𝐻‘𝐾)”⟩))    &   (𝜑 → 𝑁 ∈ 𝑈)    &   (𝜑 → (𝑀 Σg (𝐹 ++ ⟨“𝑄”⟩)) = (𝑁 · (𝑀 Σg 𝐻)))    ⇒   (𝜑 → ∃𝑐∃𝑑 ∈ (𝑈 ↑m (0..^(♯‘𝐹)))(𝑐:(0..^(♯‘𝐹))–1-1-onto→(0..^(♯‘𝐹)) ∧ ((𝐻 ∘ 𝑆) prefix ((♯‘𝐻) − 1)) = (𝑑 ∘f · (𝐹 ∘ 𝑐))))
 
29-May-2025grlimprop2 47800 Properties of a local isomorphism of graphs. (Contributed by AV, 29-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))    &   𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}    &   𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}    ⇒   (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
 
29-May-2025isgrlim2 47797 A local isomorphism of graphs is a bijection between their vertices that preserves neighborhoods. Definitions expanded. (Contributed by AV, 29-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝑁 = (𝐺 ClNeighbVtx 𝑣)    &   𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣))    &   𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}    &   𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}    ⇒   ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
 
29-May-2025clnbgrisubgrgrim 47774 Isomorphic subgraphs induced by closed neighborhoods of vertices of two graphs. (Contributed by AV, 29-May-2025.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   𝑁 = (𝐺 ClNeighbVtx 𝑋)    &   𝑀 = (𝐻 ClNeighbVtx 𝑌)    &   𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}    &   𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}    ⇒   ((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) → ((𝐺 ISubGr 𝑁) ≃𝑔𝑟 (𝐻 ISubGr 𝑀) ↔ ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
 
29-May-2025isubgrgrim 47771 Isomorphic subgraphs induced by subsets of vertices of two graphs. (Contributed by AV, 29-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}    &   𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}    ⇒   (((𝐺 ∈ 𝑈 ∧ 𝐻 ∈ 𝑇) ∧ (𝑁 ⊆ 𝑉 ∧ 𝑀 ⊆ 𝑊)) → ((𝐺 ISubGr 𝑁) ≃𝑔𝑟 (𝐻 ISubGr 𝑀) ↔ ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
 
29-May-2025dvdsruasso2 33371 A reformulation of dvdsruasso 33370. (Proposed by Gerard Lang, 28-May-2025.) (Contributed by Thiery Arnoux, 29-May-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    ∥ = (∥r‘𝑅)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝑈 = (Unit‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ IDomn)    &    1 = (1r‘𝑅)    ⇒   (𝜑 → ((𝑋 ∥ 𝑌 ∧ 𝑌 ∥ 𝑋) ↔ ∃𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑈 ((𝑢 · 𝑋) = 𝑌 ∧ (𝑣 · 𝑌) = 𝑋 ∧ (𝑢 · 𝑣) = 1 )))
 
28-May-2025n0nsnel 32536 If a class with one element is not a singleton, there is at least another element in this class. (Contributed by AV, 6-Mar-2025.) (Revised by Thierry Arnoux, 28-May-2025.)
((𝐶 ∈ 𝐵 ∧ 𝐵 ≠ {𝐴}) → ∃𝑥 ∈ 𝐵 𝑥 ≠ 𝐴)
 
28-May-2025elex 3509 If a class is a member of another class, then it is a set. Theorem 6.12 of [Quine] p. 44. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Wolf Lammen, 28-May-2025.)
(𝐴 ∈ 𝐵 → 𝐴 ∈ V)
 
27-May-2025sn-tz6.12-2 42628 tz6.12-2 6903 without ax-10 2141, ax-11 2158, ax-12 2178. Improves 118 theorems. (Contributed by SN, 27-May-2025.)
(¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)
 
27-May-2025euabsn2w 42627 Replace ax-10 2141, ax-11 2158, ax-12 2178 in euabsn2 4750 with substitution hypotheses. (Contributed by SN, 27-May-2025.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   (𝑥 = 𝑧 → (𝜑 ↔ 𝜃))    ⇒   (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})
 
27-May-2025absnw 42626 Replace ax-10 2141, ax-11 2158, ax-12 2178 in absn 4667 with a substitution hypothesis. (Contributed by SN, 27-May-2025.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ({𝑥 ∣ 𝜑} = {𝑌} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑌))
 
27-May-2025abbibw 42625 Replace ax-10 2141, ax-11 2158, ax-12 2178 in abbib 2814 with substitution hypotheses. (Contributed by SN, 27-May-2025.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜃))    &   (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))    ⇒   ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 ↔ 𝜓))
 
27-May-2025eu6w 42624 Replace ax-10 2141, ax-12 2178 in eu6 2577 with substitution hypotheses. (Contributed by SN, 27-May-2025.)
(𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜃))    ⇒   (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
 
27-May-20251arithidom 33522 Uniqueness of prime factorizations in an integral domain 𝑅. Given two equal products 𝐹 and 𝐺 of prime elements, 𝐹 and 𝐺 are equal up to a renumbering 𝑤 and a multiplication by units 𝑢. See also 1arith 16968. Chapter VII, Paragraph 3, Section 3, Proposition 2 of [BourbakiCAlg2], p. 228. (Contributed by Thierry Arnoux, 27-May-2025.)
𝑈 = (Unit‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &    · = (.r‘𝑅)    &   𝐽 = (0..^(♯‘𝐹))    &   (𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → 𝐹 ∈ Word 𝑃)    &   (𝜑 → 𝐺 ∈ Word 𝑃)    &   (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg 𝐺))    ⇒   (𝜑 → ∃𝑤∃𝑢 ∈ (𝑈 ↑m 𝐽)(𝑤:𝐽–1-1-onto→𝐽 ∧ 𝐺 = (𝑢 ∘f · (𝐹 ∘ 𝑤))))
 
27-May-20251arithidomlem2 33521 Lemma for 1arithidom 33522: induction step. (Contributed by Thierry Arnoux, 27-May-2025.)
𝑈 = (Unit‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &    · = (.r‘𝑅)    &   𝐽 = (0..^(♯‘𝐹))    &   (𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → 𝐹 ∈ Word 𝑃)    &   (𝜑 → 𝐺 ∈ Word 𝑃)    &   (𝜑 → (𝑀 Σg 𝐹) = (𝑀 Σg 𝐺))    &   (𝜑 → 𝑄 ∈ 𝑃)    &   (𝜑 → ∀𝑔 ∈ Word 𝑃(∃𝑘 ∈ 𝑈 (𝑀 Σg 𝐹) = (𝑘 · (𝑀 Σg 𝑔)) → ∃𝑤∃𝑢 ∈ (𝑈 ↑m (0..^(♯‘𝐹)))(𝑤:(0..^(♯‘𝐹))–1-1-onto→(0..^(♯‘𝐹)) ∧ 𝑔 = (𝑢 ∘f · (𝐹 ∘ 𝑤)))))    &   (𝜑 → 𝐻 ∈ Word 𝑃)    &   (𝜑 → ∃𝑘 ∈ 𝑈 (𝑀 Σg (𝐹 ++ ⟨“𝑄”⟩)) = (𝑘 · (𝑀 Σg 𝐻)))    &   (𝜑 → 𝐾 ∈ (0..^(♯‘𝐻)))    &   (𝜑 → 𝑄(∥r‘𝑅)(𝐻‘𝐾))    &   (𝜑 → 𝑇 ∈ 𝑈)    &   (𝜑 → (𝑇 · 𝑄) = (𝐻‘𝐾))    &   (𝜑 → 𝑆:(0..^(♯‘𝐻))–1-1-onto→(0..^(♯‘𝐻)))    &   (𝜑 → (𝐻 ∘ 𝑆) = (((𝐻 ∘ 𝑆) prefix ((♯‘𝐻) − 1)) ++ ⟨“(𝐻‘𝐾)”⟩))    &   (𝜑 → 𝑁 ∈ 𝑈)    &   (𝜑 → (𝑀 Σg (𝐹 ++ ⟨“𝑄”⟩)) = (𝑁 · (𝑀 Σg 𝐻)))    &   (𝜑 → 𝐷 ∈ (𝑈 ↑m (0..^(♯‘𝐹))))    &   (𝜑 → 𝐶:(0..^(♯‘𝐹))–1-1-onto→(0..^(♯‘𝐹)))    &   (𝜑 → ((𝐻 ∘ 𝑆) prefix ((♯‘𝐻) − 1)) = (𝐷 ∘f · (𝐹 ∘ 𝐶)))    ⇒   (𝜑 → (((𝐶 ++ ⟨“(♯‘𝐹)”⟩) ∘ ◡𝑆):(0..^(♯‘(𝐹 ++ ⟨“𝑄”⟩)))–1-1-onto→(0..^(♯‘(𝐹 ++ ⟨“𝑄”⟩))) ∧ 𝐻 = (((𝐷 ++ ⟨“𝑇”⟩) ∘ ◡𝑆) ∘f · ((𝐹 ++ ⟨“𝑄”⟩) ∘ ((𝐶 ++ ⟨“(♯‘𝐹)”⟩) ∘ ◡𝑆)))))
 
27-May-2025rprmndvdsru 33514 A ring prime element does not divide any ring unit. (Contributed by Thierry Arnoux, 27-May-2025.)
𝑈 = (Unit‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &    ∥ = (∥r‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑄 ∈ 𝑃)    &   (𝜑 → 𝑇 ∈ 𝑈)    ⇒   (𝜑 → ¬ 𝑄 ∥ 𝑇)
 
27-May-2025rprmasso3 33512 In an integral domain, if a prime element divides another, they are associates. (Contributed by Thierry Arnoux, 27-May-2025.)
𝐵 = (Base‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &    ∥ = (∥r‘𝑅)    &   (𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑋 ∥ 𝑌)    &   (𝜑 → 𝑌 ∈ 𝑃)    &    · = (.r‘𝑅)    &   𝑈 = (Unit‘𝑅)    ⇒   (𝜑 → ∃𝑡 ∈ 𝑈 (𝑡 · 𝑋) = 𝑌)
 
27-May-2025unitprodclb 33374 A finite product is a unit iff all factors are units. (Contributed by Thierry Arnoux, 27-May-2025.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝐹 ∈ Word 𝐵)    ⇒   (𝜑 → ((𝑀 Σg 𝐹) ∈ 𝑈 ↔ ran 𝐹 ⊆ 𝑈))
 
27-May-2025wrdpmtrlast 33078 Reorder a word, so that the symbol given at index 𝐼 is at the end. (Contributed by Thierry Arnoux, 27-May-2025.)
𝐽 = (0..^(♯‘𝑊))    &   (𝜑 → 𝐼 ∈ 𝐽)    &   (𝜑 → 𝑊 ∈ Word 𝑆)    &   𝑈 = ((𝑊 ∘ 𝑠) prefix ((♯‘𝑊) − 1))    ⇒   (𝜑 → ∃𝑠(𝑠:𝐽–1-1-onto→𝐽 ∧ (𝑊 ∘ 𝑠) = (𝑈 ++ ⟨“(𝑊‘𝐼)”⟩)))
 
27-May-2025fzo0pmtrlast 33077 Reorder a half-open integer range based at 0, so that the given index 𝐼 is at the end. (Contributed by Thierry Arnoux, 27-May-2025.)
𝐽 = (0..^𝑁)    &   (𝜑 → 𝐼 ∈ 𝐽)    ⇒   (𝜑 → ∃𝑠(𝑠:𝐽–1-1-onto→𝐽 ∧ (𝑠‘(𝑁 − 1)) = 𝐼))
 
27-May-2025ccatws1f1olast 32911 Two ways to reorder symbols in a word 𝑊 according to permutation 𝑇, and add a last symbol 𝑋. (Contributed by Thierry Arnoux, 27-May-2025.)
𝑁 = (♯‘𝑊)    &   (𝜑 → 𝑊 ∈ Word 𝑆)    &   (𝜑 → 𝑋 ∈ 𝑆)    &   (𝜑 → 𝑇:(0..^𝑁)–1-1-onto→(0..^𝑁))    ⇒   (𝜑 → ((𝑊 ++ ⟨“𝑋”⟩) ∘ (𝑇 ++ ⟨“𝑁”⟩)) = ((𝑊 ∘ 𝑇) ++ ⟨“𝑋”⟩))
 
27-May-2025ccatws1f1o 32910 Conditions for the concatenation of a word and a singleton word to be bijective. (Contributed by Thierry Arnoux, 27-May-2025.)
𝑁 = (♯‘𝑇)    &   𝐽 = (0..^(𝑁 + 1))    &   (𝜑 → 𝑇:(0..^𝑁)–1-1-onto→(0..^𝑁))    ⇒   (𝜑 → (𝑇 ++ ⟨“𝑁”⟩):𝐽–1-1-onto→𝐽)
 
27-May-2025wrdpmcl 32896 Closure of a word with permuted symbols. (Contributed by Thierry Arnoux, 27-May-2025.)
𝐽 = (0..^(♯‘𝑊))    &   (𝜑 → 𝐸:𝐽–1-1-onto→𝐽)    &   (𝜑 → 𝑊 ∈ Word 𝑆)    ⇒   (𝜑 → (𝑊 ∘ 𝐸) ∈ Word 𝑆)
 
27-May-2025wrdfsupp 32895 A word has finite support. (Contributed by Thierry Arnoux, 27-May-2025.)
(𝜑 → 𝑍 ∈ 𝑉)    &   (𝜑 → 𝑊 ∈ Word 𝑆)    ⇒   (𝜑 → 𝑊 finSupp 𝑍)
 
27-May-2025fzo0opth 32802 Equality for a half open integer range starting at zero is the same as equality of its upper bound, analogous to fzopth 13615 and fzoopth 13806. (Contributed by Thierry Arnoux, 27-May-2025.)
(𝜑 → 𝑀 ∈ ℕ0)    &   (𝜑 → 𝑁 ∈ ℕ0)    ⇒   (𝜑 → ((0..^𝑀) = (0..^𝑁) ↔ 𝑀 = 𝑁))
 
27-May-2025of0r 32688 Function operation with the empty function. (Contributed by Thierry Arnoux, 27-May-2025.)
(𝐹 ∘f 𝑅∅) = ∅
 
27-May-2025feq3dd 32635 Equality deduction for functions. (Contributed by Thierry Arnoux, 27-May-2025.)
(𝜑 → 𝐵 = 𝐶)    &   (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   (𝜑 → 𝐹:𝐴⟶𝐶)
 
27-May-2025feq2dd 32634 Equality deduction for functions. (Contributed by Thierry Arnoux, 27-May-2025.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → 𝐹:𝐴⟶𝐶)    ⇒   (𝜑 → 𝐹:𝐵⟶𝐶)
 
27-May-20251p1e2s 28410 One plus one is two. Surreal version. (Contributed by Scott Fenton, 27-May-2025.)
( 1s +s 1s ) = 2s
 
27-May-2025df-zs12 28409 Define the set of dyadic rationals. This is the set of rationals whose denominator is a power of two. Later we will prove that this is precisely the set of surreals with a finite birthday. (Contributed by Scott Fenton, 27-May-2025.)
ℤs[1/2] = {𝑥 ∣ ∃𝑦 ∈ ℤs ∃𝑧 ∈ ℕ0s 𝑥 = (𝑦 /su (2s↑s𝑧))}
 
27-May-2025df-exps 28407 Define surreal exponentiation. Compare df-exp 14107. (Contributed by Scott Fenton, 27-May-2025.)
↑s = (𝑥 ∈ No , 𝑦 ∈ ℤs ↦ if(𝑦 = 0s , 1s , if( 0s <s 𝑦, (seqs 1s ( ·s , (ℕs × {𝑥}))‘𝑦), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝑥}))‘( -us ‘𝑦))))))
 
27-May-2025df-2s 28405 Define surreal two. This is the simplest number greater than one. See 1p1e2s 28410 for its addition version. (Contributed by Scott Fenton, 27-May-2025.)
2s = ({ 1s } |s ∅)
 
26-May-2025zsbday 28402 A surreal integer has a finite birthday. (Contributed by Scott Fenton, 26-May-2025.)
(𝐴 ∈ ℤs → ( bday ‘𝐴) ∈ ω)
 
26-May-2025elzn0s 28394 A surreal integer is a surreal that is a non-negative integer or whose negative is a non-negative integer. (Contributed by Scott Fenton, 26-May-2025.)
(𝐴 ∈ ℤs ↔ (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))
 
26-May-2025znegscld 28389 The surreal integers are closed under negation. (Contributed by Scott Fenton, 26-May-2025.)
(𝜑 → 𝐴 ∈ ℤs)    ⇒   (𝜑 → ( -us ‘𝐴) ∈ ℤs)
 
26-May-2025znegscl 28388 The surreal integers are closed under negation. (Contributed by Scott Fenton, 26-May-2025.)
(𝐴 ∈ ℤs → ( -us ‘𝐴) ∈ ℤs)
 
26-May-2025n0zsd 28386 A non-negative surreal integer is a surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
(𝜑 → 𝐴 ∈ ℕ0s)    ⇒   (𝜑 → 𝐴 ∈ ℤs)
 
26-May-2025n0zs 28385 A non-negative surreal integer is a surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
(𝐴 ∈ ℕ0s → 𝐴 ∈ ℤs)
 
26-May-20250zs 28384 Zero is a surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
0s ∈ ℤs
 
26-May-2025nnzsd 28383 A positive surreal integer is a surreal integer. Deduction form. (Contributed by Scott Fenton, 26-May-2025.)
(𝜑 → 𝐴 ∈ ℕs)    ⇒   (𝜑 → 𝐴 ∈ ℤs)
 
26-May-2025n0p1nns 28371 One plus a non-negative surreal integer is a positive surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
(𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕs)
 
26-May-2025n0subs 28370 Subtraction of non-negative surreal integers. (Contributed by Scott Fenton, 26-May-2025.)
((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 ≤s 𝑁 ↔ (𝑁 -s 𝑀) ∈ ℕ0s))
 
26-May-2025n0s0m1 28369 Every non-negative surreal integer is either zero or a successor. (Contributed by Scott Fenton, 26-May-2025.)
(𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ (𝐴 -s 1s ) ∈ ℕ0s))
 
26-May-2025eln0s 28368 A non-negative surreal integer is zero or a positive surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
(𝐴 ∈ ℕ0s ↔ (𝐴 ∈ ℕs ∨ 𝐴 = 0s ))
 
26-May-2025nnn0sd 28343 A positive surreal integer is a non-negative surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
(𝜑 → 𝐴 ∈ ℕs)    ⇒   (𝜑 → 𝐴 ∈ ℕ0s)
 
26-May-2025nnn0s 28342 A positive surreal integer is a non-negative surreal integer. (Contributed by Scott Fenton, 26-May-2025.)
(𝐴 ∈ ℕs → 𝐴 ∈ ℕ0s)
 
26-May-2025subsge0d 28139 Non-negative subtraction. (Contributed by Scott Fenton, 26-May-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → ( 0s ≤s (𝐴 -s 𝐵) ↔ 𝐵 ≤s 𝐴))
 
26-May-2025slesubaddd 28133 Surreal less-than or equal relationship between subtraction and addition. (Contributed by Scott Fenton, 26-May-2025.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    &   (𝜑 → 𝐶 ∈ No )    ⇒   (𝜑 → ((𝐴 -s 𝐵) ≤s 𝐶 ↔ 𝐴 ≤s (𝐶 +s 𝐵)))
 
26-May-2025rabid2im 3477 One direction of rabid2 3478 is based on fewer axioms. (Contributed by Wolf Lammen, 26-May-2025.)
(∀𝑥 ∈ 𝐴 𝜑 → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝜑})
 
21-May-2025grlimf1o 47799 A local isomorphism of graphs is a bijection between their vertices. (Contributed by AV, 21-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    ⇒   (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → 𝐹:𝑉–1-1-onto→𝑊)
 
21-May-2025grlimprop 47798 Properties of a local isomorphism of graphs. (Contributed by AV, 21-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    ⇒   (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣)))))
 
21-May-20252mos 2652 Double "there exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.) (Proof shortened by Wolf Lammen, 21-May-2025.)
((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
 
20-May-2025isgrlim 47796 A local isomorphism of graphs is a bijection between their vertices that preserves neighborhoods. (Contributed by AV, 20-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    ⇒   ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 (𝐺 ISubGr (𝐺 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (𝐻 ISubGr (𝐻 ClNeighbVtx (𝐹‘𝑣))))))
 
20-May-2025grlimdmrel 47794 The domain of the graph local isomorphism function is a relation. (Contributed by AV, 20-May-2025.)
Rel dom GraphLocIso
 
20-May-2025grlimfn 47793 The graph local isomorphism function is a well-defined function. (Contributed by AV, 20-May-2025.)
GraphLocIso Fn (V × V)
 
20-May-2025rhmply1mon 22406 Apply a ring homomorphism between two univariate polynomial algebras to a scaled monomial, as in ply1coe 22315. (Contributed by SN, 20-May-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑄 = (Poly1‘𝑆)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))    &   𝑋 = (var1‘𝑅)    &   𝑌 = (var1‘𝑆)    &    · = ( ·𝑠 ‘𝑃)    &    ∙ = ( ·𝑠 ‘𝑄)    &   𝑀 = (mulGrp‘𝑃)    &   𝑁 = (mulGrp‘𝑄)    &    ↑ = (.g‘𝑀)    &    ∧ = (.g‘𝑁)    &   (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))    &   (𝜑 → 𝐶 ∈ 𝐾)    &   (𝜑 → 𝐸 ∈ ℕ0)    ⇒   (𝜑 → (𝐹‘(𝐶 · (𝐸 ↑ 𝑋))) = ((𝐻‘𝐶) ∙ (𝐸 ∧ 𝑌)))
 
20-May-2025rhmply1vsca 22405 Apply a ring homomorphism between two univariate polynomial algebras to a scaled polynomial. (Contributed by SN, 20-May-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑄 = (Poly1‘𝑆)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))    &    · = ( ·𝑠 ‘𝑃)    &    ∙ = ( ·𝑠 ‘𝑄)    &   (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))    &   (𝜑 → 𝐶 ∈ 𝐾)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝐹‘(𝐶 · 𝑋)) = ((𝐻‘𝐶) ∙ (𝐹‘𝑋)))
 
20-May-2025rhmply1vr1 22404 A ring homomorphism between two univariate polynomial algebras sends one variable to the other. (Contributed by SN, 20-May-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑄 = (Poly1‘𝑆)    &   𝐵 = (Base‘𝑃)    &   𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))    &   𝑋 = (var1‘𝑅)    &   𝑌 = (var1‘𝑆)    &   (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))    ⇒   (𝜑 → (𝐹‘𝑋) = 𝑌)
 
20-May-2025rhmply1 22403 Provide a ring homomorphism between two univariate polynomial algebras over their respective base rings given a ring homomorphism between the two base rings. (Contributed by SN, 20-May-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑄 = (Poly1‘𝑆)    &   𝐵 = (Base‘𝑃)    &   𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))    &   (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))    ⇒   (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑄))
 
20-May-2025mhmcoply1 22402 The composition of a monoid homomorphism and a polynomial is a polynomial. (Contributed by SN, 20-May-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑄 = (Poly1‘𝑆)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Base‘𝑄)    &   (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆))    &   (𝜑 → 𝐹 ∈ 𝐵)    ⇒   (𝜑 → (𝐻 ∘ 𝐹) ∈ 𝐶)
 
20-May-2025ply1vscl 22401 Closure of scalar multiplication for univariate polynomials. (Contributed by SN, 20-May-2025.)
𝑃 = (Poly1‘𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠 ‘𝑃)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐶 ∈ 𝐾)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝐶 · 𝑋) ∈ 𝐵)
 
20-May-2025ply1bas 22209 The value of the base set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.) Remove hypothesis. (Revised by SN, 20-May-2025.)
𝑃 = (Poly1‘𝑅)    &   𝑈 = (Base‘𝑃)    ⇒   𝑈 = (Base‘(1o mPoly 𝑅))
 
20-May-2025fabexd 7969 Existence of a set of functions. In contrast to fabex 7972 or fabexg 7970, the condition in the class abstraction does not contain the function explicitly, but the function can be derived from it. Therefore, this theorem is also applicable for more special functions like one-to-one, onto or one-to-one onto functions. (Contributed by AV, 20-May-2025.)
((𝜑 ∧ 𝜓) → 𝑓:𝑋⟶𝑌)    &   (𝜑 → 𝑋 ∈ 𝑉)    &   (𝜑 → 𝑌 ∈ 𝑊)    ⇒   (𝜑 → {𝑓 ∣ 𝜓} ∈ V)
 
20-May-2025coof 7731 The composition of a homomorphism with a function operation. (Contributed by SN, 20-May-2025.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   (𝜑 → 𝐺:𝐴⟶𝐵)    &   (𝜑 → 𝐻 Fn 𝐵)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑏𝑅𝑐) ∈ 𝐵)    &   ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝐻‘(𝑏𝑅𝑐)) = ((𝐻‘𝑏)𝑆(𝐻‘𝑐)))    ⇒   (𝜑 → (𝐻 ∘ (𝐹 ∘f 𝑅𝐺)) = ((𝐻 ∘ 𝐹) ∘f 𝑆(𝐻 ∘ 𝐺)))
 
19-May-2025evl1maprhm 22396 The function 𝐹 mapping polynomials 𝑝 to their evaluation at a given point 𝑋 is a ring homomorphism. (Contributed by metakunt, 19-May-2025.)
𝑂 = (eval1‘𝑅)    &   𝑃 = (Poly1‘𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   𝐹 = (𝑝 ∈ 𝑈 ↦ ((𝑂‘𝑝)‘𝑋))    ⇒   (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑅))
 
19-May-2025ssrexv 4078 Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.) Avoid axioms. (Revised by GG, 19-May-2025.)
(𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑))
 
19-May-2025ssralv 4077 Quantification restricted to a subclass. (Contributed by NM, 11-Mar-2006.) Avoid axioms. (Revised by GG, 19-May-2025.)
(𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 𝜑))
 
19-May-2025sstr2 4015 Transitivity of subclass relationship. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 24-Jun-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) Avoid axioms. (Revised by GG, 19-May-2025.)
(𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝐶 → 𝐴 ⊆ 𝐶))
 
18-May-2025rhmcomulpsr 42499 Show that the ring homomorphism in rhmpsr 42500 preserves multiplication. (Contributed by SN, 18-May-2025.)
𝑃 = (𝐼 mPwSer 𝑅)    &   𝑄 = (𝐼 mPwSer 𝑆)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Base‘𝑄)    &    · = (.r‘𝑃)    &    ∙ = (.r‘𝑄)    &   (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐺 ∈ 𝐵)    ⇒   (𝜑 → (𝐻 ∘ (𝐹 · 𝐺)) = ((𝐻 ∘ 𝐹) ∙ (𝐻 ∘ 𝐺)))
 
18-May-2025mhmcoaddpsr 42498 Show that the ring homomorphism in rhmpsr 42500 preserves addition. (Contributed by SN, 18-May-2025.)
𝑃 = (𝐼 mPwSer 𝑅)    &   𝑄 = (𝐼 mPwSer 𝑆)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Base‘𝑄)    &    + = (+g‘𝑃)    &    ✚ = (+g‘𝑄)    &   (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆))    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐺 ∈ 𝐵)    ⇒   (𝜑 → (𝐻 ∘ (𝐹 + 𝐺)) = ((𝐻 ∘ 𝐹) ✚ (𝐻 ∘ 𝐺)))
 
18-May-2025mhmcopsr 42497 The composition of a monoid homomorphism and a power series is a power series. (Contributed by SN, 18-May-2025.)
𝑃 = (𝐼 mPwSer 𝑅)    &   𝑄 = (𝐼 mPwSer 𝑆)    &   𝐵 = (Base‘𝑃)    &   𝐶 = (Base‘𝑄)    &   (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆))    &   (𝜑 → 𝐹 ∈ 𝐵)    ⇒   (𝜑 → (𝐻 ∘ 𝐹) ∈ 𝐶)
 
18-May-2025intnanrt 42193 Introduction of conjunct inside of a contradiction. Would be used in elfvov1 7485. (Contributed by SN, 18-May-2025.)
(¬ 𝜑 → ¬ (𝜑 ∧ 𝜓))
 
18-May-2025dfprm3 33538 The (positive) prime elements of the integer ring are the prime numbers. (Contributed by Thierry Arnoux, 18-May-2025.)
ℙ = (ℕ ∩ (RPrime‘ℤring))
 
18-May-2025zringpid 33537 The ring of integers is a principal ideal domain. (Contributed by Thierry Arnoux, 18-May-2025.)
ℤring ∈ PID
 
18-May-2025rprmdvdsprod 33519 If a prime element 𝑄 divides a product, then it divides one term. (Contributed by Thierry Arnoux, 18-May-2025.)
𝐵 = (Base‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &    ∥ = (∥r‘𝑅)    &    1 = (1r‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑄 ∈ 𝑃)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝐹 finSupp 1 )    &   (𝜑 → 𝐹:𝐼⟶𝐵)    &   (𝜑 → 𝑄 ∥ (𝑀 Σg 𝐹))    ⇒   (𝜑 → ∃𝑥 ∈ (𝐹 supp 1 )𝑄 ∥ (𝐹‘𝑥))
 
18-May-2025rprmdvdspow 33518 If a prime element divides a ring "power", it divides the term. (Contributed by Thierry Arnoux, 18-May-2025.)
𝐵 = (Base‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &    ∥ = (∥r‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &    ↑ = (.g‘𝑀)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑄 ∈ 𝑃)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑄 ∥ (𝑁 ↑ 𝑋))    ⇒   (𝜑 → 𝑄 ∥ 𝑋)
 
18-May-2025rprmirredb 33517 In a principal ideal domain, the converse of rprmirred 33516 holds, i.e. irreducible elements are prime. (Contributed by Thierry Arnoux, 18-May-2025.)
𝑃 = (RPrime‘𝑅)    &   𝐼 = (Irred‘𝑅)    &   (𝜑 → 𝑅 ∈ PID)    ⇒   (𝜑 → 𝐼 = 𝑃)
 
18-May-2025rprmirred 33516 In an integral domain, ring primes are irreducible. (Contributed by Thierry Arnoux, 18-May-2025.)
𝑃 = (RPrime‘𝑅)    &   𝐼 = (Irred‘𝑅)    &   (𝜑 → 𝑄 ∈ 𝑃)    &   (𝜑 → 𝑅 ∈ IDomn)    ⇒   (𝜑 → 𝑄 ∈ 𝐼)
 
18-May-2025rprmirredlem 33515 Lemma for rprmirred 33516. (Contributed by Thierry Arnoux, 18-May-2025.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &    ∥ = (∥r‘𝑅)    &   (𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → 𝑄 ≠ 0 )    &   (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑈))    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑄 = (𝑋 · 𝑌))    &   (𝜑 → 𝑄 ∥ 𝑋)    ⇒   (𝜑 → 𝑌 ∈ 𝑈)
 
18-May-2025rprmasso2 33511 In an integral domain, if a prime element divides another, they are associates. (Contributed by Thierry Arnoux, 18-May-2025.)
𝐵 = (Base‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &    ∥ = (∥r‘𝑅)    &   (𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑋 ∥ 𝑌)    &   (𝜑 → 𝑌 ∈ 𝑃)    ⇒   (𝜑 → 𝑌 ∥ 𝑋)
 
18-May-2025rprmasso 33510 In an integral domain, the associate of a prime is a prime. (Contributed by Thierry Arnoux, 18-May-2025.)
𝐵 = (Base‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &    ∥ = (∥r‘𝑅)    &   (𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → 𝑋 ∈ 𝑃)    &   (𝜑 → 𝑋 ∥ 𝑌)    &   (𝜑 → 𝑌 ∥ 𝑋)    ⇒   (𝜑 → 𝑌 ∈ 𝑃)
 
18-May-2025rprmndvdsr1 33509 A ring prime element does not divide the ring multiplicative identity. (Contributed by Thierry Arnoux, 18-May-2025.)
1 = (1r‘𝑅)    &    ∥ = (∥r‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑄 ∈ 𝑃)    ⇒   (𝜑 → ¬ 𝑄 ∥ 1 )
 
18-May-2025rsprprmprmidlb 33508 In an integral domain, an ideal generated by a single element is a prime iff that element is prime. (Contributed by Thierry Arnoux, 18-May-2025.)
0 = (0g‘𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑋 ≠ 0 )    ⇒   (𝜑 → (𝑋 ∈ 𝑃 ↔ (𝐾‘{𝑋}) ∈ (PrmIdeal‘𝑅)))
 
18-May-2025rsprprmprmidl 33507 In a commutative ring, ideals generated by prime elements are prime ideals. (Contributed by Thierry Arnoux, 18-May-2025.)
𝐾 = (RSpan‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑃 ∈ (RPrime‘𝑅))    ⇒   (𝜑 → (𝐾‘{𝑃}) ∈ (PrmIdeal‘𝑅))
 
18-May-2025rprmnunit 33506 A ring prime is not a unit. (Contributed by Thierry Arnoux, 18-May-2025.)
𝑃 = (RPrime‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   (𝜑 → 𝑅 ∈ 𝑉)    &   (𝜑 → 𝑄 ∈ 𝑃)    ⇒   (𝜑 → ¬ 𝑄 ∈ 𝑈)
 
18-May-2025rprmnz 33505 A ring prime is nonzero. (Contributed by Thierry Arnoux, 18-May-2025.)
𝑃 = (RPrime‘𝑅)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ 𝑉)    &   (𝜑 → 𝑄 ∈ 𝑃)    ⇒   (𝜑 → 𝑄 ≠ 0 )
 
18-May-2025rprmdvds 33504 If a ring prime 𝑄 divides a product 𝑋 · 𝑌, then it divides either 𝑋 or 𝑌. (Contributed by Thierry Arnoux, 18-May-2025.)
𝐵 = (Base‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &    ∥ = (∥r‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ 𝑉)    &   (𝜑 → 𝑄 ∈ 𝑃)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑄 ∥ (𝑋 · 𝑌))    ⇒   (𝜑 → (𝑄 ∥ 𝑋 ∨ 𝑄 ∥ 𝑌))
 
18-May-2025rprmcl 33503 A ring prime is an element of the base set. (Contributed by Thierry Arnoux, 18-May-2025.)
𝐵 = (Base‘𝑅)    &   𝑃 = (RPrime‘𝑅)    &   (𝜑 → 𝑅 ∈ 𝑉)    &   (𝜑 → 𝑋 ∈ 𝑃)    ⇒   (𝜑 → 𝑋 ∈ 𝐵)
 
18-May-2025ellpi 33358 Elementhood in a left principal ideal in terms of the "divides" relation. (Contributed by Thierry Arnoux, 18-May-2025.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    ∥ = (∥r‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑌 ∈ (𝐾‘{𝑋}) ↔ 𝑋 ∥ 𝑌))
 
18-May-2025subrfld 33248 A subring of a field is an integral domain. (Contributed by Thierry Arnoux, 18-May-2025.)
(𝜑 → 𝑅 ∈ Field)    &   (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    ⇒   (𝜑 → (𝑅 ↾s 𝑆) ∈ IDomn)
 
18-May-2025subridom 33247 A subring of an integral domain is an integral domain. (Contributed by Thierry Arnoux, 18-May-2025.)
(𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    ⇒   (𝜑 → (𝑅 ↾s 𝑆) ∈ IDomn)
 
18-May-2025subrdom 33246 A subring of a domain is a domain. (Contributed by Thierry Arnoux, 18-May-2025.)
(𝜑 → 𝑅 ∈ Domn)    &   (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    ⇒   (𝜑 → (𝑅 ↾s 𝑆) ∈ Domn)
 
18-May-20250ringcring 33216 The zero ring is commutative. (Contributed by Thierry Arnoux, 18-May-2025.)
𝐵 = (Base‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → (♯‘𝐵) = 1)    ⇒   (𝜑 → 𝑅 ∈ CRing)
 
18-May-2025irrednzr 33214 A ring with an irreducible element cannot be the zero ring. (Contributed by Thierry Arnoux, 18-May-2025.)
𝐼 = (Irred‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐼)    ⇒   (𝜑 → 𝑅 ∈ NzRing)
 
18-May-2025orim12da 32479 Deduce a disjunction from another one. Variation on orim12d 965. (Contributed by Thierry Arnoux, 18-May-2025.)
((𝜑 ∧ 𝜓) → 𝜃)    &   ((𝜑 ∧ 𝜒) → 𝜏)    &   (𝜑 → (𝜓 ∨ 𝜒))    ⇒   (𝜑 → (𝜃 ∨ 𝜏))
 
18-May-2025mhpvscacl 22173 Homogeneous polynomials are closed under scalar multiplication. (Contributed by SN, 25-Sep-2023.) Remove sethood hypothesis. (Revised by SN, 18-May-2025.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &    · = ( ·𝑠 ‘𝑃)    &   𝐾 = (Base‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ 𝐾)    &   (𝜑 → 𝐹 ∈ (𝐻‘𝑁))    ⇒   (𝜑 → (𝑋 · 𝐹) ∈ (𝐻‘𝑁))
 
18-May-2025mhpinvcl 22171 Homogeneous polynomials are closed under taking the opposite. (Contributed by SN, 12-Sep-2023.) Remove sethood hypothesis. (Revised by SN, 18-May-2025.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝑀 = (invg‘𝑃)    &   (𝜑 → 𝑅 ∈ Grp)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    ⇒   (𝜑 → (𝑀‘𝑋) ∈ (𝐻‘𝑁))
 
18-May-2025mhpaddcl 22170 Homogeneous polynomials are closed under addition. (Contributed by SN, 26-Aug-2023.) Remove sethood hypothesis. (Revised by SN, 18-May-2025.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &    + = (+g‘𝑃)    &   (𝜑 → 𝑅 ∈ Grp)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    &   (𝜑 → 𝑌 ∈ (𝐻‘𝑁))    ⇒   (𝜑 → (𝑋 + 𝑌) ∈ (𝐻‘𝑁))
 
18-May-2025mhppwdeg 22169 Degree of a homogeneous polynomial raised to a power. General version of deg1pw 26172. (Contributed by SN, 26-Jul-2024.) Remove sethood hypothesis. (Revised by SN, 18-May-2025.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝑇 = (mulGrp‘𝑃)    &    ↑ = (.g‘𝑇)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑀 ∈ ℕ0)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ (𝐻‘𝑀))    ⇒   (𝜑 → (𝑁 ↑ 𝑋) ∈ (𝐻‘(𝑀 · 𝑁)))
 
18-May-2025mhpmulcl 22168 A product of homogeneous polynomials is a homogeneous polynomial whose degree is the sum of the degrees of the factors. Compare mdegmulle2 26130 (which shows less-than-or-equal instead of equal). (Contributed by SN, 22-Jul-2024.) Remove sethood hypothesis. (Revised by SN, 18-May-2025.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑌 = (𝐼 mPoly 𝑅)    &    · = (.r‘𝑌)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑀 ∈ ℕ0)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑃 ∈ (𝐻‘𝑀))    &   (𝜑 → 𝑄 ∈ (𝐻‘𝑁))    ⇒   (𝜑 → (𝑃 · 𝑄) ∈ (𝐻‘(𝑀 + 𝑁)))
 
18-May-2025reldmmhp 22157 The domain of the homogeneous polynomial operator is a relation. (Contributed by SN, 18-May-2025.)
Rel dom mHomP
 
18-May-2025psrascl 22015 Value of the scalar injection into the power series algebra. (Contributed by SN, 18-May-2025.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &    0 = (0g‘𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑆)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐾)    ⇒   (𝜑 → (𝐴‘𝑋) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝐼 × {0}), 𝑋, 0 )))
 
18-May-2025elfvov1 7485 Utility theorem: reverse closure for any operation that results in a function. (Contributed by SN, 18-May-2025.)
Rel dom 𝑂    &   𝑆 = (𝐼𝑂𝑅)    &   (𝜑 → 𝑋 ∈ (𝑆‘𝑌))    ⇒   (𝜑 → 𝐼 ∈ V)
 
17-May-2025dfsclnbgr6 47720 Alternate definition of a semiclosed neighborhood of a vertex as a union of a semiopen neighborhood and the vertex itself if there is a loop at this vertex. (Contributed by AV, 17-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    &   𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    ⇒   (𝑁 ∈ 𝑉 → 𝑆 = (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}))
 
17-May-2025dfnbgr6 47719 Alternate definition of the (open) neighborhood of a vertex as a difference of its semiopen neighborhood and the singleton of itself. (Contributed by AV, 17-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    ⇒   (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑈 ∖ {𝑁}))
 
17-May-2025dfclnbgr6 47718 Alternate definition of the closed neighborhood of a vertex as union of the vertex with its semiopen neighborhood. (Contributed by AV, 17-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    ⇒   (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ 𝑈))
 
17-May-2025nnzs 28382 A positive surreal integer is a surreal integer. (Contributed by Scott Fenton, 17-May-2025.)
(𝐴 ∈ ℕs → 𝐴 ∈ ℤs)
 
17-May-2025elzs 28380 Membership in the set of surreal integers. (Contributed by Scott Fenton, 17-May-2025.)
(𝐴 ∈ ℤs ↔ ∃𝑥 ∈ ℕs ∃𝑦 ∈ ℕs 𝐴 = (𝑥 -s 𝑦))
 
17-May-2025znod 28379 A surreal integer is a surreal. Deduction form. (Contributed by Scott Fenton, 17-May-2025.)
(𝜑 → 𝐴 ∈ ℤs)    ⇒   (𝜑 → 𝐴 ∈ No )
 
17-May-2025zno 28378 A surreal integer is a surreal. (Contributed by Scott Fenton, 17-May-2025.)
(𝐴 ∈ ℤs → 𝐴 ∈ No )
 
17-May-2025zssno 28377 The surreal integers are a subset of the surreals. (Contributed by Scott Fenton, 17-May-2025.)
ℤs ⊆ No
 
17-May-2025zsex 28376 The surreal integers form a set. (Contributed by Scott Fenton, 17-May-2025.)
ℤs ∈ V
 
17-May-2025df-zs 28375 Define the surreal integers. Compare dfz2 12652. (Contributed by Scott Fenton, 17-May-2025.)
ℤs = ( -s “ (ℕs × ℕs))
 
17-May-2025subsfo 28105 Surreal subtraction is an onto function. (Contributed by Scott Fenton, 17-May-2025.)
-s :( No × No )–onto→ No
 
17-May-2025subsf 28104 Function statement for surreal subtraction. (Contributed by Scott Fenton, 17-May-2025.)
-s :( No × No )⟶ No
 
17-May-2025rabsneq 4666 Equality of class abstractions restricted to a singleton. (Contributed by AV, 17-May-2025.)
(𝑁 ∈ 𝑉 → {𝑥 ∈ {𝑁} ∣ 𝜓} = {𝑥 ∈ 𝑉 ∣ (𝑥 = 𝑁 ∧ 𝜓)})
 
16-May-2025dfnbgrss2 47721 Subset chain for different kinds of neighborhoods of a vertex. (Contributed by AV, 16-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    &   𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    ⇒   (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑈 ∧ 𝑈 ⊆ 𝑆 ∧ 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁)))
 
16-May-2025vopnbgrelself 47717 A vertex 𝑁 is a member of its semiopen neighborhood iff there is a loop joining the vertex with itself. (Contributed by AV, 16-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    ⇒   (𝑁 ∈ 𝑉 → (𝑁 ∈ 𝑈 ↔ ∃𝑒 ∈ 𝐸 𝑒 = {𝑁}))
 
16-May-2025vopnbgrel 47716 Characterization of a member 𝑋 of the semiopen neighborhood of a vertex 𝑁 in a graph 𝐺. (Contributed by AV, 16-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    ⇒   (𝑁 ∈ 𝑉 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒) ∨ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋})))))
 
16-May-2025dfnbgrss 47714 Subset chain for different kinds of neighborhoods of a vertex. (Contributed by AV, 16-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   𝐸 = (Edg‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑆 ∧ 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁)))
 
16-May-2025dfnbgr5 47713 Alternate definition of the (open) neighborhood of a vertex as a semiclosed neighborhood without itself. (Contributed by AV, 16-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   𝐸 = (Edg‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑆 ∖ {𝑁}))
 
16-May-2025dfclnbgr5 47712 Alternate definition of the closed neighborhood of a vertex as union of the vertex with its semiclosed neighborhood. (Contributed by AV, 16-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   𝐸 = (Edg‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ 𝑆))
 
16-May-2025sclnbgrisvtx 47711 Every member 𝑋 of the semiclosed neighborhood of a vertex 𝑁 is a vertex. (Contributed by AV, 16-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   𝐸 = (Edg‘𝐺)    ⇒   (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝑉)
 
16-May-2025sclnbgrelself 47710 A vertex 𝑁 is a member of its semiclosed neighborhood iff there is an edge joining the vertex with a vertex. (Contributed by AV, 16-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   𝐸 = (Edg‘𝐺)    ⇒   (𝑁 ∈ 𝑆 ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 𝑁 ∈ 𝑒))
 
16-May-2025sclnbgrel 47709 Characterization of a member 𝑋 of the semiclosed neighborhood of a vertex 𝑁 in a graph 𝐺. (Contributed by AV, 16-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   𝐸 = (Edg‘𝐺)    ⇒   (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒))
 
16-May-2025dfsclnbgr2 47708 Alternate definition of the semiclosed neighborhood of a vertex breaking up the subset relationship of an unordered pair. A semiclosed neighborhood 𝑆 of a vertex 𝑁 is the set of all vertices incident with edges which join the vertex 𝑁 with a vertex. Therefore, a vertex is contained in its semiclosed neighborhood if it is connected with any vertex by an edge (see sclnbgrelself 47710), even only with itself (i.e., by a loop). (Contributed by AV, 16-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   𝐸 = (Edg‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
 
16-May-2025clnbgrn0 47695 The closed neighborhood of a vertex is never empty. (Contributed by AV, 16-May-2025.)
𝑉 = (Vtx‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) ≠ ∅)
 
16-May-2025aks6d1c7 42134 𝑁 is a prime power if the hypotheses of the AKS algorithm hold. Claim 7 of Theorem 6.1 https://www3.nd.edu/%7eandyp/notes/AKS.pdf. (Contributed by metakunt, 16-May-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    &   (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    ⇒   (𝜑 → 𝑁 = (𝑃↑(𝑃 pCnt 𝑁)))
 
16-May-2025aks6d1c7lem4 42133 In the AKS algorithm there exists a unique prime number 𝑝 that divides 𝑁. (Contributed by metakunt, 16-May-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    &   (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    ⇒   (𝜑 → ∃!𝑝 ∈ ℙ 𝑝 ∥ 𝑁)
 
16-May-2025aks6d1c7lem3 42132 Remove lots of hypotheses now that we have the AKS contradiction. (Contributed by metakunt, 16-May-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    &   (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    &   (𝜑 → (𝑄 ∈ ℙ ∧ 𝑄 ∥ 𝑁))    ⇒   (𝜑 → 𝑃 = 𝑄)
 
16-May-2025aks6d1c7lem2 42131 Contradiction to Claim 2 and Claim 7. We assumed in Claim 2 that there are two different prime numbers 𝑃 and 𝑄. (Contributed by metakunt, 16-May-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))    &   𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))    &   𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    &   (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))    &   𝐵 = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))    &   𝐶 = (𝐸 “ ((0...𝐵) × (0...𝐵)))    &   (𝜑 → (𝑄 ∈ ℙ ∧ 𝑄 ∥ 𝑁))    &   (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    &   𝑆 = {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}    ⇒   (𝜑 → 𝑃 = 𝑄)
 
16-May-2025imaco 6277 Image of the composition of two classes. (Contributed by Jason Orendorff, 12-Dec-2006.) (Proof shortened by Wolf Lammen, 16-May-2025.)
((𝐴 ∘ 𝐵) “ 𝐶) = (𝐴 “ (𝐵 “ 𝐶))
 
16-May-2025difxp 6190 Difference of Cartesian products, expressed in terms of a union of Cartesian products of differences. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 26-Jun-2014.) (Proof shortened by Wolf Lammen, 16-May-2025.)
((𝐶 × 𝐷) ∖ (𝐴 × 𝐵)) = (((𝐶 ∖ 𝐴) × 𝐷) ∪ (𝐶 × (𝐷 ∖ 𝐵)))
 
15-May-2025isubgrusgr 47732 An induced subgraph of a simple graph is a simple graph. (Contributed by AV, 15-May-2025.)
𝑉 = (Vtx‘𝐺)    ⇒   ((𝐺 ∈ USGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ USGraph)
 
15-May-2025isubgrumgr 47731 An induced subgraph of a multigraph is a multigraph. (Contributed by AV, 15-May-2025.)
𝑉 = (Vtx‘𝐺)    ⇒   ((𝐺 ∈ UMGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UMGraph)
 
15-May-2025dfvopnbgr2 47715 Alternate definition of the semiopen neighborhood of a vertex breaking up the subset relationship of an unordered pair. A semiopen neighborhood 𝑈 of a vertex 𝑁 is its open neighborhood together with itself if there is a loop at this vertex. (Contributed by AV, 15-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    ⇒   (𝑁 ∈ 𝑉 → 𝑈 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))})
 
15-May-2025aks6d1c6lem5 42127 Eliminate the size hypothesis. Claim 6. (Contributed by metakunt, 15-May-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))    &   𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    &   (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))    &   𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))    &   𝑆 = {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}    &   𝐽 = (𝑗 ∈ ℤ ↦ (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))    &   𝑈 = {𝑚 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑛 ∈ (Base‘(mulGrp‘𝐾))(𝑛(+g‘(mulGrp‘𝐾))𝑚) = (0g‘(mulGrp‘𝐾))}    &   𝑋 = (𝑏 ∈ (Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))) ↦ ∪ (𝐽 “ 𝑏))    ⇒   (𝜑 → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))))
 
15-May-2025aks6d1c6isolem3 42126 The preimage of a map sending a primitive root to its powers of zero is equal to the set of integers that divide 𝑅. (Contributed by metakunt, 15-May-2025.)
(𝜑 → 𝑅 ∈ CMnd)    &   (𝜑 → 𝐾 ∈ ℕ)    &   𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)}    &   𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀))    &   (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))    &   𝑆 = (RSpan‘ℤring)    ⇒   (𝜑 → (𝑆‘{𝐾}) = (◡𝐹 “ {(0g‘(𝑅 ↾s 𝑈))}))
 
15-May-2025primrootspoweq0 42056 The power of a 𝑅-th primitive root is zero if and only if it divides 𝑅. (Contributed by metakunt, 15-May-2025.)
(𝜑 → 𝑅 ∈ CMnd)    &   (𝜑 → 𝐾 ∈ ℕ)    &   (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))    &   𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)}    &   (𝜑 → 𝑁 ∈ ℤ)    ⇒   (𝜑 → ((𝑁(.g‘(𝑅 ↾s 𝑈))𝑀) = (0g‘(𝑅 ↾s 𝑈)) ↔ 𝐾 ∥ 𝑁))
 
15-May-2025primrootlekpowne0 42055 There is no smaller power of a primitive root that sends it to the neutral element. (Contributed by metakunt, 15-May-2025.)
(𝜑 → 𝑅 ∈ CMnd)    &   (𝜑 → 𝐾 ∈ ℕ)    &   (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))    &   (𝜑 → 𝑁 ∈ (1...(𝐾 − 1)))    ⇒   (𝜑 → (𝑁(.g‘𝑅)𝑀) ≠ (0g‘𝑅))
 
15-May-2025remexz 42054 Division with rest. (Contributed by metakunt, 15-May-2025.)
(𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐴 ∈ ℕ)    ⇒   (𝜑 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (0...(𝐴 − 1))𝑁 = ((𝑥 · 𝐴) + 𝑦))
 
15-May-2025dmresss 6035 The domain of a restriction is a subset of the original domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.) Proof shortened and axiom usage reduced. (Proof shortened by AV, 15-May-2025.)
dom (𝐴 ↾ 𝐵) ⊆ dom 𝐴
 
15-May-2025inuni 5368 The intersection of a union ∪ 𝐴 with a class 𝐵 is equal to the union of the intersections of each element of 𝐴 with 𝐵. (Contributed by FL, 24-Mar-2007.) (Proof shortened by Wolf Lammen, 15-May-2025.)
(∪ 𝐴 ∩ 𝐵) = ∪ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝑥 = (𝑦 ∩ 𝐵)}
 
15-May-2025rexdifpr 4681 Restricted existential quantification over a set with two elements removed. (Contributed by Alexander van der Vekens, 7-Feb-2018.) (Proof shortened by Wolf Lammen, 15-May-2025.)
(∃𝑥 ∈ (𝐴 ∖ {𝐵, 𝐶})𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝑥 ≠ 𝐵 ∧ 𝑥 ≠ 𝐶 ∧ 𝜑))
 
15-May-2025reuun2 4344 Transfer uniqueness to a smaller or larger class. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Wolf Lammen, 15-May-2025.)
(¬ ∃𝑥 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ ∃!𝑥 ∈ 𝐴 𝜑))
 
15-May-2025dfss2 3994 Alternate definition of the subclass relationship between two classes. Exercise 9 of [TakeutiZaring] p. 18. This was the original definition before df-ss 3993. (Contributed by NM, 27-Apr-1994.) Revise df-ss 3993. (Revised by GG, 15-May-2025.)
(𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
 
15-May-2025df-ss 3993 Define the subclass relationship. Definition 5.9 of [TakeutiZaring] p. 17. For example, {1, 2} ⊆ {1, 2, 3} (ex-ss 30451). Note that 𝐴 ⊆ 𝐴 (proved in ssid 4031). Contrast this relationship with the relationship 𝐴 ⊊ 𝐵 (as will be defined in df-pss 3996). For an alternative definition, not requiring a dummy variable, see dfss2 3994. Other possible definitions are given by dfss3 3997, dfss4 4288, sspss 4125, ssequn1 4209, ssequn2 4212, sseqin2 4244, and ssdif0 4389.

We prefer the label "ss" ("subset") for ⊆, despite the fact that it applies to classes. It is much more common to refer to this as the subset relation than subclass, especially since most of the time the arguments are in fact sets (and for pragmatic reasons we don't want to need to use different operations for sets). The way set.mm is set up, many things are technically classes despite morally (and provably) being sets, like 1 (cf. df-1 11186 and 1ex 11280) or ℝ ( cf. df-r 11188 and reex 11269). This has to do with the fact that there are no "set expressions": classes are expressions but there are only set variables in set.mm (cf. https://us.metamath.org/downloads/grammar-ambiguity.txt 11269). This is why we use ⊆ both for subclass relations and for subset relations and call it "subset". (Contributed by NM, 8-Jan-2002.) Revised from the original definition dfss2 3994. (Revised by GG, 15-May-2025.)

(𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
 
14-May-2025isubgrupgr 47730 An induced subgraph of a pseudograph is a pseudograph. (Contributed by AV, 14-May-2025.)
𝑉 = (Vtx‘𝐺)    ⇒   ((𝐺 ∈ UPGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UPGraph)
 
14-May-2025isubgrsubgr 47729 An induced subgraph of a hypergraph is a subgraph of the hypergraph. (Contributed by AV, 14-May-2025.)
𝑉 = (Vtx‘𝐺)    ⇒   ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) SubGraph 𝐺)
 
14-May-2025aks6d1c6isolem2 42125 Lemma to construct the group homomorphism for the AKS Theorem. (Contributed by metakunt, 14-May-2025.)
(𝜑 → 𝑅 ∈ CMnd)    &   (𝜑 → 𝐾 ∈ ℕ)    &   𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)}    &   𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀))    &   (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))    ⇒   (𝜑 → 𝐹 ∈ (ℤring GrpHom ((𝑅 ↾s 𝑈) ↾s ran 𝐹)))
 
14-May-2025aks6d1c6isolem1 42124 Lemma to construct the map out of the quotient for AKS. (Contributed by metakunt, 14-May-2025.)
(𝜑 → 𝑅 ∈ CMnd)    &   (𝜑 → 𝐾 ∈ ℕ)    &   𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)}    &   𝐹 = (𝑥 ∈ ℤ ↦ (𝑥(.g‘(𝑅 ↾s 𝑈))𝑀))    &   (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))    ⇒   (𝜑 → ((𝑅 ↾s 𝑈) ↾s ran 𝐹) ∈ Grp)
 
14-May-2025aks6d1c6lem4 42123 Claim 6 of Theorem 6.1 of https://www3.nd.edu/%7eandyp/notes/AKS.pdf Add hypothesis on coprimality, lift function to the integers so that group operations may be applied. Inline definition. (Contributed by metakunt, 14-May-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)    &   𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))    &   𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    &   (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))    &   𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))    &   𝑆 = {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}    &   𝐽 = (𝑗 ∈ ℤ ↦ (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))    &   (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))))    &   𝑈 = {𝑚 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑛 ∈ (Base‘(mulGrp‘𝐾))(𝑛(+g‘(mulGrp‘𝐾))𝑚) = (0g‘(mulGrp‘𝐾))}    ⇒   (𝜑 → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))))
 
14-May-2025idomcringd 20743 An integral domain is a commutative ring with unity. (Contributed by Thierry Arnoux, 4-May-2025.) Formerly subproof of idomringd 20744. (Proof shortened by SN, 14-May-2025.)
(𝜑 → 𝑅 ∈ IDomn)    ⇒   (𝜑 → 𝑅 ∈ CRing)
 
14-May-2025ressmulgnnd 19112 Values for the group multiple function in a restricted structure, a deduction version. (Contributed by metakunt, 14-May-2025.)
𝐻 = (𝐺 ↾s 𝐴)    &   (𝜑 → 𝐴 ⊆ (Base‘𝐺))    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑁 ∈ ℕ)    ⇒   (𝜑 → (𝑁(.g‘𝐻)𝑋) = (𝑁(.g‘𝐺)𝑋))
 
14-May-2025nfiu1 5050 Bound-variable hypothesis builder for indexed union. (Contributed by NM, 12-Oct-2003.) Avoid ax-11 2158, ax-12 2178. (Revised by SN, 14-May-2025.)
Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
 
14-May-2025nfin 4245 Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.) Avoid ax-10 2141, ax-11 2158, ax-12 2178. (Revised by SN, 14-May-2025.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐵    ⇒   Ⅎ𝑥(𝐴 ∩ 𝐵)
 
14-May-2025nfun 4193 Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.) Avoid ax-10 2141, ax-11 2158, ax-12 2178. (Revised by SN, 14-May-2025.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐵    ⇒   Ⅎ𝑥(𝐴 ∪ 𝐵)
 
14-May-2025nfdif 4152 Bound-variable hypothesis builder for class difference. (Contributed by NM, 3-Dec-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) Avoid ax-10 2141, ax-11 2158, ax-12 2178. (Revised by SN, 14-May-2025.)
Ⅎ𝑥𝐴    &   Ⅎ𝑥𝐵    ⇒   Ⅎ𝑥(𝐴 ∖ 𝐵)
 
14-May-2025sbciegft 3842 Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3844.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) (Proof shortened by SN, 14-May-2025.)
((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
 
14-May-2025nfaba1 2916 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2380. See nfaba1g 2918 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) Avoid ax-6 1967, ax-7 2007, ax-12 2178. (Revised by SN, 14-May-2025.)
Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑}
 
14-May-2025hbsbw 2172 If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. Version of hbsb 2532 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 12-Aug-1993.) (Revised by GG, 23-May-2024.) (Proof shortened by Wolf Lammen, 14-May-2025.)
(𝜑 → ∀𝑧𝜑)    ⇒   ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
 
13-May-2025isubgr0uhgr 47733 The subgraph induced by an empty set of vertices of a hypergraph. (Contributed by AV, 13-May-2025.)
(𝐺 ∈ UHGraph → (𝐺 ISubGr ∅) = ⟨∅, ∅⟩)
 
13-May-2025isubgruhgr 47728 An induced subgraph of a hypergraph is a hypergraph. (Contributed by AV, 13-May-2025.)
𝑉 = (Vtx‘𝐺)    ⇒   ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UHGraph)
 
13-May-2025rhmqusnsg 21312 The mapping 𝐽 induced by a ring homomorphism 𝐹 from a subring 𝑁 of the quotient group 𝑄 over 𝐹's kernel 𝐾 is a ring homomorphism. (Contributed by Thierry Arnoux, 13-May-2025.)
0 = (0g‘𝐻)    &   (𝜑 → 𝐹 ∈ (𝐺 RingHom 𝐻))    &   𝐾 = (◡𝐹 “ { 0 })    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   (𝜑 → 𝐺 ∈ CRing)    &   (𝜑 → 𝑁 ⊆ 𝐾)    &   (𝜑 → 𝑁 ∈ (LIdeal‘𝐺))    ⇒   (𝜑 → 𝐽 ∈ (𝑄 RingHom 𝐻))
 
13-May-2025ghmqusnsg 19316 The mapping 𝐻 induced by a surjective group homomorphism 𝐹 from the quotient group 𝑄 over a normal subgroup 𝑁 of 𝐹's kernel 𝐾 is a group isomorphism. In this case, one says that 𝐹 factors through 𝑄, which is also called the factor group. (Contributed by Thierry Arnoux, 13-May-2025.)
0 = (0g‘𝐻)    &   (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   𝐾 = (◡𝐹 “ { 0 })    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   (𝜑 → 𝑁 ⊆ 𝐾)    &   (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))
 
13-May-2025ghmqusnsglem2 19315 Lemma for ghmqusnsg 19316. (Contributed by Thierry Arnoux, 13-May-2025.)
0 = (0g‘𝐻)    &   (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   𝐾 = (◡𝐹 “ { 0 })    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   (𝜑 → 𝑁 ⊆ 𝐾)    &   (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    &   (𝜑 → 𝑌 ∈ (Base‘𝑄))    ⇒   (𝜑 → ∃𝑥 ∈ 𝑌 (𝐽‘𝑌) = (𝐹‘𝑥))
 
13-May-2025ghmqusnsglem1 19314 Lemma for ghmqusnsg 19316. (Contributed by Thierry Arnoux, 13-May-2025.)
0 = (0g‘𝐻)    &   (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   𝐾 = (◡𝐹 “ { 0 })    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   (𝜑 → 𝑁 ⊆ 𝐾)    &   (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    &   (𝜑 → 𝑋 ∈ (Base‘𝐺))    ⇒   (𝜑 → (𝐽‘[𝑋](𝐺 ~QG 𝑁)) = (𝐹‘𝑋))
 
13-May-2025fssrescdmd 7155 Restriction of a function to a subclass of its domain as a function with domain and codomain. (Contributed by AV, 13-May-2025.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   (𝜑 → 𝐶 ⊆ 𝐴)    &   (𝜑 → (𝐹 “ 𝐶) ⊆ 𝐷)    ⇒   (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐷)
 
12-May-2025isubgrvtx 47727 The vertices of an induced subgraph. (Contributed by AV, 12-May-2025.)
𝑉 = (Vtx‘𝐺)    ⇒   ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) = 𝑆)
 
12-May-2025isubgrvtxuhgr 47726 The subgraph induced by the full set of vertices of a hypergraph. (Contributed by AV, 12-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    ⇒   (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = ⟨𝑉, 𝐸⟩)
 
12-May-2025isubgriedg 47725 The edges of an induced subgraph. (Contributed by AV, 12-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    ⇒   ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆}))
 
12-May-2025isisubgr 47724 The subgraph induced by a subset of vertices. (Contributed by AV, 12-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    ⇒   ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) = ⟨𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})⟩)
 
12-May-2025aks6d1c7lem1 42130 The last set of inequalities of Claim 7 of Theorem 6.1 https://www3.nd.edu/%7eandyp/notes/AKS.pdf. (Contributed by metakunt, 12-May-2025.)
(𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))    &   𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))    &   𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))    &   (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    ⇒   (𝜑 → (𝑁↑(⌊‘(√‘𝐷))) < ((𝐷 + 𝐴)C(𝐷 − 1)))
 
12-May-2025bcle2d 42129 Inequality for binomial coefficients. (Contributed by metakunt, 12-May-2025.)
(𝜑 → 𝐴 ∈ ℕ0)    &   (𝜑 → 𝐵 ∈ ℕ0)    &   (𝜑 → 𝐶 ∈ ℕ0)    &   (𝜑 → 𝐷 ∈ ℤ)    &   (𝜑 → 𝐴 ≤ 𝐵)    &   (𝜑 → 𝐷 ≤ 𝐶)    ⇒   (𝜑 → ((𝐴 + 𝐶)C(𝐴 + 𝐷)) ≤ ((𝐵 + 𝐶)C(𝐵 + 𝐷)))
 
12-May-2025bcled 42128 Inequality for binomial coefficients. (Contributed by metakunt, 12-May-2025.)
(𝜑 → 𝐴 ∈ ℕ0)    &   (𝜑 → 𝐵 ∈ ℕ0)    &   (𝜑 → 𝐶 ∈ ℤ)    &   (𝜑 → 𝐴 ≤ 𝐵)    ⇒   (𝜑 → (𝐴C𝐶) ≤ (𝐵C𝐶))
 
12-May-2025aks6d1c4 42074 Claim 4 of Theorem 6.1 of the AKS inequality lemma. https://www3.nd.edu/%7eandyp/notes/AKS.pdf (Contributed by metakunt, 12-May-2025.)
(𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))    ⇒   (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (ϕ‘𝑅))
 
10-May-2025clnbgrlevtx 47707 The size of the closed neighborhood of a vertex is at most the number of vertices of a graph. (Contributed by AV, 10-May-2025.)
𝑉 = (Vtx‘𝐺)    ⇒   (♯‘(𝐺 ClNeighbVtx 𝑈)) ≤ (♯‘𝑉)
 
10-May-2025clnbfiusgrfi 47706 The closed neighborhood of a vertex in a finite simple graph is a finite set. (Contributed by AV, 10-May-2025.)
((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx 𝑁) ∈ Fin)
 
10-May-2025clnbusgrfi 47705 The closed neighborhood of a vertex in a simple graph with a finite number of edges is a finite set. (Contributed by AV, 10-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   ((𝐺 ∈ USGraph ∧ 𝐸 ∈ Fin ∧ 𝑈 ∈ 𝑉) → (𝐺 ClNeighbVtx 𝑈) ∈ Fin)
 
10-May-2025edgusgrclnbfin 47704 The size of the closed neighborhood of a vertex in a simple graph is finite iff the number of edges having this vertex as endpoint is finite. (Contributed by AV, 10-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → ((𝐺 ClNeighbVtx 𝑈) ∈ Fin ↔ {𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒} ∈ Fin))
 
10-May-2025clnbgrsym 47700 In a graph, the closed neighborhood relation is symmetric: a vertex 𝑁 in a graph 𝐺 is a neighbor of a second vertex 𝐾 iff the second vertex 𝐾 is a neighbor of the first vertex 𝑁. (Contributed by AV, 10-May-2025.)
(𝑁 ∈ (𝐺 ClNeighbVtx 𝐾) ↔ 𝐾 ∈ (𝐺 ClNeighbVtx 𝑁))
 
10-May-2025clnbgr0edg 47699 In an empty graph (with no edges), all closed neighborhoods consists of a single vertex. (Contributed by AV, 10-May-2025.)
(((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx 𝐾) = {𝐾})
 
10-May-2025clnbupgrel 47697 A member of the closed neighborhood of a vertex in a pseudograph. (Contributed by AV, 10-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   ((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → (𝑁 ∈ (𝐺 ClNeighbVtx 𝐾) ↔ (𝑁 = 𝐾 ∨ {𝑁, 𝐾} ∈ 𝐸)))
 
10-May-2025clnbupgr 47696 The closed neighborhood of a vertex in a pseudograph. (Contributed by AV, 10-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸}))
 
10-May-2025clnbgrvtxel 47692 Every vertex 𝐾 is a member of its closed neighborhood. (Contributed by AV, 10-May-2025.)
𝑉 = (Vtx‘𝐺)    ⇒   (𝐾 ∈ 𝑉 → 𝐾 ∈ (𝐺 ClNeighbVtx 𝐾))
 
10-May-2025imadomfi 41952 An image of a function under a finite set is dominated by the set. (Contributed by SN, 10-May-2025.)
((𝐴 ∈ Fin ∧ Fun 𝐹) → (𝐹 “ 𝐴) ≼ 𝐴)
 
10-May-2025idomsubr 33268 Every integral domain is isomorphic with a subring of some field. (Proposed by Gerard Lang, 10-May-2025.) (Contributed by Thierry Arnoux, 10-May-2025.)
(𝜑 → 𝑅 ∈ IDomn)    ⇒   (𝜑 → ∃𝑓 ∈ Field ∃𝑠 ∈ (SubRing‘𝑓)𝑅 ≃𝑟 (𝑓 ↾s 𝑠))
 
10-May-2025fracf1 33266 The embedding of a commutative ring 𝑅 into its field of fractions. (Contributed by Thierry Arnoux, 10-May-2025.)
𝐵 = (Base‘𝑅)    &   𝐸 = (RLReg‘𝑅)    &    1 = (1r‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &    ∼ = (𝑅 ~RL 𝐸)    &   𝐹 = (𝑥 ∈ 𝐵 ↦ [⟨𝑥, 1 ⟩] ∼ )    ⇒   (𝜑 → (𝐹:𝐵–1-1→((𝐵 × 𝐸) / ∼ ) ∧ 𝐹 ∈ (𝑅 RingHom ( Frac ‘𝑅))))
 
10-May-2025fracbas 33264 The base of the field of fractions. (Contributed by Thierry Arnoux, 10-May-2025.)
𝐵 = (Base‘𝑅)    &   𝐸 = (RLReg‘𝑅)    &   𝐹 = ( Frac ‘𝑅)    &    ∼ = (𝑅 ~RL 𝐸)    ⇒   ((𝐵 × 𝐸) / ∼ ) = (Base‘𝐹)
 
10-May-2025rrgsubm 33245 The left regular elements of a ring form a submonoid of the multiplicative group. (Contributed by Thierry Arnoux, 10-May-2025.)
𝐸 = (RLReg‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → 𝐸 ∈ (SubMnd‘𝑀))
 
10-May-2025reldmrloc 33221 Ring localization is a proper operator, so it can be used with ovprc1 7482. (Contributed by Thierry Arnoux, 10-May-2025.)
Rel dom RLocal
 
10-May-2025nbgr0vtx 29382 In a null graph (with no vertices), all neighborhoods are empty. (Contributed by AV, 15-Nov-2020.) (Proof shortened by AV, 10-May-2025.)
((Vtx‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)
 
10-May-2025unfib 9369 A union is finite if and only if the operands are finite. (Contributed by AV, 10-May-2025.)
((𝐴 ∪ 𝐵) ∈ Fin ↔ (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin))
 
9-May-2025clnbgrssvtx 47694 The closed neighborhood of a vertex 𝐾 in a graph is a subset of all vertices of the graph. (Contributed by AV, 9-May-2025.)
𝑉 = (Vtx‘𝐺)    ⇒   (𝐺 ClNeighbVtx 𝐾) ⊆ 𝑉
 
9-May-2025clnbgrisvtx 47693 Every member 𝑁 of the closed neighborhood of a vertex 𝐾 is a vertex. (Contributed by AV, 9-May-2025.)
𝑉 = (Vtx‘𝐺)    ⇒   (𝑁 ∈ (𝐺 ClNeighbVtx 𝐾) → 𝑁 ∈ 𝑉)
 
9-May-2025clnbgrel 47691 Characterization of a member 𝑁 of the closed neighborhood of a vertex 𝑋 in a graph 𝐺. (Contributed by AV, 9-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
 
8-May-2025clnbgrnvtx0 47690 If a class 𝑋 is not a vertex of a graph 𝐺, then it has an empty closed neighborhood in 𝐺. (Contributed by AV, 8-May-2025.)
𝑉 = (Vtx‘𝐺)    ⇒   (𝑋 ∉ 𝑉 → (𝐺 ClNeighbVtx 𝑋) = ∅)
 
8-May-2025dfclnbgr3 47689 Alternate definition of the closed neighborhood of a vertex using the edge function instead of the edges themselves (see also clnbgrval 47686). (Contributed by AV, 8-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    ⇒   ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)}))
 
8-May-2025dfclnbgr4 47688 Alternate definition of the closed neighborhood of a vertex as union of the vertex with its open neighborhood. (Contributed by AV, 8-May-2025.)
𝑉 = (Vtx‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
 
8-May-2025aks6d1c6lem3 42122 Claim 6 of Theorem 6.1 of https://www3.nd.edu/%7eandyp/notes/AKS.pdf TODO, eliminate hypothesis. (Contributed by metakunt, 8-May-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   (𝜑 → 𝐴 < 𝑃)    &   𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))    &   (𝜑 → 𝐴 ∈ ℕ0)    &   𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    &   (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))    &   𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))    &   𝑆 = {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}    &   𝐽 = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))    &   (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(𝐽 “ (ℕ0 × ℕ0))))    ⇒   (𝜑 → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))))
 
8-May-2025aks6d1c6lem2 42121 Every primitive root is root of G(u)-G(v). (Contributed by metakunt, 8-May-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   (𝜑 → 𝐴 < 𝑃)    &   𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))    &   (𝜑 → 𝐴 ∈ ℕ0)    &   𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    &   (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))    &   𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))    &   𝑆 = {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}    &   (𝜑 → 𝑈 ∈ 𝑆)    &   (𝜑 → 𝑉 ∈ 𝑆)    &   (𝜑 → ((𝐻 ↾ 𝑆)‘𝑈) = ((𝐻 ↾ 𝑆)‘𝑉))    &   (𝜑 → 𝑈 ≠ 𝑉)    &   𝐽 = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))    &   (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(𝐽 “ (ℕ0 × ℕ0))))    ⇒   (𝜑 → 𝐷 ≤ (♯‘(◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)})))
 
7-May-2025dfclnbgr2 47687 Alternate definition of the closed neighborhood of a vertex breaking up the subset relationship of an unordered pair. (Contributed by AV, 7-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}))
 
7-May-2025clnbgrval 47686 The closed neighborhood of a vertex 𝑉 in a graph 𝐺. (Contributed by AV, 7-May-2025.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    ⇒   (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
 
7-May-2025clnbgrcl 47685 If a class 𝑋 has at least one element in its closed neighborhood, this class must be a vertex. (Contributed by AV, 7-May-2025.)
𝑉 = (Vtx‘𝐺)    ⇒   (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) → 𝑋 ∈ 𝑉)
 
7-May-2025clnbgrprc0 47684 The closed neighborhood is empty if the graph 𝐺 or the vertex 𝑁 are proper classes. (Contributed by AV, 7-May-2025.)
(¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 ClNeighbVtx 𝑁) = ∅)
 
7-May-2025df-clnbgr 47683 Define the closed neighborhood resp. the class of all neighbors of a vertex (in a graph) and the vertex itself, see definition in section I.1 of [Bollobas] p. 3. The closed neighborhood of a vertex are all vertices which are connected with this vertex by an edge and the vertex itself (in contrast to an open neighborhood, see df-nbgr 29360). Alternatively, a closed neighborhood of a vertex could have been defined as its open neighborhood enhanced by the vertex itself, see dfclnbgr4 47688. This definition is applicable even for arbitrary hypergraphs. (Contributed by AV, 7-May-2025.)
ClNeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}))
 
7-May-2025aks6d1c6lem1 42120 Lemma for claim 6, deduce exact degree of the polynomial. (Contributed by metakunt, 7-May-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   (𝜑 → 𝐴 < 𝑃)    &   𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))    &   (𝜑 → 𝐴 ∈ ℕ0)    &   𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    &   (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))    &   𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))    &   𝑆 = {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}    &   (𝜑 → 𝑈 ∈ (ℕ0 ↑m (0...𝐴)))    ⇒   (𝜑 → ((deg1‘𝐾)‘(𝐺‘𝑈)) = Σ𝑡 ∈ (0...𝐴)(𝑈‘𝑡))
 
7-May-2025sticksstones23 42119 Non-exhaustive sticks and stones. (Contributed by metakunt, 7-May-2025.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑆 ∈ Fin)    &   (𝜑 → 𝑆 ≠ ∅)    &   𝐴 = {𝑓 ∈ (ℕ0 ↑m 𝑆) ∣ Σ𝑖 ∈ 𝑆 (𝑓‘𝑖) ≤ 𝑁}    ⇒   (𝜑 → (♯‘𝐴) = ((𝑁 + (♯‘𝑆))C(♯‘𝑆)))
 
7-May-202519.28 2229 Theorem 19.28 of [Margaris] p. 90. See 19.28v 1990 for a version requiring fewer axioms. (Contributed by NM, 1-Aug-1993.) (Proof shortened by Wolf Lammen, 7-May-2025.)
Ⅎ𝑥𝜑    ⇒   (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
 
7-May-2025anbiim 640 Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.) (Proof shortened by Wolf Lammen, 7-May-2025.)
(𝜑 → (𝜒 → 𝜃))    &   (𝜓 → (𝜃 → 𝜒))    ⇒   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
 
6-May-2025deg1pow 42091 Exact degree of a power of a polynomial in an integral domain. (Contributed by metakunt, 6-May-2025.)
(𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → 𝐹 ∈ (Base‘(Poly1‘𝑅)))    &   (𝜑 → 𝐹 ≠ (0g‘(Poly1‘𝑅)))    &   (𝜑 → 𝐴 ∈ ℕ0)    &    ↑ = (.g‘(mulGrp‘(Poly1‘𝑅)))    &   𝐷 = (deg1‘𝑅)    ⇒   (𝜑 → (𝐷‘(𝐴 ↑ 𝐹)) = (𝐴 · (𝐷‘𝐹)))
 
6-May-2025deg1gprod 42090 Degree multiplication is a homomorphism. (Contributed by metakunt, 6-May-2025.)
(𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → 𝑁 ∈ Fin)    &   (𝜑 → ∀𝑥 ∈ 𝑁 (𝐶 ∈ (Base‘(Poly1‘𝑅)) ∧ 𝐶 ≠ (0g‘(Poly1‘𝑅))))    ⇒   (𝜑 → (((deg1‘𝑅)‘((mulGrp‘(Poly1‘𝑅)) Σg (𝑥 ∈ 𝑁 ↦ 𝐶))) = Σ𝑛 ∈ 𝑁 ((deg1‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg1‘𝑅)‘((mulGrp‘(Poly1‘𝑅)) Σg (𝑥 ∈ 𝑁 ↦ 𝐶)))))
 
6-May-20251rrg 33244 The multiplicative identity is a left-regular element. (Contributed by Thierry Arnoux, 6-May-2025.)
1 = (1r‘𝑅)    &   𝐸 = (RLReg‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    ⇒   (𝜑 → 1 ∈ 𝐸)
 
6-May-2025deg1mul 26166 Degree of multiplication of two nonzero polynomials in a domain. (Contributed by metakunt, 6-May-2025.)
𝐷 = (deg1‘𝑅)    &   𝑃 = (Poly1‘𝑅)    &   𝐵 = (Base‘𝑃)    &    · = (.r‘𝑃)    &    0 = (0g‘𝑃)    &   (𝜑 → 𝑅 ∈ Domn)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐹 ≠ 0 )    &   (𝜑 → 𝐺 ∈ 𝐵)    &   (𝜑 → 𝐺 ≠ 0 )    ⇒   (𝜑 → (𝐷‘(𝐹 · 𝐺)) = ((𝐷‘𝐹) + (𝐷‘𝐺)))
 
6-May-2025isdomn6 20730 A ring is a domain iff the regular elements are the nonzero elements. Compare isdomn2 20727, domnrrg 20729. (Contributed by Thierry Arnoux, 6-May-2025.)
𝐵 = (Base‘𝑅)    &   𝐸 = (RLReg‘𝑅)    &    0 = (0g‘𝑅)    ⇒   (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) = 𝐸))
 
6-May-2025rrgnz 20720 In a nonzero ring, the zero is a left zero divisor (that is, not a left-regular element). (Contributed by Thierry Arnoux, 6-May-2025.)
𝐸 = (RLReg‘𝑅)    &    0 = (0g‘𝑅)    ⇒   (𝑅 ∈ NzRing → ¬ 0 ∈ 𝐸)
 
5-May-2025gricrel 47762 The "is isomorphic to" relation for graphs is a relation. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 5-May-2025.)
Rel ≃𝑔𝑟
 
5-May-2025gricuspgr 47761 The "is isomorphic to" relation for two simple pseudographs. This corresponds to the definition in [Bollobas] p. 3. (Contributed by AV, 1-Dec-2022.) (Proof shortened by AV, 5-May-2025.)
𝑉 = (Vtx‘𝐴)    &   𝑊 = (Vtx‘𝐵)    &   𝐸 = (Edg‘𝐴)    &   𝐾 = (Edg‘𝐵)    ⇒   ((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) → (𝐴 ≃𝑔𝑟 𝐵 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾))))
 
5-May-2025dfgric2 47758 Alternate, explicit definition of the "is isomorphic to" relation for two graphs. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 5-May-2025.)
𝑉 = (Vtx‘𝐴)    &   𝑊 = (Vtx‘𝐵)    &   𝐼 = (iEdg‘𝐴)    &   𝐽 = (iEdg‘𝐵)    ⇒   ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴 ≃𝑔𝑟 𝐵 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
 
5-May-2025aks6d1c5 42089 Claim 5 of Theorem 6.1 https://www3.nd.edu/%7eandyp/notes/AKS.pdf. The mapping defined by 𝐺 is injective. (Contributed by metakunt, 5-May-2025.)
(𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐴 ∈ ℕ0)    &   (𝜑 → 𝐴 < 𝑃)    &   𝑋 = (var1‘𝐾)    &    ↑ = (.g‘(mulGrp‘(Poly1‘𝐾)))    &   𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))    ⇒   (𝜑 → 𝐺:(ℕ0 ↑m (0...𝐴))–1-1→(Base‘(Poly1‘𝐾)))
 
5-May-2025aks6d1c5lem2 42088 Lemma for Claim 5, contradiction of different evaluations that map to the same. (Contributed by metakunt, 5-May-2025.)
(𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐴 ∈ ℕ0)    &   (𝜑 → 𝐴 < 𝑃)    &   𝑋 = (var1‘𝐾)    &    ↑ = (.g‘(mulGrp‘(Poly1‘𝐾)))    &   𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))    &   (𝜑 → 𝑌 ∈ (ℕ0 ↑m (0...𝐴)))    &   (𝜑 → 𝑍 ∈ (ℕ0 ↑m (0...𝐴)))    &   (𝜑 → (𝐺‘𝑌) = (𝐺‘𝑍))    &   (𝜑 → 𝑊 ∈ (0...𝐴))    &   (𝜑 → (𝑌‘𝑊) < (𝑍‘𝑊))    ⇒   (𝜑 → (0g‘𝐾) ≠ (0g‘𝐾))
 
5-May-2025aks6d1c5lem3 42087 Lemma for Claim 5, polynomial division with a linear power. (Contributed by metakunt, 5-May-2025.)
(𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐴 ∈ ℕ0)    &   (𝜑 → 𝐴 < 𝑃)    &   𝑋 = (var1‘𝐾)    &    ↑ = (.g‘(mulGrp‘(Poly1‘𝐾)))    &   𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))    &   (𝜑 → 𝑌 ∈ (ℕ0 ↑m (0...𝐴)))    &   (𝜑 → 𝑊 ∈ (0...𝐴))    &   (𝜑 → 𝐶 ∈ ℕ0)    &   (𝜑 → 𝐶 ≤ (𝑌‘𝑊))    &   𝑄 = (quot1p‘𝐾)    &   𝑆 = (algSc‘(Poly1‘𝐾))    &   𝑀 = (mulGrp‘(Poly1‘𝐾))    ⇒   (𝜑 → ((𝐺‘𝑌)𝑄(𝐶 ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))) = ((((𝑌‘𝑊) − 𝐶) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑊))))(+g‘𝑀)(𝑀 Σg (𝑖 ∈ ((0...𝐴) ∖ {𝑊}) ↦ ((𝑌‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))(𝑆‘((ℤRHom‘𝐾)‘𝑖))))))))
 
5-May-2025aks6d1c5lem1 42086 Lemma for claim 5, evaluate the linear factor at -c to get a root. (Contributed by metakunt, 5-May-2025.)
(𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐴 ∈ ℕ0)    &   (𝜑 → 𝐴 < 𝑃)    &   𝑋 = (var1‘𝐾)    &    ↑ = (.g‘(mulGrp‘(Poly1‘𝐾)))    &   𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))    &   (𝜑 → 𝐵 ∈ (0...𝐴))    &   (𝜑 → 𝐶 ∈ (0...𝐴))    ⇒   (𝜑 → (𝐵 = 𝐶 ↔ (((eval1‘𝐾)‘(𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝐵))))‘((ℤRHom‘𝐾)‘(0 − 𝐶))) = (0g‘𝐾)))
 
5-May-2025aks6d1c5lem0 42085 Lemma for Claim 5 of Theorem 6.1, G defines a map into the polynomials. (Contributed by metakunt, 5-May-2025.)
(𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐴 ∈ ℕ0)    &   (𝜑 → 𝐴 < 𝑃)    &   𝑋 = (var1‘𝐾)    &    ↑ = (.g‘(mulGrp‘(Poly1‘𝐾)))    &   𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖) ↑ (𝑋(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))    ⇒   (𝜑 → 𝐺:(ℕ0 ↑m (0...𝐴))⟶(Base‘(Poly1‘𝐾)))
 
5-May-2025ringexp0nn 42084 Zero to the power of a positive integer is zero. (Contributed by metakunt, 5-May-2025.)
(𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑁 ∈ ℕ)    &    ↑ = (.g‘(mulGrp‘𝑅))    ⇒   (𝜑 → (𝑁 ↑ (0g‘𝑅)) = (0g‘𝑅))
 
5-May-2025idomnnzgmulnz 42083 A finite product of non-zero elements in an integral domain is non-zero. (Contributed by metakunt, 5-May-2025.)
𝐺 = (mulGrp‘𝑅)    &   (𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → 𝑁 ∈ Fin)    &   ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅))    &   ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐴 ≠ (0g‘𝑅))    ⇒   (𝜑 → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ 𝐴)) ≠ (0g‘𝑅))
 
5-May-2025idomnnzpownz 42082 A non-zero power in an integral domain is non-zero. (Contributed by metakunt, 5-May-2025.)
(𝜑 → 𝑅 ∈ IDomn)    &   (𝜑 → 𝐴 ∈ (Base‘𝑅))    &   (𝜑 → 𝐴 ≠ (0g‘𝑅))    &   (𝜑 → 𝑁 ∈ ℕ0)    &    ↑ = (.g‘(mulGrp‘𝑅))    ⇒   (𝜑 → (𝑁 ↑ 𝐴) ≠ (0g‘𝑅))
 
5-May-2025rspcsbnea 42081 Special case related to rspsbc 3901. (Contributed by metakunt, 5-May-2025.)
((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝐶 ≠ 𝐷) → ⦋𝐴 / 𝑥⦌𝐶 ≠ 𝐷)
 
5-May-2025fracerl 33265 Rewrite the ring localization equivalence relation in the case of a field of fractions. (Contributed by Thierry Arnoux, 5-May-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    ∼ = (𝑅 ~RL (RLReg‘𝑅))    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝐸 ∈ 𝐵)    &   (𝜑 → 𝐺 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (RLReg‘𝑅))    &   (𝜑 → 𝐻 ∈ (RLReg‘𝑅))    ⇒   (𝜑 → (⟨𝐸, 𝐹⟩ ∼ ⟨𝐺, 𝐻⟩ ↔ (𝐸 · 𝐻) = (𝐺 · 𝐹)))
 
5-May-2025fracval 33263 Value of the field of fractions. (Contributed by Thierry Arnoux, 5-May-2025.)
( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅))
 
5-May-2025psdmullem 22184 Lemma for psdmul 22185. Transitive law for union of class difference. (Contributed by SN, 5-May-2025.)
(𝜑 → 𝐶 ⊆ 𝐵)    &   (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   (𝜑 → ((𝐴 ∖ 𝐵) ∪ (𝐵 ∖ 𝐶)) = (𝐴 ∖ 𝐶))
 
5-May-2025inxp 5851 Intersection of two Cartesian products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-10 2141, ax-12 2178. (Revised by SN, 5-May-2025.)
((𝐴 × 𝐵) ∩ (𝐶 × 𝐷)) = ((𝐴 ∩ 𝐶) × (𝐵 ∩ 𝐷))
 
4-May-2025opstrgric 47769 A graph represented as an extensible structure with vertices as base set and indexed edges is isomorphic to a hypergraph represented as ordered pair with the same vertices and edges. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 4-May-2025.)
𝐺 = ⟨𝑉, 𝐸⟩    &   𝐻 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩}    ⇒   ((𝐺 ∈ UHGraph ∧ 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐺 ≃𝑔𝑟 𝐻)
 
4-May-2025grimidvtxedg 47750 The identity relation restricted to the set of vertices of a graph is a graph isomorphism between the graph and a graph with the same vertices and edges. (Contributed by AV, 4-May-2025.)
(𝜑 → 𝐺 ∈ UHGraph)    &   (𝜑 → 𝐻 ∈ 𝑉)    &   (𝜑 → (Vtx‘𝐺) = (Vtx‘𝐻))    &   (𝜑 → (iEdg‘𝐺) = (iEdg‘𝐻))    ⇒   (𝜑 → ( I ↾ (Vtx‘𝐺)) ∈ (𝐺 GraphIso 𝐻))
 
4-May-2025zringfrac 33539 The field of fractions of the ring of integers is isomorphic to the field of the rational numbers. (Contributed by Thierry Arnoux, 4-May-2025.)
𝑄 = (ℂfld ↾s ℚ)    &    ∼ = (ℤring ~RL (ℤ ∖ {0}))    &   𝐹 = (𝑞 ∈ ℚ ↦ [⟨(numer‘𝑞), (denom‘𝑞)⟩] ∼ )    ⇒   𝐹 ∈ (𝑄 RingIso ( Frac ‘ℤring))
 
4-May-2025zringidom 33536 The ring of integers is an integral domain. (Contributed by Thierry Arnoux, 4-May-2025.)
ℤring ∈ IDomn
 
4-May-2025fracfld 33267 The field of fractions of an integral domain is a field. (Contributed by Thierry Arnoux, 4-May-2025.)
(𝜑 → 𝑅 ∈ IDomn)    ⇒   (𝜑 → ( Frac ‘𝑅) ∈ Field)
 
4-May-2025rlocf1 33237 The embedding 𝐹 of a ring 𝑅 into its localization 𝐿. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    1 = (1r‘𝑅)    &   𝐿 = (𝑅 RLocal 𝑆)    &    ∼ = (𝑅 ~RL 𝑆)    &   𝐹 = (𝑥 ∈ 𝐵 ↦ [⟨𝑥, 1 ⟩] ∼ )    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   (𝜑 → 𝑆 ⊆ (RLReg‘𝑅))    ⇒   (𝜑 → (𝐹:𝐵–1-1→((𝐵 × 𝑆) / ∼ ) ∧ 𝐹 ∈ (𝑅 RingHom 𝐿)))
 
4-May-2025rloc1r 33236 The multiplicative identity of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.)
0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    &   𝐿 = (𝑅 RLocal 𝑆)    &    ∼ = (𝑅 ~RL 𝑆)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   𝐼 = [⟨ 1 , 1 ⟩] ∼    ⇒   (𝜑 → 𝐼 = (1r‘𝐿))
 
4-May-2025rloc0g 33235 The zero of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.)
0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    &   𝐿 = (𝑅 RLocal 𝑆)    &    ∼ = (𝑅 ~RL 𝑆)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   𝑂 = [⟨ 0 , 1 ⟩] ∼    ⇒   (𝜑 → 𝑂 = (0g‘𝐿))
 
4-May-2025rloccring 33234 The ring localization 𝐿 of a commutative ring 𝑅 by a multiplicatively closed set 𝑆 is itself a commutative ring. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    + = (+g‘𝑅)    &   𝐿 = (𝑅 RLocal 𝑆)    &    ∼ = (𝑅 ~RL 𝑆)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    ⇒   (𝜑 → 𝐿 ∈ CRing)
 
4-May-2025rlocmulval 33233 Value of the addition in the ring localization, given two representatives. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    + = (+g‘𝑅)    &   𝐿 = (𝑅 RLocal 𝑆)    &    ∼ = (𝑅 ~RL 𝑆)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   (𝜑 → 𝐸 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐺 ∈ 𝑆)    &   (𝜑 → 𝐻 ∈ 𝑆)    &    ⊗ = (.r‘𝐿)    ⇒   (𝜑 → ([⟨𝐸, 𝐺⟩] ∼ ⊗ [⟨𝐹, 𝐻⟩] ∼ ) = [⟨(𝐸 · 𝐹), (𝐺 · 𝐻)⟩] ∼ )
 
4-May-2025rlocaddval 33232 Value of the addition in the ring localization, given two representatives. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    + = (+g‘𝑅)    &   𝐿 = (𝑅 RLocal 𝑆)    &    ∼ = (𝑅 ~RL 𝑆)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &   (𝜑 → 𝐸 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐺 ∈ 𝑆)    &   (𝜑 → 𝐻 ∈ 𝑆)    &    ⊕ = (+g‘𝐿)    ⇒   (𝜑 → ([⟨𝐸, 𝐺⟩] ∼ ⊕ [⟨𝐹, 𝐻⟩] ∼ ) = [⟨((𝐸 · 𝐻) + (𝐹 · 𝐺)), (𝐺 · 𝐻)⟩] ∼ )
 
4-May-2025rlocbas 33231 The base set of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &    − = (-g‘𝑅)    &   𝑊 = (𝐵 × 𝑆)    &   𝐿 = (𝑅 RLocal 𝑆)    &    ∼ = (𝑅 ~RL 𝑆)    &   (𝜑 → 𝑅 ∈ 𝑉)    &   (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   (𝜑 → (𝑊 / ∼ ) = (Base‘𝐿))
 
4-May-2025elrlocbasi 33230 Membership in the basis of a ring localization. (Contributed by Thierry Arnoux, 4-May-2025.)
(𝜑 → 𝑋 ∈ ((𝐵 × 𝑆) / ∼ ))    ⇒   (𝜑 → ∃𝑎 ∈ 𝐵 ∃𝑏 ∈ 𝑆 𝑋 = [⟨𝑎, 𝑏⟩] ∼ )
 
4-May-2025erler 33229 The relation used to build the ring localization is an equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    1 = (1r‘𝑅)    &    · = (.r‘𝑅)    &    − = (-g‘𝑅)    &   𝑊 = (𝐵 × 𝑆)    &    ∼ = (𝑅 ~RL 𝑆)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    ⇒   (𝜑 → ∼ Er 𝑊)
 
4-May-2025erlbr2d 33228 Deduce the ring localization equivalence relation. Pairs ⟨𝐸, 𝐺⟩ and ⟨𝑇 · 𝐸, 𝑇 · 𝐺⟩ for 𝑇 ∈ 𝑆 are equivalent under the localization relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    ∼ = (𝑅 ~RL 𝑆)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))    &    · = (.r‘𝑅)    &   (𝜑 → 𝑈 = ⟨𝐸, 𝐺⟩)    &   (𝜑 → 𝑉 = ⟨𝐹, 𝐻⟩)    &   (𝜑 → 𝐸 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐺 ∈ 𝑆)    &   (𝜑 → 𝐻 ∈ 𝑆)    &   (𝜑 → 𝑇 ∈ 𝑆)    &   (𝜑 → 𝐹 = (𝑇 · 𝐸))    &   (𝜑 → 𝐻 = (𝑇 · 𝐺))    ⇒   (𝜑 → 𝑈 ∼ 𝑉)
 
4-May-2025erlbrd 33227 Deduce the ring localization equivalence relation. If for some 𝑇 ∈ 𝑆 we have 𝑇 · (𝐸 · 𝐻 − 𝐹 · 𝐺) = 0, then pairs ⟨𝐸, 𝐺⟩ and ⟨𝐹, 𝐻⟩ are equivalent under the localization relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    ∼ = (𝑅 ~RL 𝑆)    &   (𝜑 → 𝑆 ⊆ 𝐵)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &    − = (-g‘𝑅)    &   (𝜑 → 𝑈 = ⟨𝐸, 𝐺⟩)    &   (𝜑 → 𝑉 = ⟨𝐹, 𝐻⟩)    &   (𝜑 → 𝐸 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐺 ∈ 𝑆)    &   (𝜑 → 𝐻 ∈ 𝑆)    &   (𝜑 → 𝑇 ∈ 𝑆)    &   (𝜑 → (𝑇 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 )    ⇒   (𝜑 → 𝑈 ∼ 𝑉)
 
4-May-2025erldi 33226 Main property of the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    ∼ = (𝑅 ~RL 𝑆)    &   (𝜑 → 𝑆 ⊆ 𝐵)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &    − = (-g‘𝑅)    &   (𝜑 → 𝑈 ∼ 𝑉)    ⇒   (𝜑 → ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑈) · (2nd ‘𝑉)) − ((1st ‘𝑉) · (2nd ‘𝑈)))) = 0 )
 
4-May-2025erlcl2 33225 Closure for the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    ∼ = (𝑅 ~RL 𝑆)    &   (𝜑 → 𝑆 ⊆ 𝐵)    &   (𝜑 → 𝑈 ∼ 𝑉)    ⇒   (𝜑 → 𝑉 ∈ (𝐵 × 𝑆))
 
4-May-2025erlcl1 33224 Closure for the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    ∼ = (𝑅 ~RL 𝑆)    &   (𝜑 → 𝑆 ⊆ 𝐵)    &   (𝜑 → 𝑈 ∼ 𝑉)    ⇒   (𝜑 → 𝑈 ∈ (𝐵 × 𝑆))
 
4-May-2025rlocval 33223 Expand the value of the ring localization operation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &    − = (-g‘𝑅)    &    + = (+g‘𝑅)    &    ≤ = (le‘𝑅)    &   𝐹 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝐹)    &   𝐶 = ( ·𝑠 ‘𝑅)    &   𝑊 = (𝐵 × 𝑆)    &    ∼ = (𝑅 ~RL 𝑆)    &   𝐽 = (TopSet‘𝑅)    &   𝐷 = (dist‘𝑅)    &    ⊕ = (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨(((1st ‘𝑎) · (2nd ‘𝑏)) + ((1st ‘𝑏) · (2nd ‘𝑎))), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)    &    ⊗ = (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ ⟨((1st ‘𝑎) · (1st ‘𝑏)), ((2nd ‘𝑎) · (2nd ‘𝑏))⟩)    &    × = (𝑘 ∈ 𝐾, 𝑎 ∈ 𝑊 ↦ ⟨(𝑘𝐶(1st ‘𝑎)), (2nd ‘𝑎)⟩)    &    ≲ = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ((1st ‘𝑎) · (2nd ‘𝑏)) ≤ ((1st ‘𝑏) · (2nd ‘𝑎)))}    &   𝐸 = (𝑎 ∈ 𝑊, 𝑏 ∈ 𝑊 ↦ (((1st ‘𝑎) · (2nd ‘𝑏))𝐷((1st ‘𝑏) · (2nd ‘𝑎))))    &   (𝜑 → 𝑅 ∈ 𝑉)    &   (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   (𝜑 → (𝑅 RLocal 𝑆) = ((({⟨(Base‘ndx), 𝑊⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), ⊗ ⟩} ∪ {⟨(Scalar‘ndx), 𝐹⟩, ⟨( ·𝑠 ‘ndx), × ⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), (𝐽 ×t (𝐽 ↾t 𝑆))⟩, ⟨(le‘ndx), ≲ ⟩, ⟨(dist‘ndx), 𝐸⟩}) /s ∼ ))
 
4-May-2025erlval 33222 Value of the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &    − = (-g‘𝑅)    &   𝑊 = (𝐵 × 𝑆)    &    ∼ = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )}    &   (𝜑 → 𝑆 ⊆ 𝐵)    ⇒   (𝜑 → (𝑅 ~RL 𝑆) = ∼ )
 
4-May-2025ringdird 33202 Distributive law for the multiplication operation of a ring (right-distributivity). (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))
 
4-May-2025ringdid 33201 Distributive law for the multiplication operation of a ring (left-distributivity). (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    + = (+g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 · (𝑌 + 𝑍)) = ((𝑋 · 𝑌) + (𝑋 · 𝑍)))
 
4-May-2025cringmul32d 33200 Commutative/associative law that swaps the last two factors in a triple product in a commutative ring. See also mul32 11450. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋 · 𝑌) · 𝑍) = ((𝑋 · 𝑍) · 𝑌))
 
4-May-2025submcld 33013 Submonoids are closed under the monoid operation. (Contributed by Thierry Arnoux, 4-May-2025.)
+ = (+g‘𝑀)    &   (𝜑 → 𝑆 ∈ (SubMnd‘𝑀))    &   (𝜑 → 𝑋 ∈ 𝑆)    &   (𝜑 → 𝑌 ∈ 𝑆)    ⇒   (𝜑 → (𝑋 + 𝑌) ∈ 𝑆)
 
4-May-2025cmn145236 33012 Rearrange terms in a commutative monoid sum. Lemma for rlocaddval 33232. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝑈 ∈ 𝐵)    &   (𝜑 → 𝑉 ∈ 𝐵)    &   (𝜑 → 𝑊 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋 + 𝑌) + ((𝑍 + 𝑈) + (𝑉 + 𝑊))) = ((𝑋 + (𝑈 + 𝑉)) + (𝑌 + (𝑍 + 𝑊))))
 
4-May-2025cmn246135 33011 Rearrange terms in a commutative monoid sum. Lemma for rlocaddval 33232. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝑈 ∈ 𝐵)    &   (𝜑 → 𝑉 ∈ 𝐵)    &   (𝜑 → 𝑊 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋 + 𝑌) + ((𝑍 + 𝑈) + (𝑉 + 𝑊))) = ((𝑌 + (𝑈 + 𝑊)) + (𝑋 + (𝑍 + 𝑉))))
 
4-May-2025cmn4d 33010 Commutative/associative law for commutative monoids. (Contributed by Thierry Arnoux, 4-May-2025.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝑊 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))
 
4-May-2025zdend 32809 Denominator of an integer. (Contributed by Thierry Arnoux, 4-May-2025.)
(𝜑 → 𝑍 ∈ ℤ)    ⇒   (𝜑 → (denom‘𝑍) = 1)
 
4-May-2025znumd 32808 Numerator of an integer. (Contributed by Thierry Arnoux, 4-May-2025.)
(𝜑 → 𝑍 ∈ ℤ)    ⇒   (𝜑 → (numer‘𝑍) = 𝑍)
 
4-May-2025brab2d 32621 Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by Thierry Arnoux, 4-May-2025.)
(𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑉) ∧ 𝜓)})    &   ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (𝐴𝑅𝐵 ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))
 
4-May-2025copsex2dv 32620 Implicit substitution deduction for ordered pairs. (Contributed by Thierry Arnoux, 4-May-2025.)
(𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
 
4-May-2025bibiad 32478 Eliminate an hypothesis 𝜃 in a biconditional. (Contributed by Thierry Arnoux, 4-May-2025.)
((𝜑 ∧ 𝜓) → 𝜃)    &   ((𝜑 ∧ 𝜒) → 𝜃)    &   ((𝜑 ∧ 𝜃) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (𝜓 ↔ 𝜒))
 
4-May-2025eldifsnd 4812 Membership in a set with an element removed : deduction version. (Contributed by Thierry Arnoux, 4-May-2025.)
(𝜑 → 𝐴 ∈ 𝐵)    &   (𝜑 → 𝐴 ≠ 𝐶)    ⇒   (𝜑 → 𝐴 ∈ (𝐵 ∖ {𝐶}))
 
3-May-2025gricen 47768 Isomorphic graphs have equinumerous sets of vertices. (Contributed by AV, 3-May-2025.)
𝐵 = (Vtx‘𝑅)    &   𝐶 = (Vtx‘𝑆)    ⇒   (𝑅 ≃𝑔𝑟 𝑆 → 𝐵 ≈ 𝐶)
 
3-May-2025grictr 47766 Graph isomorphism is transitive. (Contributed by AV, 5-Dec-2022.) (Revised by AV, 3-May-2025.)
((𝑅 ≃𝑔𝑟 𝑆 ∧ 𝑆 ≃𝑔𝑟 𝑇) → 𝑅 ≃𝑔𝑟 𝑇)
 
3-May-2025gricsymb 47765 Graph isomorphism is symmetric in both directions for hypergraphs. (Contributed by AV, 11-Nov-2022.) (Proof shortened by AV, 3-May-2025.)
((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴 ≃𝑔𝑟 𝐵 ↔ 𝐵 ≃𝑔𝑟 𝐴))
 
3-May-2025gricsym 47764 Graph isomorphism is symmetric for hypergraphs. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 3-May-2025.)
(𝐺 ∈ UHGraph → (𝐺 ≃𝑔𝑟 𝑆 → 𝑆 ≃𝑔𝑟 𝐺))
 
3-May-2025grimco 47754 The composition of graph isomorphisms is a graph isomorphism. (Contributed by AV, 3-May-2025.)
((𝐹 ∈ (𝑇 GraphIso 𝑈) ∧ 𝐺 ∈ (𝑆 GraphIso 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 GraphIso 𝑈))
 
3-May-2025wl-sb8motv 37528 Substitution of variable in universal quantifier. Closed form of sb8mo 2604 without ax-13 2380, but requiring 𝑥 and 𝑦 being disjoint.

This theorem relates to wl-mo3t 37523, since replacing 𝜑 with [𝑦 / 𝑥]𝜑 in the latter yields subexpressions like [𝑥 / 𝑦][𝑦 / 𝑥]𝜑, which can be reduced to 𝜑 via sbft 2271 and sbco 2515. So ∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑 is provable from wl-mo3t 37523 in a simple fashion. From an educational standpoint, one would assume wl-mo3t 37523 to be more fundamental, as it hints how the "at most one" objects on both sides of the biconditional correlate (they are the same), if they exist at all, and then prove this theorem from it. (Contributed by Wolf Lammen, 3-May-2025.)

(∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃*𝑦[𝑦 / 𝑥]𝜑))
 
3-May-2025wl-sb8eutv 37526 Substitution of variable in universal quantifier. Closed form of sb8euv 2602. (Contributed by Wolf Lammen, 3-May-2025.)
(∀𝑥Ⅎ𝑦𝜑 → (∃!𝑥𝜑 ↔ ∃!𝑦[𝑦 / 𝑥]𝜑))
 
2-May-2025grimuhgr 47752 If there is a graph isomorphism between a hypergraph and a class with an edge function, the class is also a hypergraph. (Contributed by AV, 2-May-2025.)
((𝑆 ∈ UHGraph ∧ 𝐹 ∈ (𝑆 GraphIso 𝑇) ∧ Fun (iEdg‘𝑇)) → 𝑇 ∈ UHGraph)
 
2-May-2025aks6d1c2 42080 Claim 2 of Theorem 6.1 of https://www3.nd.edu/%7eandyp/notes/AKS.pdf (Contributed by metakunt, 2-May-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))    &   (𝜑 → 𝐴 ∈ ℕ0)    &   𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    &   (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))    &   𝐵 = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))    &   𝐶 = (𝐸 “ ((0...𝐵) × (0...𝐵)))    &   (𝜑 → (𝑄 ∈ ℙ ∧ 𝑄 ∥ 𝑁 ∧ 𝑃 ≠ 𝑄))    ⇒   (𝜑 → (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))) ≤ (𝑁↑𝐵))
 
2-May-2025hashnexinjle 42079 If the number of elements of the domain are greater than the number of elements in a codomain, then there are two different values that map to the same. Also we introduce a one sided inequality to simplify a duplicateable proof. (Contributed by metakunt, 2-May-2025.)
(𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝐵 ∈ Fin)    &   (𝜑 → (♯‘𝐵) < (♯‘𝐴))    &   (𝜑 → 𝐹:𝐴⟶𝐵)    &   (𝜑 → 𝐴 ⊆ ℝ)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 < 𝑦))
 
2-May-2025hashnexinj 42078 If the number of elements of the domain are greater than the number of elements in a codomain, then there are two different values that map to the same. (Contributed by metakunt, 2-May-2025.)
(𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝐵 ∈ Fin)    &   (𝜑 → (♯‘𝐵) < (♯‘𝐴))    &   (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦))
 
2-May-2025wl-nfsbtv 37524 Closed form of nfsbv 2334. (Contributed by Wolf Lammen, 2-May-2025.)
(∀𝑥Ⅎ𝑧𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
 
2-May-2025psrbagleadd1 21964 The analogue of "𝑋 ≤ 𝐹 implies 𝑋 + 𝐺 ≤ 𝐹 + 𝐺 " (compare leadd1d 11878) for bags. (Contributed by SN, 2-May-2025.)
𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹}    &   𝑇 = {𝑧 ∈ 𝐷 ∣ 𝑧 ∘r ≤ (𝐹 ∘f + 𝐺)}    ⇒   ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → (𝑋 ∘f + 𝐺) ∈ 𝑇)
 
2-May-2025sbnf 2316 Move nonfree predicate in and out of substitution; see sbal 2170 and sbex 2285. (Contributed by BJ, 2-May-2019.) (Proof shortened by Wolf Lammen, 2-May-2025.)
([𝑧 / 𝑦]Ⅎ𝑥𝜑 ↔ Ⅎ𝑥[𝑧 / 𝑦]𝜑)
 
1-May-2025grimcnv 47753 The converse of a graph isomorphism is a graph isomorphism. (Contributed by AV, 1-May-2025.)
(𝑆 ∈ UHGraph → (𝐹 ∈ (𝑆 GraphIso 𝑇) → ◡𝐹 ∈ (𝑇 GraphIso 𝑆)))
 
1-May-2025aks6d1c2lem4 42077 Claim 2 of Theorem 6.1 AKS, Preparation for injectivity proof. (Contributed by metakunt, 1-May-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   (𝜑 → 𝐹:(0...𝐴)⟶ℕ0)    &   𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))    &   (𝜑 → 𝐴 ∈ ℕ0)    &   𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    &   (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))    &   𝐵 = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))    &   𝐶 = (𝐸 “ ((0...𝐵) × (0...𝐵)))    &   (𝜑 → 𝐼 ∈ 𝐶)    &   (𝜑 → 𝐽 ∈ 𝐶)    &   (𝜑 → 𝐼 < 𝐽)    &    ↑ = (.g‘(mulGrp‘(Poly1‘𝐾)))    &   𝑋 = (var1‘𝐾)    &   𝑆 = ((𝐽 ↑ 𝑋)(-g‘(Poly1‘𝐾))(𝐼 ↑ 𝑋))    &   (𝜑 → 𝑈 ∈ ℕ)    &   (𝜑 → 𝐽 = (𝐼 + (𝑈 · 𝑅)))    ⇒   (𝜑 → (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))) ≤ (𝑁↑𝐵))
 
1-May-2025aks6d1c2lem3 42076 Lemma for aks6d1c2 42080 to simplify context. (Contributed by metakunt, 1-May-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   (𝜑 → 𝐹:(0...𝐴)⟶ℕ0)    &   𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))    &   (𝜑 → 𝐴 ∈ ℕ0)    &   𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    &   (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    &   (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))    &   𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))    &   𝐵 = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))    &   𝐶 = (𝐸 “ ((0...𝐵) × (0...𝐵)))    &   (𝜑 → 𝐼 ∈ 𝐶)    &   (𝜑 → 𝐽 ∈ 𝐶)    &   (𝜑 → 𝐼 < 𝐽)    &    ↑ = (.g‘(mulGrp‘(Poly1‘𝐾)))    &   𝑋 = (var1‘𝐾)    &   𝑆 = ((𝐽 ↑ 𝑋)(-g‘(Poly1‘𝐾))(𝐼 ↑ 𝑋))    &   (𝜑 → 𝑈 ∈ ℕ)    &   (𝜑 → 𝐽 = (𝐼 + (𝑈 · 𝑅)))    &   (𝜑 → 𝑠 ∈ (ℕ0 ↑m (0...𝐴)))    &   (𝜑 → 𝑟 ∈ (0...𝐵))    &   (𝜑 → 𝑜 ∈ (0...𝐵))    &   (𝜑 → 𝐽 = (𝑟𝐸𝑜))    &   (𝜑 → 𝑝 ∈ (0...𝐵))    &   (𝜑 → 𝑞 ∈ (0...𝐵))    &   (𝜑 → 𝐼 = (𝑝𝐸𝑞))    &   (𝜑 → 𝐼 ∈ ℕ0)    ⇒   (𝜑 → (𝐽(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (𝐼(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)))
 
1-May-2025aks6d1c1rh 42075 Claim 1 of AKS primality proof with collapsed definitions since their ease of use is no longer needed. (Contributed by metakunt, 1-May-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}    &   𝑃 = (chr‘𝐾)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   (𝜑 → 𝐹:(0...𝐴)⟶ℕ0)    &   𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))    &   (𝜑 → 𝐴 ∈ ℕ0)    &   (𝜑 → 𝑈 ∈ ℕ0)    &   (𝜑 → 𝐿 ∈ ℕ0)    &   𝐸 = ((𝑃↑𝑈) · ((𝑁 / 𝑃)↑𝐿))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))    &   (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))    ⇒   (𝜑 → 𝐸 ∼ (𝐺‘𝐹))
 
30-Apr-2025aks6d1c1 42066 Claim 1 of Theorem 6.1 https://www3.nd.edu/%7eandyp/notes/AKS.pdf. (Contributed by metakunt, 30-Apr-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}    &   𝑆 = (Poly1‘𝐾)    &   𝐵 = (Base‘𝑆)    &   𝑋 = (var1‘𝐾)    &   𝑊 = (mulGrp‘𝑆)    &   𝑉 = (mulGrp‘𝐾)    &    ↑ = (.g‘𝑉)    &   𝐶 = (algSc‘𝑆)    &   𝐷 = (.g‘𝑊)    &   𝑃 = (chr‘𝐾)    &   𝑂 = (eval1‘𝐾)    &    + = (+g‘𝑆)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   (𝜑 → 𝐹:(0...𝐴)⟶ℕ0)    &   𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))    &   (𝜑 → 𝐴 ∈ ℕ0)    &   (𝜑 → 𝑈 ∈ ℕ0)    &   (𝜑 → 𝐿 ∈ ℕ0)    &   𝐸 = ((𝑃↑𝑈) · ((𝑁 / 𝑃)↑𝐿))    &   (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))    &   (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingIso 𝐾))    ⇒   (𝜑 → 𝐸 ∼ (𝐺‘𝐹))
 
30-Apr-2025aks6d1c1p8 42065 If a number 𝐸 is introspective to 𝐹, then so are its powers. (Contributed by metakunt, 30-Apr-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}    &   𝑆 = (Poly1‘𝐾)    &   𝐵 = (Base‘𝑆)    &   𝑋 = (var1‘𝐾)    &   𝑊 = (mulGrp‘𝑆)    &   𝑉 = (mulGrp‘𝐾)    &    ↑ = (.g‘𝑉)    &   𝐶 = (algSc‘𝑆)    &   𝐷 = (.g‘𝑊)    &   𝑃 = (chr‘𝐾)    &   𝑂 = (eval1‘𝐾)    &    + = (+g‘𝑆)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   (𝜑 → 𝐸 ∼ 𝐹)    &   (𝜑 → 𝐿 ∈ ℕ0)    &   (𝜑 → (𝐸 gcd 𝑅) = 1)    ⇒   (𝜑 → (𝐸↑𝐿) ∼ 𝐹)
 
30-Apr-2025aks6d1c1p6 42064 If a polynomials 𝐹 is introspective to 𝐸, then so are its powers. (Contributed by metakunt, 30-Apr-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}    &   𝑆 = (Poly1‘𝐾)    &   𝐵 = (Base‘𝑆)    &   𝑋 = (var1‘𝐾)    &   𝑊 = (mulGrp‘𝑆)    &   𝑉 = (mulGrp‘𝐾)    &    ↑ = (.g‘𝑉)    &   𝐶 = (algSc‘𝑆)    &   𝐷 = (.g‘𝑊)    &   𝑃 = (chr‘𝐾)    &   𝑂 = (eval1‘𝐾)    &    + = (+g‘𝑆)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   (𝜑 → 𝐸 ∼ 𝐹)    &   (𝜑 → 𝐿 ∈ ℕ0)    ⇒   (𝜑 → 𝐸 ∼ (𝐿𝐷𝐹))
 
30-Apr-2025aks6d1c1p7 42063 𝑋 is introspective to all positive integers. (Contributed by metakunt, 30-Apr-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}    &   𝑆 = (Poly1‘𝐾)    &   𝐵 = (Base‘𝑆)    &   𝑋 = (var1‘𝐾)    &   𝑉 = (mulGrp‘𝐾)    &    ↑ = (.g‘𝑉)    &   𝑃 = (chr‘𝐾)    &   𝑂 = (eval1‘𝐾)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   (𝜑 → 𝐿 ∈ ℕ)    ⇒   (𝜑 → 𝐿 ∼ 𝑋)
 
30-Apr-2025taylply2 26419 The Taylor polynomial is a polynomial of degree (at most) 𝑁. This version of taylply 26421 shows that the coefficients of 𝑇 are in a subring of the complex numbers. (Contributed by Mario Carneiro, 1-Jan-2017.) Avoid ax-mulf 11258. (Revised by GG, 30-Apr-2025.)
(𝜑 → 𝑆 ∈ {ℝ, ℂ})    &   (𝜑 → 𝐹:𝐴⟶ℂ)    &   (𝜑 → 𝐴 ⊆ 𝑆)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁))    &   𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)    &   (𝜑 → 𝐷 ∈ (SubRing‘ℂfld))    &   (𝜑 → 𝐵 ∈ 𝐷)    &   ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ 𝐷)    ⇒   (𝜑 → (𝑇 ∈ (Poly‘𝐷) ∧ (deg‘𝑇) ≤ 𝑁))
 
30-Apr-2025dvply2g 26336 The derivative of a polynomial with coefficients in a subring is a polynomial with coefficients in the same ring. (Contributed by Mario Carneiro, 1-Jan-2017.) Avoid ax-mulf 11258. (Revised by GG, 30-Apr-2025.)
((𝑆 ∈ (SubRing‘ℂfld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℂ D 𝐹) ∈ (Poly‘𝑆))
 
30-Apr-2025cnsubrglem 21451 Lemma for resubdrg 21643 and friends. (Contributed by Mario Carneiro, 4-Dec-2014.) Avoid ax-mulf 11258. (Revised by GG, 30-Apr-2025.)
(𝑥 ∈ 𝐴 → 𝑥 ∈ ℂ)    &   ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 + 𝑦) ∈ 𝐴)    &   (𝑥 ∈ 𝐴 → -𝑥 ∈ 𝐴)    &   1 ∈ 𝐴    &   ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 · 𝑦) ∈ 𝐴)    ⇒   𝐴 ∈ (SubRing‘ℂfld)
 
30-Apr-2025cnflddiv 21430 The division operation in the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) Avoid ax-mulf 11258. (Revised by GG, 30-Apr-2025.)
/ = (/r‘ℂfld)
 
30-Apr-2025cndrng 21428 The complex numbers form a division ring. (Contributed by Stefan O'Rear, 27-Nov-2014.) Avoid ax-mulf 11258. (Revised by GG, 30-Apr-2025.)
ℂfld ∈ DivRing
 
29-Apr-2025gricref 47763 Graph isomorphism is reflexive for hypergraphs. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 29-Apr-2025.)
(𝐺 ∈ UHGraph → 𝐺 ≃𝑔𝑟 𝐺)
 
29-Apr-2025grimid 47751 The identity relation restricted to the set of vertices of a graph is a graph isomorphism between the graph and itself. (Contributed by AV, 29-Apr-2025.) (Prove shortened by AV, 5-May-2025.)
(𝐺 ∈ UHGraph → ( I ↾ (Vtx‘𝐺)) ∈ (𝐺 GraphIso 𝐺))
 
29-Apr-2025grimf1o 47744 An isomorphism of graphs is a bijection between their vertices. (Contributed by AV, 29-Apr-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    ⇒   (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:𝑉–1-1-onto→𝑊)
 
29-Apr-2025grimprop 47743 Properties of an isomorphism of graphs. (Contributed by AV, 29-Apr-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐸 = (iEdg‘𝐺)    &   𝐷 = (iEdg‘𝐻)    ⇒   (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom 𝐸–1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖)))))
 
29-Apr-2025evl1gprodd 42067 Polynomial evaluation builder for a finite group product of polynomials. (Contributed by metakunt, 29-Apr-2025.)
𝑂 = (eval1‘𝑅)    &   𝑃 = (Poly1‘𝑅)    &   𝑄 = (mulGrp‘𝑃)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   𝑆 = (mulGrp‘𝑅)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → ∀𝑥 ∈ 𝑁 𝑀 ∈ 𝑈)    &   (𝜑 → 𝑁 ∈ Fin)    ⇒   (𝜑 → ((𝑂‘(𝑄 Σg (𝑥 ∈ 𝑁 ↦ 𝑀)))‘𝑌) = (𝑆 Σg (𝑥 ∈ 𝑁 ↦ ((𝑂‘𝑀)‘𝑌))))
 
28-Apr-2025brgrici 47756 Prove that two graphs are isomorphic by an explicit isomorphism. (Contributed by AV, 28-Apr-2025.)
(𝐹 ∈ (𝑅 GraphIso 𝑆) → 𝑅 ≃𝑔𝑟 𝑆)
 
28-Apr-2025brgric 47755 The relation "is isomorphic to" for graphs. (Contributed by AV, 28-Apr-2025.)
(𝑅 ≃𝑔𝑟 𝑆 ↔ (𝑅 GraphIso 𝑆) ≠ ∅)
 
28-Apr-2025grimdmrel 47740 The domain of the graph isomorphism function is a relation. (Contributed by AV, 28-Apr-2025.)
Rel dom GraphIso
 
28-Apr-2025grimfn 47739 The graph isomorphism function is a well-defined function. (Contributed by AV, 28-Apr-2025.)
GraphIso Fn (V × V)
 
28-Apr-2025aks6d1c3 42073 Claim 3 of Theorem 6.1 of the AKS inequality lemma. https://www3.nd.edu/%7eandyp/notes/AKS.pdf (Contributed by metakunt, 28-Apr-2025.)
(𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   𝐿 = (ℤRHom‘𝑌)    &   𝑌 = (ℤ/nℤ‘𝑅)    &   (𝜑 → ((2 logb 𝑁)↑2) < ((odℤ‘𝑅)‘𝑁))    ⇒   (𝜑 → ((2 logb 𝑁)↑2) < (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))
 
28-Apr-2025hashscontpow 42072 If a set contains all 𝑁-th powers, then the size of the image under the ZR homomorphism is greater than the 𝑅-th order of 𝑁. (Contributed by metakunt, 28-Apr-2025.)
(𝜑 → 𝐸 ⊆ ℤ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → ∀𝑘 ∈ ℕ0 (𝑁↑𝑘) ∈ 𝐸)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   𝐿 = (ℤRHom‘𝑌)    &   𝑌 = (ℤ/nℤ‘𝑅)    ⇒   (𝜑 → ((odℤ‘𝑅)‘𝑁) ≤ (♯‘(𝐿 “ 𝐸)))
 
28-Apr-2025hashscontpow1 42071 Helper lemma for to prove inequality in Zr. (Contributed by metakunt, 28-Apr-2025.)
(𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝐴 ∈ (1...((odℤ‘𝑅)‘𝑁)))    &   (𝜑 → 𝐵 ∈ (1...((odℤ‘𝑅)‘𝑁)))    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   𝐿 = (ℤRHom‘𝑌)    &   𝑌 = (ℤ/nℤ‘𝑅)    &   (𝜑 → 𝐴 < 𝐵)    ⇒   (𝜑 → (𝐿‘(𝑁↑𝐴)) ≠ (𝐿‘(𝑁↑𝐵)))
 
28-Apr-2025hashscontpowcl 42070 Closure of E for https://www3.nd.edu/%7eandyp/notes/AKS.pdf Theorem 6.1. (Contributed by metakunt, 28-Apr-2025.)
(𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))    &   𝐿 = (ℤRHom‘𝑌)    &   𝑌 = (ℤ/nℤ‘𝑅)    ⇒   (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℕ0)
 
28-Apr-2025df-erl 33219 Define the operation giving the equivalence relation used in the localization of a ring 𝑟 by a set 𝑠. Two pairs 𝑎 = ⟨𝑥, 𝑦⟩ and 𝑏 = ⟨𝑧, 𝑤⟩ are equivalent if there exists 𝑡 ∈ 𝑠 such that 𝑡 · (𝑥 · 𝑤 − 𝑧 · 𝑦) = 0. This corresponds to the usual comparison of fractions 𝑥 / 𝑦 and 𝑧 / 𝑤. (Contributed by Thierry Arnoux, 28-Apr-2025.)
~RL = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋(.r‘𝑟) / 𝑥⦌⦋((Base‘𝑟) × 𝑠) / 𝑤⦌{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ∧ ∃𝑡 ∈ 𝑠 (𝑡𝑥(((1st ‘𝑎)𝑥(2nd ‘𝑏))(-g‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎)))) = (0g‘𝑟))})
 
27-Apr-2025df-grlic 47795 Two graphs are said to be locally isomorphic iff they are connected by at least one local isomorphism. (Contributed by AV, 27-Apr-2025.)
≃𝑙𝑔𝑟 = (◡ GraphLocIso “ (V ∖ 1o))
 
27-Apr-2025df-grlim 47792 A local isomorphism of graphs is a bijection between the sets of vertices of two graphs that preserves local adjacency, i.e. the subgraph induced by the closed neighborhood of a vertex of the first graph and the subgraph induced by the closed neighborhood of the associated vertex of the second graph are isomorphic. See the following chat in mathoverflow: https://mathoverflow.net/questions/491133/locally-isomorphic-graphs. (Contributed by AV, 27-Apr-2025.)
GraphLocIso = (𝑔 ∈ V, ℎ ∈ V ↦ {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∀𝑣 ∈ (Vtx‘𝑔)(𝑔 ISubGr (𝑔 ClNeighbVtx 𝑣)) ≃𝑔𝑟 (ℎ ISubGr (ℎ ClNeighbVtx (𝑓‘𝑣))))})
 
27-Apr-2025isuspgrim 47749 A class is an isomorphism of simple pseudographs iff it is a bijection between their vertices that preserves adjacency, i.e. there is an edge in one graph connecting one or two vertices iff there is an edge in the other graph connecting the vertices which are the images of the vertices. This corresponds to the formal definition in [Bollobas] p. 3 and the definition in [Diestel] p. 3. (Contributed by AV, 27-Apr-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (Edg‘𝐻)    ⇒   ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷))))
 
27-Apr-2025isuspgrimlem 47748 Lemma for isuspgrim 47749. (Contributed by AV, 27-Apr-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (Edg‘𝐻)    ⇒   ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷)) → (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)
 
27-Apr-2025uspgrimprop 47747 An isomorphism of simple pseudographs is a bijection between their vertices that preserves adjacency, i.e. there is an edge in one graph connecting one or two vertices iff there is an edge in the other graph connecting the vertices which are the images of the vertices. (Contributed by AV, 27-Apr-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (Edg‘𝐻)    ⇒   ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹‘𝑥), (𝐹‘𝑦)} ∈ 𝐷))))
 
27-Apr-2025df-isubgr 47723 Define the function mapping graphs and subsets of their vertices to their induced subgraphs. A subgraph induced by a subset of vertices of a graph is a subgraph of the graph which contains all edges of the graph that join vertices of the subgraph (see section I.1 in [Bollobas] p. 2 or section 1.1 in [Diestel] p. 4). Although a graph may be given in any meaningful representation, its induced subgraphs are always ordered pairs of vertices and edges. (Contributed by AV, 27-Apr-2025.)
ISubGr = (𝑔 ∈ V, 𝑣 ∈ 𝒫 (Vtx‘𝑔) ↦ ⟨𝑣, ⦋(iEdg‘𝑔) / 𝑒⦌(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣})⟩)
 
27-Apr-2025imaexi 45118 The image of a set is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Proof shortened by SN, 27-Apr-2025.)
𝐴 ∈ 𝑉    ⇒   (𝐴 “ 𝐵) ∈ V
 
27-Apr-2025wl-sb8eft 37498 Substitution of variable in existentialal quantifier. Closed form of sb8ef 2361. (Contributed by Wolf Lammen, 27-Apr-2025.)
(∀𝑥Ⅎ𝑦𝜑 → (∃𝑥𝜑 ↔ ∃𝑦[𝑦 / 𝑥]𝜑))
 
27-Apr-2025wl-sb8ft 37497 Substitution of variable in universal quantifier. Closed form of sb8f 2359. (Contributed by Wolf Lammen, 27-Apr-2025.)
(∀𝑥Ⅎ𝑦𝜑 → (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑))
 
27-Apr-2025wl-sb9v 37496 Commutation of quantification and substitution variables based on fewer axioms than sb9 2527. (Contributed by Wolf Lammen, 27-Apr-2025.)
(∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
 
27-Apr-2025irredminply 33699 An irreducible, monic, annihilating polynomial is the minimal polynomial. (Contributed by Thierry Arnoux, 27-Apr-2025.)
𝑂 = (𝐸 evalSub1 𝐹)    &   𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   𝐵 = (Base‘𝐸)    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐴 ∈ 𝐵)    &    0 = (0g‘𝐸)    &   𝑀 = (𝐸 minPoly 𝐹)    &   𝑍 = (0g‘𝑃)    &   (𝜑 → ((𝑂‘𝐺)‘𝐴) = 0 )    &   (𝜑 → 𝐺 ∈ (Irred‘𝑃))    &   (𝜑 → 𝐺 ∈ (Monic1p‘(𝐸 ↾s 𝐹)))    ⇒   (𝜑 → 𝐺 = (𝑀‘𝐴))
 
27-Apr-2025m1pmeq 33565 If two monic polynomials 𝐼 and 𝐽 differ by a unit factor 𝐾, then they are equal. (Contributed by Thierry Arnoux, 27-Apr-2025.)
𝑃 = (Poly1‘𝐹)    &   𝑀 = (Monic1p‘𝐹)    &   𝑈 = (Unit‘𝑃)    &    · = (.r‘𝑃)    &   (𝜑 → 𝐹 ∈ Field)    &   (𝜑 → 𝐼 ∈ 𝑀)    &   (𝜑 → 𝐽 ∈ 𝑀)    &   (𝜑 → 𝐾 ∈ 𝑈)    &   (𝜑 → 𝐼 = (𝐾 · 𝐽))    ⇒   (𝜑 → 𝐼 = 𝐽)
 
27-Apr-2025df-rloc 33220 Define the operation giving the localization of a ring 𝑟 by a given set 𝑠. The localized ring 𝑟 RLocal 𝑠 is the set of equivalence classes of pairs of elements in 𝑟 over the relation 𝑟 ~RL 𝑠 with addition and multiplication defined naturally. (Contributed by Thierry Arnoux, 27-Apr-2025.)
RLocal = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋(.r‘𝑟) / 𝑥⦌⦋((Base‘𝑟) × 𝑠) / 𝑤⦌((({⟨(Base‘ndx), 𝑤⟩, ⟨(+g‘ndx), (𝑎 ∈ 𝑤, 𝑏 ∈ 𝑤 ↦ ⟨(((1st ‘𝑎)𝑥(2nd ‘𝑏))(+g‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎))), ((2nd ‘𝑎)𝑥(2nd ‘𝑏))⟩)⟩, ⟨(.r‘ndx), (𝑎 ∈ 𝑤, 𝑏 ∈ 𝑤 ↦ ⟨((1st ‘𝑎)𝑥(1st ‘𝑏)), ((2nd ‘𝑎)𝑥(2nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑟)⟩, ⟨( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑟)), 𝑎 ∈ 𝑤 ↦ ⟨(𝑘( ·𝑠 ‘𝑟)(1st ‘𝑎)), (2nd ‘𝑎)⟩)⟩, ⟨(·𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑟) ×t ((TopSet‘𝑟) ↾t 𝑠))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ∧ ((1st ‘𝑎)𝑥(2nd ‘𝑏))(le‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑤, 𝑏 ∈ 𝑤 ↦ (((1st ‘𝑎)𝑥(2nd ‘𝑏))(dist‘𝑟)((1st ‘𝑏)𝑥(2nd ‘𝑎))))⟩}) /s (𝑟 ~RL 𝑠)))
 
27-Apr-2025cnfldmul 21389 The multiplication operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21382. (Revised by GG, 27-Apr-2025.)
· = (.r‘ℂfld)
 
27-Apr-2025cnfldadd 21387 The addition operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21382. (Revised by GG, 27-Apr-2025.)
+ = (+g‘ℂfld)
 
27-Apr-2025elimampo 7581 Membership in the image of an operation. (Contributed by SN, 27-Apr-2025.)
𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   (𝜑 → 𝑋 ⊆ 𝐴)    &   (𝜑 → 𝑌 ⊆ 𝐵)    ⇒   (𝜑 → (𝐷 ∈ (𝐹 “ (𝑋 × 𝑌)) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝐷 = 𝐶))
 
26-Apr-2025aks6d1c1p5 42062 The product of exponents is introspective. (Contributed by metakunt, 26-Apr-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}    &   𝑆 = (Poly1‘𝐾)    &   𝐵 = (Base‘𝑆)    &   𝑋 = (var1‘𝐾)    &   𝑊 = (mulGrp‘𝑆)    &   𝑉 = (mulGrp‘𝐾)    &    ↑ = (.g‘𝑉)    &   𝐶 = (algSc‘𝑆)    &   𝑃 = (chr‘𝐾)    &   𝑂 = (eval1‘𝐾)    &    + = (+g‘𝑆)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → (𝐸 gcd 𝑅) = 1)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → 𝐷 ∼ 𝐹)    &   (𝜑 → 𝐸 ∼ 𝐹)    ⇒   (𝜑 → (𝐷 · 𝐸) ∼ 𝐹)
 
26-Apr-2025primrootscoprbij2 42053 A bijection between coprime powers of primitive roots and primitive roots. (Contributed by metakunt, 26-Apr-2025.)
𝐹 = (𝑚 ∈ (𝑅 PrimRoots 𝐾) ↦ (𝐼(.g‘𝑅)𝑚))    &   (𝜑 → 𝑅 ∈ CMnd)    &   (𝜑 → 𝐾 ∈ ℕ)    &   (𝜑 → 𝐼 ∈ ℕ)    &   (𝜑 → (𝐼 gcd 𝐾) = 1)    ⇒   (𝜑 → 𝐹:(𝑅 PrimRoots 𝐾)–1-1-onto→(𝑅 PrimRoots 𝐾))
 
26-Apr-2025primrootscoprbij 42052 A bijection between coprime powers of primitive roots and primitive roots. (Contributed by metakunt, 26-Apr-2025.)
𝐹 = (𝑚 ∈ (𝑅 PrimRoots 𝐾) ↦ (𝐼(.g‘𝑅)𝑚))    &   (𝜑 → 𝑅 ∈ CMnd)    &   (𝜑 → 𝐾 ∈ ℕ)    &   (𝜑 → 𝐼 ∈ ℕ)    &   (𝜑 → 𝐽 ∈ ℕ)    &   (𝜑 → 𝑍 ∈ ℤ)    &   (𝜑 → 1 = ((𝐼 · 𝐽) + (𝐾 · 𝑍)))    &   𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)}    ⇒   (𝜑 → 𝐹:(𝑅 PrimRoots 𝐾)–1-1-onto→(𝑅 PrimRoots 𝐾))
 
26-Apr-2025primrootscoprf 42051 Coprime powers of primitive roots are primitive roots, as a function. (Contributed by metakunt, 26-Apr-2025.)
𝐹 = (𝑚 ∈ (𝑅 PrimRoots 𝐾) ↦ (𝐸(.g‘𝑅)𝑚))    &   (𝜑 → 𝑅 ∈ CMnd)    &   (𝜑 → 𝐾 ∈ ℕ)    &   (𝜑 → 𝐸 ∈ ℕ)    &   (𝜑 → (𝐸 gcd 𝐾) = 1)    ⇒   (𝜑 → 𝐹:(𝑅 PrimRoots 𝐾)⟶(𝑅 PrimRoots 𝐾))
 
26-Apr-2025posbezout 42050 Bezout's identity restricted on positive integers in all but one variable. (Contributed by metakunt, 26-Apr-2025.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℤ (𝐴 gcd 𝐵) = ((𝐴 · 𝑥) + (𝐵 · 𝑦)))
 
26-Apr-2025df-frac 33262 Define the field of fractions of a given integral domain. (Contributed by Thierry Arnoux, 26-Apr-2025.)
Frac = (𝑟 ∈ V ↦ (𝑟 RLocal (RLReg‘𝑟)))
 
26-Apr-2025fsuppssov1 9447 Formula building theorem for finite support: operator with left annihilator. Finite support version of suppssov1 8232. (Contributed by SN, 26-Apr-2025.)
(𝜑 → (𝑥 ∈ 𝐷 ↦ 𝐴) finSupp 𝑌)    &   ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)    &   ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑅)    &   (𝜑 → 𝑍 ∈ 𝑊)    ⇒   (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) finSupp 𝑍)
 
25-Apr-2025psrmnd 42493 The ring of power series is a monoid. (Contributed by SN, 25-Apr-2025.)
𝑆 = (𝐼 mPwSer 𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ Mnd)    ⇒   (𝜑 → 𝑆 ∈ Mnd)
 
25-Apr-2025rxp11d 42329 Real exponentiation is one-to-one with respect to the first argument. (Contributed by SN, 25-Apr-2025.)
(𝜑 → 𝐴 ∈ ℝ+)    &   (𝜑 → 𝐵 ∈ ℝ+)    &   (𝜑 → 𝐶 ∈ ℝ)    &   (𝜑 → 𝐶 ≠ 0)    &   (𝜑 → (𝐴↑𝑐𝐶) = (𝐵↑𝑐𝐶))    ⇒   (𝜑 → 𝐴 = 𝐵)
 
25-Apr-2025rplog11d 42328 The natural logarithm is one-to-one on positive reals. (Contributed by SN, 25-Apr-2025.)
(𝜑 → 𝐴 ∈ ℝ+)    &   (𝜑 → 𝐵 ∈ ℝ+)    ⇒   (𝜑 → ((log‘𝐴) = (log‘𝐵) ↔ 𝐴 = 𝐵))
 
25-Apr-2025log11d 42327 The natural logarithm is one-to-one. (Contributed by SN, 25-Apr-2025.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐴 ≠ 0)    &   (𝜑 → 𝐵 ≠ 0)    ⇒   (𝜑 → ((log‘𝐴) = (log‘𝐵) ↔ 𝐴 = 𝐵))
 
25-Apr-2025rxp112d 42326 Real exponentiation is one-to-one with respect to the second argument. (TODO: Note that the base 𝐶 must be positive since -𝐶↑𝐴 is 𝐶↑𝐴 · e↑iπ𝐴, so in the negative case 𝐴 = 𝐵 + 2𝑘). (Contributed by SN, 25-Apr-2025.)
(𝜑 → 𝐶 ∈ ℝ+)    &   (𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐶 ≠ 1)    &   (𝜑 → (𝐶↑𝑐𝐴) = (𝐶↑𝑐𝐵))    ⇒   (𝜑 → 𝐴 = 𝐵)
 
25-Apr-2025logne0d 42325 Deduction form of logne0 26631. See logccne0d 42321 for a more general version. (Contributed by SN, 25-Apr-2025.)
(𝜑 → 𝐴 ∈ ℝ+)    &   (𝜑 → 𝐴 ≠ 1)    ⇒   (𝜑 → (log‘𝐴) ≠ 0)
 
25-Apr-2025cxpi11d 42324 i to the powers of 𝐴 and 𝐵 are equal iff 𝐴 and 𝐵 are a multiple of 4 apart. EDITORIAL: This theorem may be revised to a more convenient form. (Contributed by SN, 25-Apr-2025.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    ⇒   (𝜑 → ((i↑𝑐𝐴) = (i↑𝑐𝐵) ↔ ∃𝑛 ∈ ℤ 𝐴 = (𝐵 + (4 · 𝑛))))
 
25-Apr-2025cxp111d 42323 General condition for complex exponentiation to be one-to-one with respect to the first argument. (Contributed by SN, 25-Apr-2025.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐴 ≠ 0)    &   (𝜑 → 𝐵 ≠ 0)    &   (𝜑 → 𝐶 ≠ 0)    ⇒   (𝜑 → ((𝐴↑𝑐𝐶) = (𝐵↑𝑐𝐶) ↔ ∃𝑛 ∈ ℤ (log‘𝐴) = ((log‘𝐵) + (((i · (2 · π)) · 𝑛) / 𝐶))))
 
25-Apr-2025cxp112d 42322 General condition for complex exponentiation to be one-to-one with respect to the second argument. (Contributed by SN, 25-Apr-2025.)
(𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ≠ 0)    &   (𝜑 → 𝐶 ≠ 1)    ⇒   (𝜑 → ((𝐶↑𝑐𝐴) = (𝐶↑𝑐𝐵) ↔ ∃𝑛 ∈ ℤ 𝐴 = (𝐵 + (((i · (2 · π)) · 𝑛) / (log‘𝐶)))))
 
25-Apr-2025logccne0d 42321 The logarithm isn't 0 if its argument isn't 0 or 1, deduction form. (Contributed by SN, 25-Apr-2025.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐴 ≠ 0)    &   (𝜑 → 𝐴 ≠ 1)    ⇒   (𝜑 → (log‘𝐴) ≠ 0)
 
25-Apr-2025ef11d 42320 General condition for the exponential function to be one-to-one. efper 26531 shows that exponentiation is periodic. (Contributed by SN, 25-Apr-2025.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    ⇒   (𝜑 → ((exp‘𝐴) = (exp‘𝐵) ↔ ∃𝑛 ∈ ℤ 𝐴 = (𝐵 + ((i · (2 · π)) · 𝑛))))
 
25-Apr-2025efsubd 42319 Difference of exponents law for exponential function, deduction form. (Contributed by SN, 25-Apr-2025.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    ⇒   (𝜑 → (exp‘(𝐴 − 𝐵)) = ((exp‘𝐴) / (exp‘𝐵)))
 
25-Apr-2025efne0d 42318 The exponential of a complex number is nonzero, deduction form. EDITORIAL: Using efne0d 42318 in efne0 16139 is shorter than vice versa. (Contributed by SN, 25-Apr-2025.)
(𝜑 → 𝐴 ∈ ℂ)    ⇒   (𝜑 → (exp‘𝐴) ≠ 0)
 
25-Apr-2025zdivgd 42317 Two ways to express "𝑁 is an integer multiple of 𝑀". Originally a subproof of zdiv 12707. (Contributed by SN, 25-Apr-2025.)
(𝜑 → 𝑀 ∈ ℂ)    &   (𝜑 → 𝑁 ∈ ℂ)    &   (𝜑 → 𝑀 ≠ 0)    ⇒   (𝜑 → (∃𝑘 ∈ ℤ (𝑀 · 𝑘) = 𝑁 ↔ (𝑁 / 𝑀) ∈ ℤ))
 
25-Apr-2025retire 42301 A real times i is real iff the real is zero. (Contributed by SN, 25-Apr-2025.)
(𝑅 ∈ ℝ → ((𝑅 · i) ∈ ℝ ↔ 𝑅 = 0))
 
25-Apr-2025itrere 42300 i times a real is real iff the real is zero. (Contributed by SN, 25-Apr-2025.)
(𝑅 ∈ ℝ → ((i · 𝑅) ∈ ℝ ↔ 𝑅 = 0))
 
25-Apr-20251tiei 42299 1 times i equals i. (Contributed by SN, 25-Apr-2025.)
(1 · i) = i
 
25-Apr-2025it1ei 42298 i times 1 equals i. (Contributed by SN, 25-Apr-2025.)
(i · 1) = i
 
25-Apr-20250tie0 42297 0 times i equals 0. (Contributed by SN, 25-Apr-2025.)
(0 · i) = 0
 
25-Apr-2025ine1 42296 i is not 1. (Contributed by SN, 25-Apr-2025.)
i ≠ 1
 
25-Apr-2025pine0 42295 π is nonzero. (Contributed by SN, 25-Apr-2025.)
π ≠ 0
 
25-Apr-2025readdrcl2d 42255 Reverse closure for addition: the second addend is real if the first addend is real and the sum is real. (Contributed by SN, 25-Apr-2025.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 + 𝐵) ∈ ℝ)    ⇒   (𝜑 → 𝐵 ∈ ℝ)
 
25-Apr-2025aks6d1c1p4 42061 The product of polynomials is introspective. (Contributed by metakunt, 25-Apr-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}    &   𝑆 = (Poly1‘𝐾)    &   𝐵 = (Base‘𝑆)    &   𝑋 = (var1‘𝐾)    &   𝑊 = (mulGrp‘𝑆)    &   𝑉 = (mulGrp‘𝐾)    &    ↑ = (.g‘𝑉)    &   𝐶 = (algSc‘𝑆)    &   𝐷 = (.g‘𝑊)    &   𝑃 = (chr‘𝐾)    &   𝑂 = (eval1‘𝐾)    &    + = (+g‘𝑆)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   (𝜑 → 𝐸 ∼ 𝐹)    &   (𝜑 → 𝐸 ∼ 𝐺)    ⇒   (𝜑 → 𝐸 ∼ (𝐹(+g‘𝑊)𝐺))
 
25-Apr-2025aks6d1c1p3 42060 In a field with a Frobenius isomorphism (read: algebraic closure or finite field), 𝑁 and linear factors are introspective. (Contributed by metakunt, 25-Apr-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}    &   𝑆 = (Poly1‘𝐾)    &   𝐵 = (Base‘𝑆)    &   𝑋 = (var1‘𝐾)    &   𝑊 = (mulGrp‘𝑆)    &   𝑉 = (mulGrp‘𝐾)    &    ↑ = (.g‘𝑉)    &   𝐶 = (algSc‘𝑆)    &   𝐷 = (.g‘𝑊)    &   𝑃 = (chr‘𝐾)    &   𝑂 = (eval1‘𝐾)    &    + = (+g‘𝑆)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))    &   (𝜑 → 𝐴 ∈ ℤ)    &   (𝜑 → 𝑁 ∼ 𝐹)    &   (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 ↑ 𝑥)) ∈ (𝐾 RingIso 𝐾))    ⇒   (𝜑 → (𝑁 / 𝑃) ∼ 𝐹)
 
25-Apr-2025aks6d1c1p2 42059 𝑃 and linear factors are introspective. (Contributed by metakunt, 25-Apr-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒 ↑ 𝑦)))}    &   𝑆 = (Poly1‘𝐾)    &   𝐵 = (Base‘𝑆)    &   𝑋 = (var1‘𝐾)    &   𝑊 = (mulGrp‘𝑆)    &   𝑉 = (mulGrp‘𝐾)    &    ↑ = (.g‘𝑉)    &   𝐶 = (algSc‘𝑆)    &   𝐷 = (.g‘𝑊)    &   𝑃 = (chr‘𝐾)    &   𝑂 = (eval1‘𝐾)    &    + = (+g‘𝑆)    &   (𝜑 → 𝐾 ∈ Field)    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → (𝑁 gcd 𝑅) = 1)    &   (𝜑 → 𝑃 ∥ 𝑁)    &   𝐹 = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝐴)))    &   (𝜑 → 𝐴 ∈ ℤ)    ⇒   (𝜑 → 𝑃 ∼ 𝐹)
 
25-Apr-2025aks6d1c1p1rcl 42058 Reverse closure of the introspective relation. (Contributed by metakunt, 25-Apr-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)))}    &   (𝜑 → 𝐸 ∼ 𝐹)    ⇒   (𝜑 → (𝐸 ∈ ℕ ∧ 𝐹 ∈ 𝐵))
 
25-Apr-2025aks6d1c1p1 42057 Definition of the introspective relation. (Contributed by metakunt, 25-Apr-2025.)
∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ 𝐵 ∧ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝑒 ↑ ((𝑂‘𝑓)‘𝑦)) = ((𝑂‘𝑓)‘(𝑒𝐷𝑦)))}    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐸 ∈ ℕ)    ⇒   (𝜑 → (𝐸 ∼ 𝐹 ↔ ∀𝑦 ∈ (𝐾 PrimRoots 𝑅)(𝐸 ↑ ((𝑂‘𝐹)‘𝑦)) = ((𝑂‘𝐹)‘(𝐸𝐷𝑦))))
 
25-Apr-2025primrootscoprmpow 42049 Coprime powers of primitive roots are primitive roots. (Contributed by metakunt, 25-Apr-2025.)
(𝜑 → 𝑅 ∈ CMnd)    &   (𝜑 → 𝐾 ∈ ℕ)    &   (𝜑 → 𝐸 ∈ ℕ)    &   (𝜑 → (𝐸 gcd 𝐾) = 1)    &   (𝜑 → 𝑀 ∈ (𝑅 PrimRoots 𝐾))    &   𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)}    ⇒   (𝜑 → (𝐸(.g‘𝑅)𝑀) ∈ (𝑅 PrimRoots 𝐾))
 
25-Apr-2025primrootsunit 42048 Primitive roots have left inverses. (Contributed by metakunt, 25-Apr-2025.)
(𝜑 → 𝑅 ∈ CMnd)    &   (𝜑 → 𝐾 ∈ ℕ)    &   𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)}    ⇒   (𝜑 → ((𝑅 PrimRoots 𝐾) = ((𝑅 ↾s 𝑈) PrimRoots 𝐾) ∧ (𝑅 ↾s 𝑈) ∈ Abel))
 
25-Apr-2025primrootsunit1 42047 Primitive roots have left inverses. (Contributed by metakunt, 25-Apr-2025.)
(𝜑 → 𝑅 ∈ CMnd)    &   (𝜑 → 𝐾 ∈ ℕ)    &   𝑈 = {𝑎 ∈ (Base‘𝑅) ∣ ∃𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑎) = (0g‘𝑅)}    ⇒   (𝜑 → ((𝑅 PrimRoots 𝐾) = ((𝑅 ↾s 𝑈) PrimRoots 𝐾) ∧ (𝑅 ↾s 𝑈) ∈ Abel))
 
25-Apr-2025linvh 42046 If an element has a unique left inverse, then the value satisfies the left inverse value equation. (Contributed by metakunt, 25-Apr-2025.)
(𝜑 → 𝑋 ∈ (Base‘𝑅))    &   (𝜑 → ∃!𝑖 ∈ (Base‘𝑅)(𝑖(+g‘𝑅)𝑋) = (0g‘𝑅))    ⇒   (𝜑 → (((invg‘𝑅)‘𝑋)(+g‘𝑅)𝑋) = (0g‘𝑅))
 
25-Apr-2025mndmolinv 42045 An element of a monoid that has a right inverse has at most one left inverse. (Contributed by metakunt, 25-Apr-2025.)
𝐵 = (Base‘𝑀)    &   (𝜑 → 𝑀 ∈ Mnd)    &   (𝜑 → 𝐴 ∈ 𝐵)    &   (𝜑 → ∃𝑥 ∈ 𝐵 (𝐴(+g‘𝑀)𝑥) = (0g‘𝑀))    ⇒   (𝜑 → ∃*𝑥 ∈ 𝐵 (𝑥(+g‘𝑀)𝐴) = (0g‘𝑀))
 
25-Apr-2025isprimroot 42043 The value of a primitive root. (Contributed by metakunt, 25-Apr-2025.)
(𝜑 → 𝑅 ∈ CMnd)    &   (𝜑 → 𝐾 ∈ ℕ0)    &    ↑ = (.g‘𝑅)    ⇒   (𝜑 → (𝑀 ∈ (𝑅 PrimRoots 𝐾) ↔ (𝑀 ∈ (Base‘𝑅) ∧ (𝐾 ↑ 𝑀) = (0g‘𝑅) ∧ ∀𝑙 ∈ ℕ0 ((𝑙 ↑ 𝑀) = (0g‘𝑅) → 𝐾 ∥ 𝑙))))
 
25-Apr-2025df-primroots 42042 A 𝑟-th primitive root is a root of unity such that the exponent divides 𝑟. (Contributed by metakunt, 25-Apr-2025.)
PrimRoots = (𝑟 ∈ CMnd, 𝑘 ∈ ℕ0 ↦ ⦋(Base‘𝑟) / 𝑏⦌{𝑎 ∈ 𝑏 ∣ ((𝑘(.g‘𝑟)𝑎) = (0g‘𝑟) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘𝑟)𝑎) = (0g‘𝑟) → 𝑘 ∥ 𝑙))})
 
25-Apr-2025cprimroots 42041 Define the class of primitive roots. (Contributed by metakunt, 25-Apr-2025.)
class PrimRoots
 
25-Apr-2025minplyann 33694 The minimal polynomial for 𝐴 annihilates 𝐴 (Contributed by Thierry Arnoux, 25-Apr-2025.)
𝑂 = (𝐸 evalSub1 𝐹)    &   𝑃 = (Poly1‘(𝐸 ↾s 𝐹))    &   𝐵 = (Base‘𝐸)    &   (𝜑 → 𝐸 ∈ Field)    &   (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))    &   (𝜑 → 𝐴 ∈ 𝐵)    &    0 = (0g‘𝐸)    &   𝑀 = (𝐸 minPoly 𝐹)    ⇒   (𝜑 → ((𝑂‘(𝑀‘𝐴))‘𝐴) = 0 )
 
25-Apr-2025ply1unit 33557 In a field 𝐹, a polynomial 𝐶 is a unit iff it has degree 0. This corresponds to the nonzero scalars, see ply1asclunit 33556. (Contributed by Thierry Arnoux, 25-Apr-2025.)
𝑃 = (Poly1‘𝐹)    &   𝐴 = (algSc‘𝑃)    &   𝐵 = (Base‘𝐹)    &    0 = (0g‘𝐹)    &   (𝜑 → 𝐹 ∈ Field)    &   𝐷 = (deg1‘𝐹)    &   (𝜑 → 𝐶 ∈ (Base‘𝑃))    ⇒   (𝜑 → (𝐶 ∈ (Unit‘𝑃) ↔ (𝐷‘𝐶) = 0))
 
25-Apr-2025evls1subd 33554 Univariate polynomial evaluation of a difference of polynomials. (Contributed by Thierry Arnoux, 25-Apr-2025.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝑊 = (Poly1‘𝑈)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝐵 = (Base‘𝑊)    &   𝐷 = (-g‘𝑊)    &    − = (-g‘𝑆)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝑀 ∈ 𝐵)    &   (𝜑 → 𝑁 ∈ 𝐵)    &   (𝜑 → 𝐶 ∈ 𝐾)    ⇒   (𝜑 → ((𝑄‘(𝑀𝐷𝑁))‘𝐶) = (((𝑄‘𝑀)‘𝐶) − ((𝑄‘𝑁)‘𝐶)))
 
25-Apr-2025unitnz 33211 In a nonzero ring, a unit cannot be zero. (Contributed by Thierry Arnoux, 25-Apr-2025.)
𝑈 = (Unit‘𝑅)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ NzRing)    &   (𝜑 → 𝑋 ∈ 𝑈)    ⇒   (𝜑 → 𝑋 ≠ 0 )
 
25-Apr-2025psdascl 22187 The derivative of a constant polynomial is zero. (Contributed by SN, 25-Apr-2025.)
𝑆 = (𝐼 mPwSer 𝑅)    &    0 = (0g‘𝑆)    &   𝐴 = (algSc‘𝑆)    &   𝐵 = (Base‘𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑋 ∈ 𝐼)    &   (𝜑 → 𝐶 ∈ 𝐵)    ⇒   (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝐴‘𝐶)) = 0 )
 
25-Apr-2025psd1 22186 The derivative of one is zero. (Contributed by SN, 25-Apr-2025.)
𝑆 = (𝐼 mPwSer 𝑅)    &    1 = (1r‘𝑆)    &    0 = (0g‘𝑆)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑋 ∈ 𝐼)    ⇒   (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘ 1 ) = 0 )
 
25-Apr-2025psdmul 22185 Product rule for power series. An outline is available at https://github.com/icecream17/Stuff/blob/main/math/psdmul.pdf. (Contributed by SN, 25-Apr-2025.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    + = (+g‘𝑆)    &    · = (.r‘𝑆)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ CRing)    &   (𝜑 → 𝑋 ∈ 𝐼)    &   (𝜑 → 𝐹 ∈ 𝐵)    &   (𝜑 → 𝐺 ∈ 𝐵)    ⇒   (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘(𝐹 · 𝐺)) = (((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) · 𝐺) + (𝐹 · (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐺))))
 
25-Apr-2025psrasclcl 22016 A scalar is lifted into a member of the power series. (Contributed by SN, 25-Apr-2025.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑆)    &   (𝜑 → 𝐼 ∈ 𝑊)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐶 ∈ 𝐾)    ⇒   (𝜑 → (𝐴‘𝐶) ∈ 𝐵)
 
23-Apr-2025mptcnfimad 8021 The converse of a mapping of subsets to their image of a bijection. (Contributed by AV, 23-Apr-2025.)
𝑀 = (𝑥 ∈ 𝐴 ↦ (𝐹 “ 𝑥))    &   (𝜑 → 𝐹:𝑉–1-1-onto→𝑊)    &   (𝜑 → 𝐴 ⊆ 𝒫 𝑉)    &   (𝜑 → ran 𝑀 ⊆ 𝒫 𝑊)    &   (𝜑 → 𝑉 ∈ 𝑈)    ⇒   (𝜑 → ◡𝑀 = (𝑦 ∈ ran 𝑀 ↦ (◡𝐹 “ 𝑦)))
 
22-Apr-2025feldmfvelcdm 7115 A class is an element of the domain iff it's function value is an element of the codomain of a function. (Contributed by AV, 22-Apr-2025.)
((𝐹:𝐴⟶𝐵 ∧ ∅ ∉ 𝐵) → (𝑋 ∈ 𝐴 ↔ (𝐹‘𝑋) ∈ 𝐵))
 
21-Apr-2025isuspgrim0 47746 An isomorphism of simple pseudographs is a bijection between their vertices which induces a bijection between their edges. (Contributed by AV, 21-Apr-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (Edg‘𝐻)    ⇒   ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ (𝑒 ∈ 𝐸 ↦ (𝐹 “ 𝑒)):𝐸–1-1-onto→𝐷)))
 
21-Apr-2025isuspgrim0lem 47745 An isomorphism of simple pseudographs is a bijection between their vertices which induces a bijection between their edges. (Contributed by AV, 21-Apr-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (Edg‘𝐻)    &   𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   𝑀 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥))    &   𝑁 = (𝑥 ∈ dom 𝐼 ↦ (◡𝐽‘(𝑀‘(𝐼‘𝑥))))    ⇒   ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ 𝑋) ∧ 𝐹:𝑉–1-1-onto→𝑊) ∧ 𝑀:𝐸–1-1-onto→𝐷) → (𝑁:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝐽‘(𝑁‘𝑖)) = (𝐹 “ (𝐼‘𝑖))))
 
21-Apr-2025uspgruhgr 29211 An undirected simple pseudograph is an undirected hypergraph. (Contributed by AV, 21-Apr-2025.)
(𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
 
20-Apr-2025uspgriedgedg 29203 In a simple pseudograph, for each indexed edge there is exactly one edge. (Contributed by AV, 20-Apr-2025.)
𝐸 = (Edg‘𝐺)    &   𝐼 = (iEdg‘𝐺)    ⇒   ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ dom 𝐼) → ∃!𝑘 ∈ 𝐸 𝑘 = (𝐼‘𝑋))
 
20-Apr-2025uspgredgiedg 29202 In a simple pseudograph, for each edge there is exactly one indexed edge. (Contributed by AV, 20-Apr-2025.)
𝐸 = (Edg‘𝐺)    &   𝐼 = (iEdg‘𝐺)    ⇒   ((𝐺 ∈ USPGraph ∧ 𝐾 ∈ 𝐸) → ∃!𝑥 ∈ dom 𝐼 𝐾 = (𝐼‘𝑥))
 
20-Apr-2025elovmpod 7688 Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.) Variant of elovmpo 7689 in deduction form. (Revised by AV, 20-Apr-2025.)
𝑂 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐷 ∈ 𝑉)    &   ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → 𝐶 = 𝐷)    ⇒   (𝜑 → (𝐸 ∈ (𝑋𝑂𝑌) ↔ 𝐸 ∈ 𝐷))
 
20-Apr-2025fdmeu 6973 There is exactly one codomain element for each element of the domain of a function. (Contributed by AV, 20-Apr-2025.)
((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 (𝐹‘𝑋) = 𝑦)
 
19-Apr-2025isgrim 47742 An isomorphism of graphs is a bijection between their vertices that preserves adjacency. (Contributed by AV, 19-Apr-2025.)
𝑉 = (Vtx‘𝐺)    &   𝑊 = (Vtx‘𝐻)    &   𝐸 = (iEdg‘𝐺)    &   𝐷 = (iEdg‘𝐻)    ⇒   ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom 𝐸–1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖))))))
 
19-Apr-2025df-gric 47741 Two graphs are said to be isomorphic iff they are connected by at least one isomorphism, see definition in [Diestel] p. 3 and definition in [Bollobas] p. 3. Isomorphic graphs share all global graph properties like order and size. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 19-Apr-2025.)
≃𝑔𝑟 = (◡ GraphIso “ (V ∖ 1o))
 
19-Apr-2025df-grim 47738 An isomorphism between two graphs is a bijection between the sets of vertices of the two graphs that preserves adjacency, see definition in [Diestel] p. 3. (Contributed by AV, 19-Apr-2025.)
GraphIso = (𝑔 ∈ V, ℎ ∈ V ↦ {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∃𝑗[(iEdg‘𝑔) / 𝑒][(iEdg‘ℎ) / 𝑑](𝑗:dom 𝑒–1-1-onto→dom 𝑑 ∧ ∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖))))})
 
19-Apr-2025fprodcnlem 45510 A finite product of functions to complex numbers from a common topological space is continuous. Induction step. (Contributed by Glauco Siliprandi, 8-Apr-2021.) Avoid ax-mulf 11258. (Revised by GG, 19-Apr-2025.)
Ⅎ𝑘𝜑    &   𝐾 = (TopOpen‘ℂfld)    &   (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   (𝜑 → 𝐴 ∈ Fin)    &   ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))    &   (𝜑 → 𝑍 ⊆ 𝐴)    &   (𝜑 → 𝑊 ∈ (𝐴 ∖ 𝑍))    &   (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑍 𝐵) ∈ (𝐽 Cn 𝐾))    ⇒   (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵) ∈ (𝐽 Cn 𝐾))
 
19-Apr-2025knoppcnlem10 36461 Lemma for knoppcn 36463. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) Avoid ax-mulf 11258. (Revised by GG, 19-Apr-2025.)
𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))    &   𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℝ)    &   (𝜑 → 𝑀 ∈ ℕ0)    ⇒   (𝜑 → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑀)) ∈ ((topGen‘ran (,)) Cn (TopOpen‘ℂfld)))
 
19-Apr-2025cvxsconn 35203 A convex subset of the complex numbers is simply connected. (Contributed by Mario Carneiro, 12-Feb-2015.) Avoid ax-mulf 11258. (Revised by GG, 19-Apr-2025.)
(𝜑 → 𝑆 ⊆ ℂ)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)    &   𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽 ↾t 𝑆)    ⇒   (𝜑 → 𝐾 ∈ SConn)
 
19-Apr-2025cvxpconn 35202 A convex subset of the complex numbers is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.) Avoid ax-mulf 11258. (Revised by GG, 19-Apr-2025.)
(𝜑 → 𝑆 ⊆ ℂ)    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑡 ∈ (0[,]1))) → ((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦)) ∈ 𝑆)    &   𝐽 = (TopOpen‘ℂfld)    &   𝐾 = (𝐽 ↾t 𝑆)    ⇒   (𝜑 → 𝐾 ∈ PConn)
 
19-Apr-2025seqn0sfn 28367 The surreal sequence builder is a function over ℕ0s when started from zero. (Contributed by Scott Fenton, 19-Apr-2025.)
(𝜑 → seqs 0s ( + , 𝐹) Fn ℕ0s)
 
19-Apr-2025nnsrecgt0d 28366 The reciprocal of a positive surreal integer is positive. (Contributed by Scott Fenton, 19-Apr-2025.)
(𝜑 → 𝐴 ∈ ℕs)    ⇒   (𝜑 → 0s <s ( 1s /su 𝐴))
 
19-Apr-2025seqsp1 28327 The value of the surreal sequence builder at a successor. (Contributed by Scott Fenton, 19-Apr-2025.)
(𝜑 → 𝑀 ∈ No )    &   (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω))    &   (𝜑 → 𝑁 ∈ 𝑍)    ⇒   (𝜑 → (seqs𝑀( + , 𝐹)‘(𝑁 +s 1s )) = ((seqs𝑀( + , 𝐹)‘𝑁) + (𝐹‘(𝑁 +s 1s ))))
 
19-Apr-2025seqs1 28326 The value of the surreal sequence bulder function at its initial value. (Contributed by Scott Fenton, 19-Apr-2025.)
(𝜑 → 𝑀 ∈ No )    ⇒   (𝜑 → (seqs𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀))
 
19-Apr-2025seqsfn 28325 The surreal sequence builder is a function. (Contributed by Scott Fenton, 19-Apr-2025.)
(𝜑 → 𝑀 ∈ No )    &   (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝑀) “ ω))    ⇒   (𝜑 → seqs𝑀( + , 𝐹) Fn 𝑍)
 
19-Apr-2025noseqrdgsuc 28324 Successor value of a recursive definition generator on surreal sequences. (Contributed by Scott Fenton, 19-Apr-2025.)
(𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))    &   (𝜑 → 𝑆 = ran 𝑅)    ⇒   ((𝜑 ∧ 𝐵 ∈ 𝑍) → (𝑆‘(𝐵 +s 1s )) = (𝐵𝐹(𝑆‘𝐵)))
 
19-Apr-2025efrlim 27022 The limit of the sequence (1 + 𝐴 / 𝑘)↑𝑘 is the exponential function. This is often taken as an alternate definition of the exponential function (see also dfef2 27024). (Contributed by Mario Carneiro, 1-Mar-2015.) Avoid ax-mulf 11258. (Revised by GG, 19-Apr-2025.)
𝑆 = (0(ball‘(abs ∘ − ))(1 / ((abs‘𝐴) + 1)))    ⇒   (𝐴 ∈ ℂ → (𝑘 ∈ ℝ+ ↦ ((1 + (𝐴 / 𝑘))↑𝑐𝑘)) ⇝𝑟 (exp‘𝐴))
 
19-Apr-2025taylthlem2 26426 Lemma for taylth 26428. (Contributed by Mario Carneiro, 1-Jan-2017.) Avoid ax-mulf 11258. (Revised by GG, 19-Apr-2025.)
(𝜑 → 𝐹:𝐴⟶ℝ)    &   (𝜑 → 𝐴 ⊆ ℝ)    &   (𝜑 → dom ((ℝ D𝑛 𝐹)‘𝑁) = 𝐴)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ 𝐴)    &   𝑇 = (𝑁(ℝ Tayl 𝐹)𝐵)    &   (𝜑 → 𝑀 ∈ (1..^𝑁))    &   (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) / ((𝑥 − 𝐵)↑𝑀))) limℂ 𝐵))    ⇒   (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑀 + 1)))) limℂ 𝐵))
 
19-Apr-2025abex 5344 Conditions for a class abstraction to be a set. Remark: This proof is shorter than a proof using abexd 5343. (Contributed by AV, 19-Apr-2025.)
(𝜑 → 𝑥 ∈ 𝐴)    &   𝐴 ∈ V    ⇒   {𝑥 ∣ 𝜑} ∈ V
 
19-Apr-2025abexd 5343 Conditions for a class abstraction to be a set, deduction form. (Contributed by AV, 19-Apr-2025.)
((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴)    &   (𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → {𝑥 ∣ 𝜓} ∈ V)
 
18-Apr-2025mpomulnzcnf 36257 Multiplication maps nonzero complex numbers to nonzero complex numbers. Version of mulnzcnf 11930 using maps-to notation, which does not require ax-mulf 11258. (Contributed by GG, 18-Apr-2025.)
(𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 · 𝑦)):((ℂ ∖ {0}) × (ℂ ∖ {0}))⟶(ℂ ∖ {0})
 
18-Apr-2025n0ssold 28365 The non-negative surreal integers are a subset of the old set of ω. (Contributed by Scott Fenton, 18-Apr-2025.)
ℕ0s ⊆ ( O ‘ω)
 
18-Apr-2025n0sbday 28364 A non-negative surreal integer has a finite birthday. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝐴 ∈ ℕ0s → ( bday ‘𝐴) ∈ ω)
 
18-Apr-2025noseqrdg0 28323 Initial value of a recursive definition generator on surreal sequences. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))    &   (𝜑 → 𝑆 = ran 𝑅)    ⇒   (𝜑 → (𝑆‘𝐶) = 𝐴)
 
18-Apr-2025noseqrdgfn 28322 The recursive definition generator on surreal sequences is a function. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))    &   (𝜑 → 𝑆 = ran 𝑅)    ⇒   (𝜑 → 𝑆 Fn 𝑍)
 
18-Apr-2025noseqrdglem 28321 A helper lemma for the value of a recursive defintion generator on surreal sequences. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))    ⇒   ((𝜑 ∧ 𝐵 ∈ 𝑍) → ⟨𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))⟩ ∈ ran 𝑅)
 
18-Apr-2025om2noseqrdg 28320 A helper lemma for the value of a recursive definition generator on a surreal sequence with characteristic function 𝐹(𝑥, 𝑦) and initial value 𝐴. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩) ↾ ω))    ⇒   ((𝜑 ∧ 𝐵 ∈ ω) → (𝑅‘𝐵) = ⟨(𝐺‘𝐵), (2nd ‘(𝑅‘𝐵))⟩)
 
18-Apr-2025om2noseqoi 28319 An alternative definition of 𝐺 in terms of df-oi 9573. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    ⇒   (𝜑 → 𝐺 = OrdIso( <s , 𝑍))
 
18-Apr-2025om2noseqiso 28318 𝐺 is an isomorphism from the finite ordinals to a surreal sequence. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    ⇒   (𝜑 → 𝐺 Isom E , <s (ω, 𝑍))
 
18-Apr-2025om2noseqf1o 28317 𝐺 is a bijection. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    ⇒   (𝜑 → 𝐺:ω–1-1-onto→𝑍)
 
18-Apr-2025om2noseqlt2 28316 The mapping 𝐺 preserves order. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    ⇒   ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 ∈ 𝐵 ↔ (𝐺‘𝐴) <s (𝐺‘𝐵)))
 
18-Apr-2025om2noseqlt 28315 Surreal less-than relation for 𝐺. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    ⇒   ((𝜑 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐴 ∈ 𝐵 → (𝐺‘𝐴) <s (𝐺‘𝐵)))
 
18-Apr-2025om2noseqfo 28314 Function statement for 𝐺. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) “ ω))    ⇒   (𝜑 → 𝐺:ω–onto→𝑍)
 
18-Apr-2025om2noseqsuc 28313 The value of 𝐺 at a successor. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    &   (𝜑 → 𝐴 ∈ ω)    ⇒   (𝜑 → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) +s 1s ))
 
18-Apr-2025om2noseq0 28312 The mapping 𝐺 is a one-to-one mapping from ω onto a countable sequence of surreals that will be used to show the properties of seqs. This theorem shows the value of 𝐺 at ordinal zero. Compare the series of theorems starting at om2uz0i 13992. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝐶 ∈ No )    &   (𝜑 → 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐶) ↾ ω))    ⇒   (𝜑 → (𝐺‘∅) = 𝐶)
 
18-Apr-2025noseqno 28311 An element of a surreal sequence is a surreal. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))    &   (𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ 𝑍)    ⇒   (𝜑 → 𝐵 ∈ No )
 
18-Apr-2025noseqssno 28310 A surreal sequence is a subset of the surreals. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))    &   (𝜑 → 𝐴 ∈ No )    ⇒   (𝜑 → 𝑍 ⊆ No )
 
18-Apr-2025noseqinds 28309 Induction schema for surreal sequences. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))    &   (𝜑 → 𝐴 ∈ No )    &   (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))    &   (𝑦 = 𝑧 → (𝜓 ↔ 𝜃))    &   (𝑦 = (𝑧 +s 1s ) → (𝜓 ↔ 𝜏))    &   (𝑦 = 𝐵 → (𝜓 ↔ 𝜂))    &   (𝜑 → 𝜒)    &   ((𝜑 ∧ 𝑧 ∈ 𝑍) → (𝜃 → 𝜏))    ⇒   ((𝜑 ∧ 𝐵 ∈ 𝑍) → 𝜂)
 
18-Apr-2025noseqind 28308 Peano's inductive postulate for surreal sequences. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))    &   (𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐴 ∈ 𝐵)    &   ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦 +s 1s ) ∈ 𝐵)    ⇒   (𝜑 → 𝑍 ⊆ 𝐵)
 
18-Apr-2025noseqp1 28307 One plus an element of 𝑍 is an element of 𝑍. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))    &   (𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ 𝑍)    ⇒   (𝜑 → (𝐵 +s 1s ) ∈ 𝑍)
 
18-Apr-2025noseq0 28306 The surreal 𝐴 is a member of the sequence starting at 𝐴. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))    &   (𝜑 → 𝐴 ∈ No )    ⇒   (𝜑 → 𝐴 ∈ 𝑍)
 
18-Apr-2025noseqex 28305 The next several theorems develop the concept of a countable sequence of surreals. This set is denoted by 𝑍 here and is the analogue of the upper integer sets for surreal numbers. However, we do not require the starting point to be an integer so we can accommodate infinite numbers. This first theorem establishes that 𝑍 is a set. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝑍 = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 𝐴) “ ω))    ⇒   (𝜑 → 𝑍 ∈ V)
 
18-Apr-2025seqsval 28304 The value of the surreal sequence builder. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) ↾ ω))    ⇒   (𝜑 → seqs𝑀( + , 𝐹) = ran 𝑅)
 
18-Apr-2025nfseqs 28303 Hypothesis builder for the surreal sequence builder. (Contributed by Scott Fenton, 18-Apr-2025.)
Ⅎ𝑥𝑀    &   Ⅎ𝑥 +    &   Ⅎ𝑥𝐹    ⇒   Ⅎ𝑥seqs𝑀( + , 𝐹)
 
18-Apr-2025seqseq123d 28302 Equality deduction for the surreal sequence builder. (Contributed by Scott Fenton, 18-Apr-2025.)
(𝜑 → 𝑀 = 𝑁)    &   (𝜑 → + = 𝑄)    &   (𝜑 → 𝐹 = 𝐺)    ⇒   (𝜑 → seqs𝑀( + , 𝐹) = seqs𝑁(𝑄, 𝐺))
 
18-Apr-2025seqsex 28301 Existence of the surreal sequence builder operation. (Contributed by Scott Fenton, 18-Apr-2025.)
seqs𝑀( + , 𝐹) ∈ V
 
18-Apr-2025df-seqs 28300 Define a general-purpose sequence builder for surreal numbers. Compare df-seq 14047. Note that in the theorems we develop here, we do not require 𝑀 to be an integer. This is because there are infinite surreal numbers and we may want to start our sequence there. (Contributed by Scott Fenton, 18-Apr-2025.)
seqs𝑀( + , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ ⟨(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))⟩), ⟨𝑀, (𝐹‘𝑀)⟩) “ ω)
 
18-Apr-2025fsumdvdsmul 27248 Product of two divisor sums. (This is also the main part of the proof that "Σ𝑘 ∥ 𝑁𝐹(𝑘) is a multiplicative function if 𝐹 is".) (Contributed by Mario Carneiro, 2-Jul-2015.) Avoid ax-mulf 11258. (Revised by GG, 18-Apr-2025.)
(𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → (𝑀 gcd 𝑁) = 1)    &   𝑋 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}    &   𝑌 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}    &   𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}    &   ((𝜑 ∧ 𝑗 ∈ 𝑋) → 𝐴 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐵 ∈ ℂ)    &   ((𝜑 ∧ (𝑗 ∈ 𝑋 ∧ 𝑘 ∈ 𝑌)) → (𝐴 · 𝐵) = 𝐷)    &   (𝑖 = (𝑗 · 𝑘) → 𝐶 = 𝐷)    ⇒   (𝜑 → (Σ𝑗 ∈ 𝑋 𝐴 · Σ𝑘 ∈ 𝑌 𝐵) = Σ𝑖 ∈ 𝑍 𝐶)
 
18-Apr-2025mpodvdsmulf1o 27247 If 𝑀 and 𝑁 are two coprime integers, multiplication forms a bijection from the set of pairs ⟨𝑗, 𝑘⟩ where 𝑗 ∥ 𝑀 and 𝑘 ∥ 𝑁, to the set of divisors of 𝑀 · 𝑁. Version of dvdsmulf1o 27249 using maps-to notation, which does not require ax-mulf 11258. (Contributed by GG, 18-Apr-2025.)
(𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → (𝑀 gcd 𝑁) = 1)    &   𝑋 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑀}    &   𝑌 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}    &   𝑍 = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑀 · 𝑁)}    ⇒   (𝜑 → ((𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–1-1-onto→𝑍)
 
18-Apr-2025df2idl2 21284 Alternate (the usual textbook) definition of a two-sided ideal of a ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.) (Proof shortened by AV, 18-Apr-2025.)
𝑈 = (2Ideal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ Ring → (𝐼 ∈ 𝑈 ↔ (𝐼 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐼 ((𝑥 · 𝑦) ∈ 𝐼 ∧ (𝑦 · 𝑥) ∈ 𝐼))))
 
18-Apr-2025lidl1 21260 Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Proof shortened by AV, 18-Apr-2025.)
𝑈 = (LIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)    ⇒   (𝑅 ∈ Ring → 𝐵 ∈ 𝑈)

Older news:

(29-Jul-2020) Mario Carneiro presented MM0 at the CICM conference. See this Google Group post which includes a YouTube link.

(20-Jul-2020) Rohan Ridenour found 5 shorter D-proofs in our Shortest known proofs... file. In particular, he reduced *4.39 from 901 to 609 steps. A note on the Metamath Solitaire page mentions a tool that he worked with.

(19-Jul-2020) David A. Wheeler posted a video (https://youtu.be/3R27Qx69jHc) on how to (re)prove Schwabh�user 4.6 for the Metamath Proof Explorer. See also his older videos.

(19-Jul-2020) In version 0.184 of the metamath program, "verify markup" now checks that mathboxes are independent i.e. do not cross-reference each other. To turn off this check, use "/mathbox_skip"

(30-Jun-2020) In version 0.183 of the metamath program, (1) "verify markup" now has checking for (i) underscores in labels, (ii) that *ALT and *OLD theorems have both discouragement tags, and (iii) that lines don't have trailing spaces. (2) "save proof.../rewrap" no longer left-aligns $p/$a comments that contain the string "<HTML>"; see this note.

(5-Apr-2020) Glauco Siliprandi added a new proof to the 100 theorem list, e is Transcendental etransc, bringing the Metamath total to 74.

(12-Feb-2020) A bug in the 'minimize' command of metamath.exe versions 0.179 (29-Nov-2019) and 0.180 (10-Dec-2019) may incorrectly bring in the use of new axioms. Version 0.181 fixes it.

(20-Jan-2020) David A. Wheeler created a video called Walkthrough of the tutorial in mmj2. See the Google Group announcement for more details. (All of his videos are listed on the Other Metamath-Related Topics page.)

(18-Jan-2020) The FOMM 2020 talks are on youtube now. Mario Carneiro's talk is Metamath Zero, or: How to Verify a Verifier. Since they are washed out in the video, the PDF slides are available separately.

(14-Dec-2019) Glauco Siliprandi added a new proof to the 100 theorem list, Fourier series convergence fourier, bringing the Metamath total to 73.

(25-Nov-2019) Alexander van der Vekens added a new proof to the 100 theorem list, The Cayley-Hamilton Theorem cayleyhamilton, bringing the Metamath total to 72.

(25-Oct-2019) Mario Carneiro's paper "Metamath Zero: The Cartesian Theorem Prover" (submitted to CPP 2020) is now available on arXiv: https://arxiv.org/abs/1910.10703. There is a related discussion on Hacker News.

(30-Sep-2019) Mario Carneiro's talk about MM0 at ITP 2019 is available on YouTube: x86 verification from scratch (24 minutes). Google Group discussion: Metamath Zero.

(29-Sep-2019) David Wheeler created a fascinating Gource video that animates the construction of set.mm, available on YouTube: Metamath set.mm contributions viewed with Gource through 2019-09-26 (4 minutes). Google Group discussion: Gource video of set.mm contributions.

(24-Sep-2019) nLab added a page for Metamath. It mentions Stefan O'Rear's Busy Beaver work using the set.mm axiomatization (and fails to mention Mario's definitional soundness checker)

(1-Sep-2019) Xuanji Li published a Visual Studio Code extension to support metamath syntax highlighting.

(10-Aug-2019) (revised 21-Sep-2019) Version 0.178 of the metamath program has the following changes: (1) "minimize_with" will now prevent dependence on new $a statements unless the new qualifier "/allow_new_axioms" is specified. For routine usage, it is suggested that you use "minimize_with * /allow_new_axioms * /no_new_axioms_from ax-*" instead of just "minimize_with *". See "help minimize_with" and this Google Group post. Also note that the qualifier "/allow_growth" has been renamed to "/may_grow". (2) "/no_versioning" was added to "write theorem_list".

(8-Jul-2019) Jon Pennant announced the creation of a Metamath search engine. Try it and feel free to comment on it at https://groups.google.com/d/msg/metamath/cTeU5AzUksI/5GesBfDaCwAJ.

(16-May-2019) Set.mm now has a major new section on elementary geometry. This begins with definitions that implement Tarski's axioms of geometry (including concepts such as congruence and betweenness). This uses set.mm's extensible structures, making them easier to use for many circumstances. The section then connects Tarski geometry with geometry in Euclidean places. Most of the work in this section is due to Thierry Arnoux, with earlier work by Mario Carneiro and Scott Fenton. [Reported by DAW.]

(9-May-2019) We are sad to report that long-time contributor Alan Sare passed away on Mar. 23. There is some more information at the top of his mathbox (click on "Mathbox for Alan Sare") and his obituary. We extend our condolences to his family.

(10-Mar-2019) Jon Pennant and Mario Carneiro added a new proof to the 100 theorem list, Heron's formula heron, bringing the Metamath total to 71.

(22-Feb-2019) Alexander van der Vekens added a new proof to the 100 theorem list, Cramer's rule cramer, bringing the Metamath total to 70.

(6-Feb-2019) David A. Wheeler has made significant improvements and updates to the Metamath book. Any comments, errors found, or suggestions are welcome and should be turned into an issue or pull request at https://github.com/metamath/metamath-book (or sent to me if you prefer).

(26-Dec-2018) I added Appendix 8 to the MPE Home Page that cross-references new and old axiom numbers.

(20-Dec-2018) The axioms have been renumbered according to this Google Groups post.

(24-Nov-2018) Thierry Arnoux created a new page on topological structures. The page along with its SVG files are maintained on GitHub.

(11-Oct-2018) Alexander van der Vekens added a new proof to the 100 theorem list, the Friendship Theorem friendship, bringing the Metamath total to 69.

(1-Oct-2018) Naip Moro has written gramm, a Metamath proof verifier written in Antlr4/Java.

(16-Sep-2018) The definition df-riota has been simplified so that it evaluates to the empty set instead of an Undef value. This change affects a significant part of set.mm.

(2-Sep-2018) Thierry Arnoux added a new proof to the 100 theorem list, Euler's partition theorem eulerpart, bringing the Metamath total to 68.

(1-Sep-2018) The Kate editor now has Metamath syntax highlighting built in. (Communicated by Wolf Lammen.)

(15-Aug-2018) The Intuitionistic Logic Explorer now has a Most Recent Proofs page.

(4-Aug-2018) Version 0.163 of the metamath program now indicates (with an asterisk) which Table of Contents headers have associated comments.

(10-May-2018) George Szpiro, journalist and author of several books on popular mathematics such as Poincare's Prize and Numbers Rule, used a genetic algorithm to find shorter D-proofs of "*3.37" and "meredith" in our Shortest known proofs... file.

(19-Apr-2018) The EMetamath Eclipse plugin has undergone many improvements since its initial release as the change log indicates. Thierry uses it as his main proof assistant and writes, "I added support for mmj2's auto-transformations, which allows it to infer several steps when building proofs. This added a lot of comfort for writing proofs.... I can now switch back and forth between the proof assistant and editing the Metamath file.... I think no other proof assistant has this feature."

(11-Apr-2018) Benoît Jubin solved an open problem about the "Axiom of Twoness," showing that it is necessary for completeness. See item 14 on the "Open problems and miscellany" page.

(25-Mar-2018) Giovanni Mascellani has announced mmpp, a new proof editing environment for the Metamath language.

(27-Feb-2018) Bill Hale has released an app for the Apple iPad and desktop computer that allows you to browse Metamath theorems and their proofs.

(17-Jan-2018) Dylan Houlihan has kindly provided a new mirror site. He has also provided an rsync server; type "rsync uk.metamath.org::" in a bash shell to check its status (it should return "metamath metamath").

(15-Jan-2018) The metamath program, version 0.157, has been updated to implement the file inclusion conventions described in the 21-Dec-2017 entry of mmnotes.txt.

(11-Dec-2017) I added a paragraph, suggested by Gérard Lang, to the distinct variable description here.

(10-Dec-2017) Per FL's request, his mathbox will be removed from set.mm. If you wish to export any of his theorems, today's version (master commit 1024a3a) is the last one that will contain it.

(11-Nov-2017) Alan Sare updated his completeusersproof program.

(3-Oct-2017) Sean B. Palmer created a web page that runs the metamath program under emulated Linux in JavaScript. He also wrote some programs to work with our shortest known proofs of the PM propositional calculus theorems.

(28-Sep-2017) Ivan Kuckir wrote a tutorial blog entry, Introduction to Metamath, that summarizes the language syntax. (It may have been written some time ago, but I was not aware of it before.)

(26-Sep-2017) The default directory for the Metamath Proof Explorer (MPE) has been changed from the GIF version (mpegif) to the Unicode version (mpeuni) throughout the site. Please let me know if you find broken links or other issues.

(24-Sep-2017) Saveliy Skresanov added a new proof to the 100 theorem list, Ceva's Theorem cevath, bringing the Metamath total to 67.

(3-Sep-2017) Brendan Leahy added a new proof to the 100 theorem list, Area of a Circle areacirc, bringing the Metamath total to 66.

(7-Aug-2017) Mario Carneiro added a new proof to the 100 theorem list, Principle of Inclusion/Exclusion incexc, bringing the Metamath total to 65.

(1-Jul-2017) Glauco Siliprandi added a new proof to the 100 theorem list, Stirling's Formula stirling, bringing the Metamath total to 64. Related theorems include 2 versions of Wallis' formula for π (wallispi and wallispi2).

(7-May-2017) Thierry Arnoux added a new proof to the 100 theorem list, Betrand's Ballot Problem ballotth, bringing the Metamath total to 63.

(20-Apr-2017) Glauco Siliprandi added a new proof in the supplementary list on the 100 theorem list, Stone-Weierstrass Theorem stowei.

(28-Feb-2017) David Moews added a new proof to the 100 theorem list, Product of Segments of Chords chordthm, bringing the Metamath total to 62.

(1-Jan-2017) Saveliy Skresanov added a new proof to the 100 theorem list, Isosceles triangle theorem isosctr, bringing the Metamath total to 61.

(1-Jan-2017) Mario Carneiro added 2 new proofs to the 100 theorem list, L'Hôpital's Rule lhop and Taylor's Theorem taylth, bringing the Metamath total to 60.

(28-Dec-2016) David A. Wheeler is putting together a page on Metamath (specifically set.mm) conventions. Comments are welcome on the Google Group thread.

(24-Dec-2016) Mario Carneiro introduced the abbreviation "F/ x ph" (symbols: turned F, x, phi) in df-nf to represent the "effectively not free" idiom "A. x ( ph -> A. x ph )". Theorem nf2 shows a version without nested quantifiers.

(22-Dec-2016) Naip Moro has developed a Metamath database for G. Spencer-Brown's Laws of Form. You can follow the Google Group discussion here.

(20-Dec-2016) In metamath program version 0.137, 'verify markup *' now checks that ax-XXX $a matches axXXX $p when the latter exists, per the discussion at https://groups.google.com/d/msg/metamath/Vtz3CKGmXnI/Fxq3j1I_EQAJ.

(24-Nov-2016) Mingl Yuan has kindly provided a mirror site in Beijing, China. He has also provided an rsync server; type "rsync cn.metamath.org::" in a bash shell to check its status (it should return "metamath metamath").

(14-Aug-2016) All HTML pages on this site should now be mobile-friendly and pass the Mobile-Friendly Test. If you find one that does not, let me know.

(14-Aug-2016) Daniel Whalen wrote a paper describing the use of using deep learning to prove 14% of test theorems taken from set.mm: Holophrasm: a neural Automated Theorem Prover for higher-order logic. The associated program is called Holophrasm.

(14-Aug-2016) David A. Wheeler created a video called Metamath Proof Explorer: A Modern Principia Mathematica

(12-Aug-2016) A Gitter chat room has been created for Metamath.

(9-Aug-2016) Mario Carneiro wrote a Metamath proof verifier in the Scala language as part of the ongoing Metamath -> MMT import project

(9-Aug-2016) David A. Wheeler created a GitHub project called metamath-test (last execution run) to check that different verifiers both pass good databases and detect errors in defective ones.

(4-Aug-2016) Mario gave two presentations at CICM 2016.

(17-Jul-2016) Thierry Arnoux has written EMetamath, a Metamath plugin for the Eclipse IDE.

(16-Jul-2016) Mario recovered Chris Capel's collapsible proof demo.

(13-Jul-2016) FL sent me an updated version of PDF (LaTeX source) developed with Lamport's pf2 package. See the 23-Apr-2012 entry below.

(12-Jul-2016) David A. Wheeler produced a new video for mmj2 called "Creating functions in Metamath". It shows a more efficient approach than his previous recent video "Creating functions in Metamath" (old) but it can be of interest to see both approaches.

(10-Jul-2016) Metamath program version 0.132 changes the command 'show restricted' to 'show discouraged' and adds a new command, 'set discouragement'. See the mmnotes.txt entry of 11-May-2016 (updated 10-Jul-2016).

(12-Jun-2016) Dan Getz has written Metamath.jl, a Metamath proof verifier written in the Julia language.

(10-Jun-2016) If you are using metamath program versions 0.128, 0.129, or 0.130, please update to version 0.131. (In the bad versions, 'minimize_with' ignores distinct variable violations.)

(1-Jun-2016) Mario Carneiro added new proofs to the 100 theorem list, the Prime Number Theorem pnt and the Perfect Number Theorem perfect, bringing the Metamath total to 58.

(12-May-2016) Mario Carneiro added a new proof to the 100 theorem list, Dirichlet's theorem dirith, bringing the Metamath total to 56. (Added 17-May-2016) An informal exposition of the proof can be found at http://metamath-blog.blogspot.com/2016/05/dirichlets-theorem.html

(10-Mar-2016) Metamath program version 0.125 adds a new qualifier, /fast, to 'save proof'. See the mmnotes.txt entry of 10-Mar-2016.

(6-Mar-2016) The most recent set.mm has a large update converting variables from letters to symbols. See this Google Groups post.

(16-Feb-2016) Mario Carneiro's new paper "Models for Metamath" can be found here and on arxiv.org.

(6-Feb-2016) There are now 22 math symbols that can be used as variable names. See mmascii.html near the 50th table row, starting with "./\".

(29-Jan-2016) Metamath program version 0.123 adds /packed and /explicit qualifiers to 'save proof' and 'show proof'. See this Google Groups post.

(13-Jan-2016) The Unicode math symbols now provide for external CSS and use the XITS web font. Thanks to David A. Wheeler, Mario Carneiro, Cris Perdue, Jason Orendorff, and Frédéric Liné for discussions on this topic. Two commands, htmlcss and htmlfont, were added to the $t comment in set.mm and are recognized by Metamath program version 0.122.

(21-Dec-2015) Axiom ax-12, now renamed ax-12o, was replaced by a new shorter equivalent, ax-12. The equivalence is provided by theorems ax12o and ax12.

(13-Dec-2015) A new section on the theory of classes was added to the MPE Home Page. Thanks to Gérard Lang for suggesting this section and improvements to it.

(17-Nov-2015) Metamath program version 0.121: 'verify markup' was added to check comment markup consistency; see 'help verify markup'. You are encouraged to make sure 'verify markup */f' has no warnings prior to mathbox submissions. The date consistency rules are given in this Google Groups post.

(23-Sep-2015) Drahflow wrote, "I am currently working on yet another proof assistant, main reason being: I understand stuff best if I code it. If anyone is interested: https://github.com/Drahflow/Igor (but in my own programming language, so expect a complicated build process :P)"

(23-Aug-2015) Ivan Kuckir created MM Tool, a Metamath proof verifier and editor written in JavaScript that runs in a browser.

(25-Jul-2015) Axiom ax-10 is shown to be redundant by theorem ax10 , so it was removed from the predicate calculus axiom list.

(19-Jul-2015) Mario Carneiro gave two talks related to Metamath at CICM 2015, which are linked to at Other Metamath-Related Topics.

(18-Jul-2015) The metamath program has been updated to version 0.118. 'show trace_back' now has a '/to' qualifier to show the path back to a specific axiom such as ax-ac. See 'help show trace_back'.

(12-Jul-2015) I added the HOL Explorer for Mario Carneiro's hol.mm database. Although the home page needs to be filled out, the proofs can be accessed.

(11-Jul-2015) I started a new page, Other Metamath-Related Topics, that will hold miscellaneous material that doesn't fit well elsewhere (or is hard to find on this site). Suggestions welcome.

(23-Jun-2015) Metamath's mascot, Penny the cat (2007 photo), passed away today. She was 18 years old.

(21-Jun-2015) Mario Carneiro added 3 new proofs to the 100 theorem list: All Primes (1 mod 4) Equal the Sum of Two Squares 2sq, The Law of Quadratic Reciprocity lgsquad and the AM-GM theorem amgm, bringing the Metamath total to 55.

(13-Jun-2015) Stefan O'Rear's smm, written in JavaScript, can now be used as a standalone proof verifier. This brings the total number of independent Metamath verifiers to 8, written in just as many languages (C, Java. JavaScript, Python, Haskell, Lua, C#, C++).

(12-Jun-2015) David A. Wheeler added 2 new proofs to the 100 theorem list: The Law of Cosines lawcos and Ptolemy's Theorem ptolemy, bringing the Metamath total to 52.

(30-May-2015) The metamath program has been updated to version 0.117. (1) David A. Wheeler provided an enhancement to speed up the 'improve' command by 28%; see README.TXT for more information. (2) In web pages with proofs, local hyperlinks on step hypotheses no longer clip the Expression cell at the top of the page.

(9-May-2015) Stefan O'Rear has created an archive of older set.mm releases back to 1998: https://github.com/sorear/set.mm-history/.

(7-May-2015) The set.mm dated 7-May-2015 is a major revision, updated by Mario, that incorporates the new ordered pair definition df-op that was agreed upon. There were 700 changes, listed at the top of set.mm. Mathbox users are advised to update their local mathboxes. As usual, if any mathbox user has trouble incorporating these changes into their mathbox in progress, Mario or I will be glad to do them for you.

(7-May-2015) Mario has added 4 new theorems to the 100 theorem list: Ramsey's Theorem ramsey, The Solution of a Cubic cubic, The Solution of the General Quartic Equation quart, and The Birthday Problem birthday. In the Supplementary List, Stefan O'Rear added the Hilbert Basis Theorem hbt.

(28-Apr-2015) A while ago, Mario Carneiro wrote up a proof of the unambiguity of set.mm's grammar, which has now been added to this site: grammar-ambiguity.txt.

(22-Apr-2015) The metamath program has been updated to version 0.114. In MM-PA, 'show new_proof/unknown' now shows the relative offset (-1, -2,...) used for 'assign' arguments, suggested by Stefan O'Rear.

(20-Apr-2015) I retrieved an old version of the missing "Metamath 100" page from archive.org and updated it to what I think is the current state: mm_100.html. Anyone who wants to edit it can email updates to this page to me.

(19-Apr-2015) The metamath program has been updated to version 0.113, mostly with patches provided by Stefan O'Rear. (1) 'show statement %' (or any command allowing label wildcards) will select statements whose proofs were changed in current session. ('help search' will show all wildcard matching rules.) (2) 'show statement =' will select the statement being proved in MM-PA. (3) The proof date stamp is now created only if the proof is complete.

(18-Apr-2015) There is now a section for Scott Fenton's NF database: New Foundations Explorer.

(16-Apr-2015) Mario describes his recent additions to set.mm at https://groups.google.com/forum/#!topic/metamath/VAGNmzFkHCs. It include 2 new additions to the Formalizing 100 Theorems list, Leibniz' series for pi (leibpi) and the Konigsberg Bridge problem (konigsberg)

(10-Mar-2015) Mario Carneiro has written a paper, "Arithmetic in Metamath, Case Study: Bertrand's Postulate," for CICM 2015. A preprint is available at arXiv:1503.02349.

(23-Feb-2015) Scott Fenton has created a Metamath formalization of NF set theory: https://github.com/sctfn/metamath-nf/. For more information, see the Metamath Google Group posting.

(28-Jan-2015) Mario Carneiro added Wilson's Theorem (wilth), Ascending or Descending Sequences (erdsze, erdsze2), and Derangements Formula (derangfmla, subfaclim), bringing the Metamath total for Formalizing 100 Theorems to 44.

(19-Jan-2015) Mario Carneiro added Sylow's Theorem (sylow1, sylow2, sylow2b, sylow3), bringing the Metamath total for Formalizing 100 Theorems to 41.

(9-Jan-2015) The hypothesis order of mpbi*an* was changed. See the Notes entry of 9-Jan-2015.

(1-Jan-2015) Mario Carneiro has written a paper, "Conversion of HOL Light proofs into Metamath," that has been submitted to the Journal of Formalized Reasoning. A preprint is available on arxiv.org.

(22-Nov-2014) Stefan O'Rear added the Solutions to Pell's Equation (rmxycomplete) and Liouville's Theorem and the Construction of Transcendental Numbers (aaliou), bringing the Metamath total for Formalizing 100 Theorems to 40.

(22-Nov-2014) The metamath program has been updated with version 0.111. (1) Label wildcards now have a label range indicator "~" so that e.g. you can show or search all of the statements in a mathbox. See 'help search'. (Stefan O'Rear added this to the program.) (2) A qualifier was added to 'minimize_with' to prevent the use of any axioms not already used in the proof e.g. 'minimize_with * /no_new_axioms_from ax-*' will prevent the use of ax-ac if the proof doesn't already use it. See 'help minimize_with'.

(10-Oct-2014) Mario Carneiro has encoded the axiomatic basis for the HOL theorem prover into a Metamath source file, hol.mm.

(24-Sep-2014) Mario Carneiro added the Sum of the Angles of a Triangle (ang180), bringing the Metamath total for Formalizing 100 Theorems to 38.

(15-Sep-2014) Mario Carneiro added the Fundamental Theorem of Algebra (fta), bringing the Metamath total for Formalizing 100 Theorems to 37.

(3-Sep-2014) Mario Carneiro added the Fundamental Theorem of Integral Calculus (ftc1, ftc2). This brings the Metamath total for Formalizing 100 Theorems to 35. (added 14-Sep-2014) Along the way, he added the Mean Value Theorem (mvth), bringing the total to 36.

(16-Aug-2014) Mario Carneiro started a Metamath blog at http://metamath-blog.blogspot.com/.

(10-Aug-2014) Mario Carneiro added Erdős's proof of the divergence of the inverse prime series (prmrec). This brings the Metamath total for Formalizing 100 Theorems to 34.

(31-Jul-2014) Mario Carneiro added proofs for Euler's Summation of 1 + (1/2)^2 + (1/3)^2 + .... (basel) and The Factor and Remainder Theorems (facth, plyrem). This brings the Metamath total for Formalizing 100 Theorems to 33.

(16-Jul-2014) Mario Carneiro added proofs for Four Squares Theorem (4sq), Formula for the Number of Combinations (hashbc), and Divisibility by 3 Rule (3dvds). This brings the Metamath total for Formalizing 100 Theorems to 31.

(11-Jul-2014) Mario Carneiro added proofs for Divergence of the Harmonic Series (harmonic), Order of a Subgroup (lagsubg), and Lebesgue Measure and Integration (itgcl). This brings the Metamath total for Formalizing 100 Theorems to 28.

(7-Jul-2014) Mario Carneiro presented a talk, "Natural Deduction in the Metamath Proof Language," at the 6PCM conference. Slides Audio

(25-Jun-2014) In version 0.108 of the metamath program, the 'minimize_with' command is now more automated. It now considers compressed proof length; it scans the statements in forward and reverse order and chooses the best; and it avoids $d conflicts. The '/no_distinct', '/brief', and '/reverse' qualifiers are obsolete, and '/verbose' no longer lists all statements scanned but gives more details about decision criteria.

(12-Jun-2014) To improve naming uniformity, theorems about operation values now use the abbreviation "ov". For example, df-opr, opreq1, oprabval5, and oprvres are now called df-ov, oveq1, ov5, and ovres respectively.

(11-Jun-2014) Mario Carneiro finished a major revision of set.mm. His notes are under the 11-Jun-2014 entry in the Notes

(4-Jun-2014) Mario Carneiro provided instructions and screenshots for syntax highlighting for the jEdit editor for use with Metamath and mmj2 source files.

(19-May-2014) Mario Carneiro added a feature to mmj2, in the build at https://github.com/digama0/mmj2/raw/dev-build/mmj2jar/mmj2.jar, which tests all but 5 definitions in set.mm for soundness. You can turn on the test by adding
SetMMDefinitionsCheckWithExclusions,ax-*,df-bi,df-clab,df-cleq,df-clel,df-sbc
to your RunParms.txt file.

(17-May-2014) A number of labels were changed in set.mm, listed at the top of set.mm as usual. Note in particular that the heavily-used visset, elisseti, syl11anc, syl111anc were changed respectively to vex, elexi, syl2anc, syl3anc.

(16-May-2014) Scott Fenton formalized a proof for "Sum of kth powers": fsumkthpow. This brings the Metamath total for Formalizing 100 Theorems to 25.

(9-May-2014) I (Norm Megill) presented an overview of Metamath at the "Formalization of mathematics in proof assistants" workshop at the Institut Henri Poincar� in Paris. The slides for this talk are here.

(22-Jun-2014) Version 0.107 of the metamath program adds a "PART" indention level to the Statement List table of contents, adds 'show proof ... /size' to show source file bytes used, and adds 'show elapsed_time'. The last one is helpful for measuring the run time of long commands. See 'help write theorem_list', 'help show proof', and 'help show elapsed_time' for more information.

(2-May-2014) Scott Fenton formalized a proof of Sum of the Reciprocals of the Triangular Numbers: trirecip. This brings the Metamath total for Formalizing 100 Theorems to 24.

(19-Apr-2014) Scott Fenton formalized a proof of the Formula for Pythagorean Triples: pythagtrip. This brings the Metamath total for Formalizing 100 Theorems to 23.

(11-Apr-2014) David A. Wheeler produced a much-needed and well-done video for mmj2, called "Introduction to Metamath & mmj2". Thanks, David!

(15-Mar-2014) Mario Carneiro formalized a proof of Bertrand's postulate: bpos. This brings the Metamath total for Formalizing 100 Theorems to 22.

(18-Feb-2014) Mario Carneiro proved that complex number axiom ax-cnex is redundant (theorem cnex). See also Real and Complex Numbers.

(11-Feb-2014) David A. Wheeler has created a theorem compilation that tracks those theorems in Freek Wiedijk's Formalizing 100 Theorems list that have been proved in set.mm. If you find a error or omission in this list, let me know so it can be corrected. (Update 1-Mar-2014: Mario has added eulerth and bezout to the list.)

(4-Feb-2014) Mario Carneiro writes:

The latest commit on the mmj2 development branch introduced an exciting new feature, namely syntax highlighting for mmp files in the main window. (You can pick up the latest mmj2.jar at https://github.com/digama0/mmj2/blob/develop/mmj2jar/mmj2.jar .) The reason I am asking for your help at this stage is to help with design for the syntax tokenizer, which is responsible for breaking down the input into various tokens with names like "comment", "set", and "stephypref", which are then colored according to the user's preference. As users of mmj2 and metamath, what types of highlighting would be useful to you?

One limitation of the tokenizer is that since (for performance reasons) it can be started at any line in the file, highly contextual coloring, like highlighting step references that don't exist previously in the file, is difficult to do. Similarly, true parsing of the formulas using the grammar is possible but likely to be unmanageably slow. But things like checking theorem labels against the database is quite simple to do under the current setup.

That said, how can this new feature be optimized to help you when writing proofs?

(13-Jan-2014) Mathbox users: the *19.21a*, *19.23a* series of theorems have been renamed to *alrim*, *exlim*. You can update your mathbox with a global replacement of string '19.21a' with 'alrim' and '19.23a' with 'exlim'.

(5-Jan-2014) If you downloaded mmj2 in the past 3 days, please update it with the current version, which fixes a bug introduced by the recent changes that made it unable to read in most of the proofs in the textarea properly.

(4-Jan-2014) I added a list of "Allowed substitutions" under the "Distinct variable groups" list on the theorem web pages, for example axsep. This is an experimental feature and comments are welcome.

(3-Jan-2014) Version 0.102 of the metamath program produces more space-efficient compressed proofs (still compatible with the specification in Appendix B of the Metamath book) using an algorithm suggested by Mario Carneiro. See 'help save proof' in the program. Also, mmj2 now generates proofs in the new format. The new mmj2 also has a mandatory update that fixes a bug related to the new format; you must update your mmj2 copy to use it with the latest set.mm.

(23-Dec-2013) Mario Carneiro has updated many older definitions to use the maps-to notation. If you have difficulty updating your local mathbox, contact him or me for assistance.

(1-Nov-2013) 'undo' and 'redo' commands were added to the Proof Assistant in metamath program version 0.07.99. See 'help undo' in the program.

(8-Oct-2013) Today's Notes entry describes some proof repair techniques.

(5-Oct-2013) Today's Notes entry explains some recent extensible structure improvements.

(8-Sep-2013) Mario Carneiro has revised the square root and sequence generator definitions. See today's Notes entry.

(3-Aug-2013) Mario Carneiro writes: "I finally found enough time to create a GitHub repository for development at https://github.com/digama0/mmj2. A permalink to the latest version plus source (akin to mmj2.zip) is https://github.com/digama0/mmj2/zipball/, and the jar file on its own (mmj2.jar) is at https://github.com/digama0/mmj2/blob/master/mmj2jar/mmj2.jar?raw=true. Unfortunately there is no easy way to automatically generate mmj2jar.zip, but this is available as part of the zip distribution for mmj2.zip. History tracking will be handled by the repository now. Do you have old versions of the mmj2 directory? I could add them as historical commits if you do."

(18-Jun-2013) Mario Carneiro has done a major revision and cleanup of the construction of real and complex numbers. In particular, rather than using equivalence classes as is customary for the construction of the temporary rationals, he used only "reduced fractions", so that the use of the axiom of infinity is avoided until it becomes necessary for the construction of the temporary reals.

(18-May-2013) Mario Carneiro has added the ability to produce compressed proofs to mmj2. This is not an official release but can be downloaded here if you want to try it: mmj2.jar. If you have any feedback, send it to me (NM), and I will forward it to Mario. (Disclaimer: this release has not been endorsed by Mel O'Cat. If anyone has been in contact with him, please let me know.)

(29-Mar-2013) Charles Greathouse reduced the size of our PNG symbol images using the pngout program.

(8-Mar-2013) Wolf Lammen has reorganized the theorems in the "Logical negation" section of set.mm into a more orderly, less scattered arrangement.

(27-Feb-2013) Scott Fenton has done a large cleanup of set.mm, eliminating *OLD references in 144 proofs. See the Notes entry for 27-Feb-2013.

(21-Feb-2013) *ATTENTION MATHBOX USERS* The order of hypotheses of many syl* theorems were changed, per a suggestion of Mario Carneiro. You need to update your local mathbox copy for compatibility with the new set.mm, or I can do it for you if you wish. See the Notes entry for 21-Feb-2013.

(16-Feb-2013) Scott Fenton shortened the direct-from-axiom proofs of *3.1, *3.43, *4.4, *4.41, *4.5, *4.76, *4.83, *5.33, *5.35, *5.36, and meredith in the "Shortest known proofs of the propositional calculus theorems from Principia Mathematica" (pmproofs.txt).

(27-Jan-2013) Scott Fenton writes, "I've updated Ralph Levien's mmverify.py. It's now a Python 3 program, and supports compressed proofs and file inclusion statements. This adds about fifty lines to the original program. Enjoy!"

(10-Jan-2013) A new mathbox was added for Mario Carneiro, who has contributed a number of cardinality theorems without invoking the Axiom of Choice. This is nice work, and I will be using some of these (those suffixed with "NEW") to replace the existing ones in the main part of set.mm that currently invoke AC unnecessarily.

(4-Jan-2013) As mentioned in the 19-Jun-2012 item below, Eric Schmidt discovered that the complex number axioms axaddcom (now addcom) and ax0id (now addid1) are redundant (schmidt-cnaxioms.pdf, .tex). In addition, ax1id (now mulid1) can be weakened to ax1rid. Scott Fenton has now formalized this work, so that now there are 23 instead of 25 axioms for real and complex numbers in set.mm. The Axioms for Complex Numbers page has been updated with these results. An interesting part of the proof, showing how commutativity of addition follows from other laws, is in addcomi.

(27-Nov-2012) The frequently-used theorems "an1s", "an1rs", "ancom13s", "ancom31s" were renamed to "an12s", "an32s", "an13s", "an31s" to conform to the convention for an12 etc.

(4-Nov-2012) The changes proposed in the Notes, renaming Grp to GrpOp etc., have been incorporated into set.mm. See the list of changes at the top of set.mm. If you want me to update your mathbox with these changes, send it to me along with the version of set.mm that it works with.

(20-Sep-2012) Mel O'Cat updated https://us.metamath.org/ocat/mmj2/TESTmmj2jar.zip. See the README.TXT for a description of the new features.

(21-Aug-2012) Mel O'Cat has uploaded SearchOptionsMockup9.zip, a mockup for the new search screen in mmj2. See the README.txt file for instructions. He will welcome feedback via x178g243 at yahoo.com.

(19-Jun-2012) Eric Schmidt has discovered that in our axioms for complex numbers, axaddcom and ax0id are redundant. (At some point these need to be formalized for set.mm.) He has written up these and some other nice results, including some independence results for the axioms, in schmidt-cnaxioms.pdf (schmidt-cnaxioms.tex).

(23-Apr-2012) Frédéric Liné sent me a PDF (LaTeX source) developed with Lamport's pf2 package. He wrote: "I think it works well with Metamath since the proofs are in a tree form. I use it to have a sketch of a proof. I get this way a better understanding of the proof and I can cut down its size. For instance, inpreima5 was reduced by 50% when I wrote the corresponding proof with pf2."

(5-Mar-2012) I added links to Wikiproofs and its recent changes in the "Wikis" list at the top of this page.

(12-Jan-2012) Thanks to William Hoza who sent me a ZFC T-shirt, and thanks to the ZFC models (courtesy of the Inaccessible Cardinals agency).

FrontBackDetail
ZFC T-shirt front ZFC T-shirt back ZFC T-shirt detail

(24-Nov-2011) In metamath program version 0.07.71, the 'minimize_with' command by default now scans from bottom to top instead of top to bottom, since empirically this often (although not always) results in a shorter proof. A top to bottom scan can be specified with a new qualifier '/reverse'. You can try both methods (starting from the same original proof, of course) and pick the shorter proof.

(15-Oct-2011) From Mel O'Cat:
I just uploaded mmj2.zip containing the 1-Nov-2011 (20111101) release: https://us.metamath.org/ocat/mmj2/mmj2.zip https://us.metamath.org/ocat/mmj2/mmj2.md5
A few last minute tweaks:
1. I now bless double-click starting of mmj2.bat (MacMMJ2.command in Mac OS-X)! See mmj2\QuickStart.html
2. Much improved support of Mac OS-X systems. See mmj2\QuickStart.html
3. I tweaked the Command Line Argument Options report to
a) print every time;
b) print as much as possible even if there are errors in the command line arguments -- and the last line printed corresponds to the argument in error;
c) removed Y/N argument on the command line to enable/disable the report. this simplifies things.
4) Documentation revised, including the PATutorial.
See CHGLOG.TXT for list of all changes. Good luck. And thanks for all of your help!

(15-Sep-2011) MATHBOX USERS: I made a large number of label name changes to set.mm to improve naming consistency. There is a script at the top of the current set.mm that you can use to update your mathbox or older set.mm. Or if you wish, I can do the update on your next mathbox submission - in that case, please include a .zip of the set.mm version you used.

(30-Aug-2011) Scott Fenton shortened the direct-from-axiom proofs of *3.33, *3.45, *4.36, and meredith in the "Shortest known proofs of the propositional calculus theorems from Principia Mathematica" (pmproofs.txt).

(21-Aug-2011) A post on reddit generated 60,000 hits (and a TOS violation notice from my provider...),

(18-Aug-2011) The Metamath Google Group has a discussion of my canonical conjunctions proposal. Any feedback directly to me (Norm Megill) is also welcome.

(4-Jul-2011) John Baker has provided (metamath_kindle.zip) "a modified version of [the] metamath.tex [Metamath] book source that is formatted for the Kindle. If you compile the document the resulting PDF can be loaded into into a Kindle and easily read." (Update: the PDF file is now included also.)

(3-Jul-2011) Nested 'submit' calls are now allowed, in metamath program version 0.07.68. Thus you can create or modify a command file (script) from within a command file then 'submit' it. While 'submit' cannot pass arguments (nor are there plans to add this feature), you can 'substitute' strings in the 'submit' target file before calling it in order to emulate this.

(28-Jun-2011)The metamath program version 0.07.64 adds the '/include_mathboxes' qualifier to 'minimize_with'; by default, 'minimize_with *' will now skip checking user mathboxes. Since mathboxes should be independent from each other, this will help prevent accidental cross-"contamination". Also, '/rewrap' was added to 'write source' to automatically wrap $a and $p comments so as to conform to the current formatting conventions used in set.mm. This means you no longer have to be concerned about line length < 80 etc.

(19-Jun-2011) ATTENTION MATHBOX USERS: The wff variables et, ze, si, and rh are now global. This change was made primarily to resolve some conflicts between mathboxes, but it will also let you avoid having to constantly redeclare these locally in the future. Unfortunately, this change can affect the $f hypothesis order, which can cause proofs referencing theorems that use these variables to fail. All mathbox proofs currently in set.mm have been corrected for this, and you should refresh your local copy for further development of your mathbox. You can correct your proofs that are not in set.mm as follows. Only the proofs that fail under the current set.mm (using version 0.07.62 or later of the metamath program) need to be modified.

To fix a proof that references earlier theorems using et, ze, si, and rh, do the following (using a hypothetical theorem 'abc' as an example): 'prove abc' (ignore error messages), 'delete floating', 'initialize all', 'unify all/interactive', 'improve all', 'save new_proof/compressed'. If your proof uses dummy variables, these must be reassigned manually.

To fix a proof that uses et, ze, si, and rh as local variables, make sure the proof is saved in 'compressed' format. Then delete the local declarations ($v and $f statements) and follow the same steps above to correct the proof.

I apologize for the inconvenience. If you have trouble fixing your proofs, you can contact me for assistance.

Note: Versions of the metamath program before 0.07.62 did not flag an error when global variables were redeclared locally, as it should have according to the spec. This caused these spec violations to go unnoticed in some older set.mm versions. The new error messages are in fact just informational and can be ignored when working with older set.mm versions.

(7-Jun-2011) The metamath program version 0.07.60 fixes a bug with the 'minimize_with' command found by Andrew Salmon.

(12-May-2010) Andrew Salmon shortened many proofs, shown above. For comparison, I have temporarily kept the old version, which is suffixed with OLD, such as oridmOLD for oridm.

(9-Dec-2010) Eric Schmidt has written a Metamath proof verifier in C++, called checkmm.cpp.

(3-Oct-2010) The following changes were made to the tokens in set.mm. The subset and proper subset symbol changes to C_ and C. were made to prevent defeating the parenthesis matching in Emacs. Other changes were made so that all letters a-z and A-Z are now available for variable names. One-letter constants such as _V, _e, and _i are now shown on the web pages with Roman instead of italic font, to disambiguate italic variable names. The new convention is that a prefix of _ indicates Roman font and a prefix of ~ indicates a script (curly) font. Thanks to Stefan Allan and Frédéric Liné for discussions leading to this change.

OldNewDescription
C. _C binomial coefficient
E _E epsilon relation
e _e Euler's constant
I _I identity relation
i _i imaginary unit
V _V universal class
(_ C_ subset
(. C. proper subset
P~ ~P power class
H~ ~H Hilbert space

(25-Sep-2010) The metamath program (version 0.07.54) now implements the current Metamath spec, so footnote 2 on p. 92 of the Metamath book can be ignored.

(24-Sep-2010) The metamath program (version 0.07.53) fixes bug 2106, reported by Michal Burger.

(14-Sep-2010) The metamath program (version 0.07.52) has a revamped LaTeX output with 'show statement xxx /tex', which produces the combined statement, description, and proof similar to the web page generation. Also, 'show proof xxx /lemmon/renumber' now matches the web page step numbers. ('show proof xxx/renumber' still has the indented form conforming to the actual RPN proof, with slightly different numbering.)

(9-Sep-2010) The metamath program (version 0.07.51) was updated with a modification by Stefan Allan that adds hyperlinks the the Ref column of proofs.

(12-Jun-2010) Scott Fenton contributed a D-proof (directly from axioms) of Meredith's single axiom (see the end of pmproofs.txt). A description of Meredith's axiom can be found in theorem meredith.

(11-Jun-2010) A new Metamath mirror was added in Austria, courtesy of Kinder-Enduro.

(28-Feb-2010) Raph Levien's Ghilbert project now has a new Ghilbert site and a Google Group.

(26-Jan-2010) Dmitri Vlasov writes, "I admire the simplicity and power of the metamath language, but still I see its great disadvantage - the proofs in metamath are completely non-manageable by humans without proof assistants. Therefore I decided to develop another language, which would be a higher-level superstructure language towards metamath, and which will support human-readable/writable proofs directly, without proof assistants. I call this language mdl (acronym for 'mathematics development language')." The latest version of Dmitri's translators from metamath to mdl and back can be downloaded from http://mathdevlanguage.sourceforge.net/. Currently only Linux is supported, but Dmitri says is should not be difficult to port it to other platforms that have a g++ compiler.

(11-Sep-2009) The metamath program (version 0.07.48) has been updated to enforce the whitespace requirement of the current spec.

(10-Sep-2009) Matthew Leitch has written an nice article, "How to write mathematics clearly", that briefly mentions Metamath. Overall it makes some excellent points. (I have written to him about a few things I disagree with.)

(28-May-2009) AsteroidMeta is back on-line. Note the URL change.

(12-May-2009) Charles Greathouse wrote a Greasemonkey script to reformat the axiom list on Metamath web site proof pages. This is a beta version; he will appreciate feedback.

(11-May-2009) Stefan Allan modified the metamath program to add the command "show statement xxx /mnemonics", which produces the output file Mnemosyne.txt for use with the Mnemosyne project. The current Metamath program download incorporates this command. Instructions: Create the file mnemosyne.txt with e.g. "show statement ax-* /mnemonics". In the Mnemosyne program, load the file by choosing File->Import then file format "Q and A on separate lines". Notes: (1) Don't try to load all of set.mm, it will crash the program due to a bug in Mnemosyne. (2) On my computer, the arrows in ax-1 don't display. Stefan reports that they do on his computer. (Both are Windows XP.)

(3-May-2009) Steven Baldasty wrote a Metamath syntax highlighting file for the gedit editor. Screenshot.

(1-May-2009) Users on a gaming forum discuss our 2+2=4 proof. Notable comments include "Ew math!" and "Whoever wrote this has absolutely no life."

(12-Mar-2009) Chris Capel has created a Javascript theorem viewer demo that (1) shows substitutions and (2) allows expanding and collapsing proof steps. You are invited to take a look and give him feedback at his Metablog.

(28-Feb-2009) Chris Capel has written a Metamath proof verifier in C#, available at http://pdf23ds.net/bzr/MathEditor/Verifier/Verifier.cs and weighing in at 550 lines. Also, that same URL without the file on it is a Bazaar repository.

(2-Dec-2008) A new section was added to the Deduction Theorem page, called Logic, Metalogic, Metametalogic, and Metametametalogic.

(24-Aug-2008) (From ocat): The 1-Aug-2008 version of mmj2 is ready (mmj2.zip), size = 1,534,041 bytes. This version contains the Theorem Loader enhancement which provides a "sandboxing" capability for user theorems and dynamic update of new theorems to the Metamath database already loaded in memory by mmj2. Also, the new "mmj2 Service" feature enables calling mmj2 as a subroutine, or having mmj2 call your program, and provides access to the mmj2 data structures and objects loaded in memory (i.e. get started writing those Jython programs!) See also mmj2 on AsteroidMeta.

(23-May-2008) Gérard Lang pointed me to Bob Solovay's note on AC and strongly inaccessible cardinals. One of the eventual goals for set.mm is to prove the Axiom of Choice from Grothendieck's axiom, like Mizar does, and this note may be helpful for anyone wanting to attempt that. Separately, I also came across a history of the size reduction of grothprim (viewable in Firefox and some versions of Internet Explorer).

(14-Apr-2008) A "/join" qualifier was added to the "search" command in the metamath program (version 0.07.37). This qualifier will join the $e hypotheses to the $a or $p for searching, so that math tokens in the $e's can be matched as well. For example, "search *com* +v" produces no results, but "search *com* +v /join" yields commutative laws involving vector addition. Thanks to Stefan Allan for suggesting this idea.

(8-Apr-2008) The 8,000th theorem, hlrel, was added to the Metamath Proof Explorer part of the database.

(2-Mar-2008) I added a small section to the end of the Deduction Theorem page.

(17-Feb-2008) ocat has uploaded the "1-Mar-2008" mmj2: mmj2.zip. See the description.

(16-Jan-2008) O'Cat has written mmj2 Proof Assistant Quick Tips.

(30-Dec-2007) "How to build a library of formalized mathematics".

(22-Dec-2007) The Metamath Proof Explorer was included in the top 30 science resources for 2007 by the University at Albany Science Library.

(17-Dec-2007) Metamath's Wikipedia entry says, "This article may require cleanup to meet Wikipedia's quality standards" (see its discussion page). Volunteers are welcome. :) (In the interest of objectivity, I don't edit this entry.)

(20-Nov-2007) Jeff Hoffman created nicod.mm and posted it to the Google Metamath Group.

(19-Nov-2007) Reinder Verlinde suggested adding tooltips to the hyperlinks on the proof pages, which I did for proof step hyperlinks. Discussion.

(5-Nov-2007) A Usenet challenge. :)

(4-Aug-2007) I added a "Request for comments on proposed 'maps to' notation" at the bottom of the AsteroidMeta set.mm discussion page.

(21-Jun-2007) A preprint (PDF file) describing Kurt Maes' axiom of choice with 5 quantifiers, proved in set.mm as ackm.

(20-Jun-2007) The 7,000th theorem, ifpr, was added to the Metamath Proof Explorer part of the database.

(29-Apr-2007) Blog mentions of Metamath: here and here.

(21-Mar-2007) Paul Chapman is working on a new proof browser, which has highlighting that allows you to see the referenced theorem before and after the substitution was made. Here is a screenshot of theorem 0nn0 and a screenshot of theorem 2p2e4.

(15-Mar-2007) A picture of Penny the cat guarding the us.metamath.org server and making the rounds.

(16-Feb-2007) For convenience, the program "drule.c" (pronounced "D-rule", not "drool") mentioned in pmproofs.txt can now be downloaded (drule.c) without having to ask me for it. The same disclaimer applies: even though this program works and has no known bugs, it was not intended for general release. Read the comments at the top of the program for instructions.

(28-Jan-2007) Jason Orendorff set up a new mailing list for Metamath: http://groups.google.com/group/metamath.

(20-Jan-2007) Bob Solovay provided a revised version of his Metamath database for Peano arithmetic, peano.mm.

(2-Jan-2007) Raph Levien has set up a wiki called Barghest for the Ghilbert language and software.

(26-Dec-2006) I posted an explanation of theorem ecoprass on Usenet.

(2-Dec-2006) Berislav Žarnić translated the Metamath Solitaire applet to Croatian.

(26-Nov-2006) Dan Getz has created an RSS feed for new theorems as they appear on this page.

(6-Nov-2006) The first 3 paragraphs in Appendix 2: Note on the Axioms were rewritten to clarify the connection between Tarski's axiom system and Metamath.

(31-Oct-2006) ocat asked for a do-over due to a bug in mmj2 -- if you downloaded the mmj2.zip version dated 10/28/2006, then download the new version dated 10/30.

(29-Oct-2006) ocat has announced that the long-awaited 1-Nov-2006 release of mmj2 is available now.
     The new "Unify+Get Hints" is quite useful, and any proof can be generated as follows. With "?" in the Hyp field and Ref field blank, select "Unify+Get Hints". Select a hint from the list and put it in the Ref field. Edit any $n dummy variables to become the desired wffs. Rinse and repeat for the new proof steps generated, until the proof is done.
     The new tutorial, mmj2PATutorial.bat, explains this in detail. One way to reduce or avoid dummy $n's is to fill in the Hyp field with a comma-separated list of any known hypothesis matches to earlier proof steps, keeping a "?" in the list to indicate that the remaining hypotheses are unknown. Then "Unify+Get Hints" can be applied. The tutorial page \mmj2\data\mmp\PATutorial\Page405.mmp has an example.
     Don't forget that the eimm export/import program lets you go back and forth between the mmj2 and the metamath program proof assistants, without exiting from either one, to exploit the best features of each as required.

(21-Oct-2006) Martin Kiselkov has written a Metamath proof verifier in the Lua scripting language, called verify.lua. While it is not practical as an everyday verifier - he writes that it takes about 40 minutes to verify set.mm on a a Pentium 4 - it could be useful to someone learning Lua or Metamath, and importantly it provides another independent way of verifying the correctness of Metamath proofs. His code looks like it is nicely structured and very readable. He is currently working on a faster version in C++.

(19-Oct-2006) New AsteroidMeta page by Raph, Distinctors_vs_binders.

(13-Oct-2006) I put a simple Metamath browser on my PDA (Palm Tungsten E) so that I don't have to lug around my laptop. Here is a screenshot. It isn't polished, but I'll provide the file + instructions if anyone wants it.

(3-Oct-2006) A blog entry, Principia for Reverse Mathematics.

(28-Sep-2006) A blog entry, Metamath responds.

(26-Sep-2006) A blog entry, Metamath isn't hygienic.

(11-Aug-2006) A blog entry, Metamath and the Peano Induction Axiom.

(26-Jul-2006) A new open problem in predicate calculus was added.

(18-Jun-2006) The 6,000th theorem, mt4d, was added to the Metamath Proof Explorer part of the database.

(9-May-2006) Luca Ciciriello has upgraded the t2mf program, which is a C program used to create the MIDI files on the Metamath Music Page, so that it works on MacOS X. This is a nice accomplishment, since the original program was written before C was standardized by ANSI and will not compile on modern compilers.
      Unfortunately, the original program source states no copyright terms. The main author, Tim Thompson, has kindly agreed to release his code to public domain, but two other authors have also contributed to the code, and so far I have been unable to contact them for copyright clearance. Therefore I cannot offer the MacOS X version for public download on this site until this is resolved. Update 10-May-2006: Another author, M. Czeiszperger, has released his contribution to public domain.
      If you are interested in Luca's modified source code, please contact me directly.

(18-Apr-2006) Incomplete proofs in progress can now be interchanged between the Metamath program's CLI Proof Assistant and mmj2's GUI Proof Assistant, using a new export-import program called eimm. This can be done without exiting either proof assistant, so that the strengths of each approach can be exploited during proof development. See "Use Case 5a" and "Use Case 5b" at mmj2ProofAssistantFeedback.

(28-Mar-2006) Scott Fenton updated his second version of Metamath Solitaire (the one that uses external axioms). He writes: "I've switched to making it a standalone program, as it seems silly to have an applet that can't be run in a web browser. Check the README file for further info." The download is mmsol-0.5.tar.gz.

(27-Mar-2006) Scott Fenton has updated the Metamath Solitaire Java applet to Java 1.5: (1) QSort has been stripped out: its functionality is in the Collections class that Sun ships; (2) all Vectors have been replaced by ArrayLists; (3) generic types have been tossed in wherever they fit: this cuts back drastically on casting; and (4) any warnings Eclipse spouted out have been dealt with. I haven't yet updated it officially, because I don't know if it will work with Microsoft's JVM in older versions of Internet Explorer. The current official version is compiled with Java 1.3, because it won't work with Microsoft's JVM if it is compiled with Java 1.4. (As distasteful as that seems, I will get complaints from users if it doesn't work with Microsoft's JVM.) If anyone can verify that Scott's new version runs on Microsoft's JVM, I would be grateful. Scott's new version is mm.java-1.5.gz; after uncompressing it, rename it to mm.java, use it to replace the existing mm.java file in the Metamath Solitaire download, and recompile according to instructions in the mm.java comments.
      Scott has also created a second version, mmsol-0.2.tar.gz, that reads the axioms from ASCII files, instead of having the axioms hard-coded in the program. This can be very useful if you want to play with custom axioms, and you can also add a collection of starting theorems as "axioms" to work from. However, it must be run from the local directory with appletviewer, since the default Java security model doesn't allow reading files from a browser. It works with the JDK 5 Update 6 Java download.
To compile (from Windows Command Prompt): C:\Program Files\Java\jdk1.5.0_06\bin\javac.exe mm.java
To run (from Windows Command Prompt): C:\Program Files\Java\jdk1.5.0_06\bin\appletviewer.exe mms.html

(21-Jan-2006) Juha Arpiainen proved the independence of axiom ax-11 from the others. This was published as an open problem in my 1995 paper (Remark 9.5 on PDF page 17). See Item 9a on the Workshop Miscellany for his seven-line proof. See also the Asteroid Meta metamathMathQuestions page under the heading "Axiom of variable substitution: ax-11". Congratulations, Juha!

(20-Oct-2005) Juha Arpiainen is working on a proof verifier in Common Lisp called Bourbaki. Its proof language has its roots in Metamath, with the goal of providing a more powerful syntax and definitional soundness checking. See its documentation and related discussion.

(17-Oct-2005) Marnix Klooster has written a Metamath proof verifier in Haskell, called Hmm. Also see his Announcement. The complete program (Hmm.hs, HmmImpl.hs, and HmmVerify.hs) has only 444 lines of code, excluding comments and blank lines. It verifies compressed as well as regular proofs; moreover, it transparently verifies both per-spec compressed proofs and the flawed format he uncovered (see comment below of 16-Oct-05).

(16-Oct-2005) Marnix Klooster noticed that for large proofs, the compressed proof format did not match the spec in the book. His algorithm to correct the problem has been put into the Metamath program (version 0.07.6). The program still verifies older proofs with the incorrect format, but the user will be nagged to update them with 'save proof *'. In set.mm, 285 out of 6376 proofs are affected. (The incorrect format did not affect proof correctness or verification, since the compression and decompression algorithms matched each other.)

(13-Sep-2005) Scott Fenton found an interesting axiom, ax46, which could be used to replace both ax-4 and ax-6.

(29-Jul-2005) Metamath was selected as site of the week by American Scientist Online.

(8-Jul-2005) Roy Longton has contributed 53 new theorems to the Quantum Logic Explorer. You can see them in the Theorem List starting at lem3.3.3lem1. He writes, "If you want, you can post an open challenge to see if anyone can find shorter proofs of the theorems I submitted."

(10-May-2005) A Usenet post I posted about the infinite prime proof; another one about indexed unions.

(3-May-2005) The theorem divexpt is the 5,000th theorem added to the Metamath Proof Explorer database.

(12-Apr-2005) Raph Levien solved the open problem in item 16 on the Workshop Miscellany page and as a corollary proved that axiom ax-9 is independent from the other axioms of predicate calculus and equality. This is the first such independence proof so far; a goal is to prove all of them independent (or to derive any redundant ones from the others).

(8-Mar-2005) I added a paragraph above our complex number axioms table, summarizing the construction and indicating where Dedekind cuts are defined. Thanks to Andrew Buhr for comments on this.

(16-Feb-2005) The Metamath Music Page is mentioned as a reference or resource for a university course called Math, Mind, and Music. .

(28-Jan-2005) Steven Cullinane parodied the Metamath Music Page in his blog.

(18-Jan-2005) Waldek Hebisch upgraded the Metamath program to run on the AMD64 64-bit processor.

(17-Jan-2005) A symbol list summary was added to the beginning of the Hilbert Space Explorer Home Page. Thanks to Mladen Pavicic for suggesting this.

(6-Jan-2005) Someone assembled an amazon.com list of some of the books in the Metamath Proof Explorer Bibliography.

(4-Jan-2005) The definition of ordinal exponentiation was decided on after this Usenet discussion.

(19-Dec-2004) A bit of trivia: my Erdös number is 2, as you can see from this list.

(20-Oct-2004) I started this Usenet discussion about the "reals are uncountable" proof (127 comments; last one on Nov. 12).

(12-Oct-2004) gch-kn shows the equivalence of the Generalized Continuum Hypothesis and Prof. Nambiar's Axiom of Combinatorial Sets. This proof answers his Open Problem 2 (PDF file).

(5-Aug-2004) I gave a talk on "Hilbert Lattice Equations" at the Argonne workshop.

(25-Jul-2004) The theorem nthruz is the 4,000th theorem added to the Metamath Proof Explorer database.

(27-May-2004) Josiah Burroughs contributed the proofs u1lemn1b, u1lem3var1, oi3oa3lem1, and oi3oa3 to the Quantum Logic Explorer database ql.mm.

(23-May-2004) Some minor typos found by Josh Purinton were corrected in the Metamath book. In addition, Josh simplified the definition of the closure of a pre-statement of a formal system in Appendix C.

(5-May-2004) Gregory Bush has found shorter proofs for 67 of the 193 propositional calculus theorems listed in Principia Mathematica, thus establishing 67 new records. (This was challenge #4 on the open problems page.)

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