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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 29-Mar-2024 at 5:04 AM ET.
Recent Additions to the Metamath Proof Explorer   Notes (last updated 7-Dec-2020 )
DateLabelDescription
Theorem
 
23-Mar-2024fundcmpsurbijinj 43447 Every function 𝐹:𝐴𝐵 can be decomposed into a surjective, a bijective and an injective function. (Contributed by AV, 23-Mar-2024.)
((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑖𝑝𝑞((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
 
23-Mar-2024cosq34lt1 25039 Cosine is less than one in the third and fourth quadrants. (Contributed by Jim Kingdon, 23-Mar-2024.)
(𝐴 ∈ (π[,)(2 · π)) → (cos‘𝐴) < 1)
 
23-Mar-2024cos02pilt1 25038 Cosine is less than one between zero and 2 · π. (Contributed by Jim Kingdon, 23-Mar-2024.)
(𝐴 ∈ (0(,)(2 · π)) → (cos‘𝐴) < 1)
 
23-Mar-2024o2p2e4 8155 2 + 2 = 4 for ordinal numbers. Ordinal numbers are modeled as Von Neumann ordinals; see df-suc 6190. For the usual proof using complex numbers, see 2p2e4 11760. (Contributed by NM, 18-Aug-2021.) Avoid ax-rep 5181, from a comment by Sophie. (Revised by SN, 23-Mar-2024.)
(2o +o 2o) = 4o
 
23-Mar-2024funfvima2d 6985 A function's value in a preimage belongs to the image. (Contributed by Stanislas Polu, 9-Mar-2020.) (Revised by AV, 23-Mar-2024.)
(𝜑𝐹:𝐴𝐵)       ((𝜑𝑋𝐴) → (𝐹𝑋) ∈ (𝐹𝐴))
 
23-Mar-2024elabd 3666 Explicit demonstration the class {𝑥𝜓} is not empty by the example 𝐴. (Contributed by RP, 12-Aug-2020.) (Revised by AV, 23-Mar-2024.)
(𝜑𝐴𝑉)    &   (𝜑𝜒)    &   (𝑥 = 𝐴 → (𝜓𝜒))       (𝜑𝐴 ∈ {𝑥𝜓})
 
22-Mar-2024fundcmpsurinjpreimafv 43445 Every function 𝐹:𝐴𝐵 can be decomposed into a surjective function onto 𝑃 and an injective function from 𝑃. (Contributed by AV, 12-Mar-2024.) (Proof shortened by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔(𝑔:𝐴onto𝑃:𝑃1-1𝐵𝐹 = (𝑔)))
 
22-Mar-2024fundcmpsurbijinjpreimafv 43444 Every function 𝐹:𝐴𝐵 can be decomposed into a surjective function onto 𝑃, a bijective function from 𝑃 and an injective function into the codomain of 𝐹. (Contributed by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑖((𝑔:𝐴onto𝑃:𝑃1-1-onto→(𝐹𝐴) ∧ 𝑖:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
 
22-Mar-2024imasetpreimafvbij 43443 The mapping 𝐻 is a bijective function betwen the set 𝑃 of all preimages of values of function 𝐹 and the range of 𝐹. (Contributed by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}    &   𝐻 = (𝑝𝑃 (𝐹𝑝))       ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃1-1-onto→(𝐹𝐴))
 
22-Mar-2024imasetpreimafvbijlemfo 43442 Lemma for imasetpreimafvbij 43443: the mapping 𝐻 is a function onto the range of function 𝐹. (Contributed by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}    &   𝐻 = (𝑝𝑃 (𝐹𝑝))       ((𝐹 Fn 𝐴𝐴𝑉) → 𝐻:𝑃onto→(𝐹𝐴))
 
22-Mar-2024imasetpreimafvbijlemf1 43441 Lemma for imasetpreimafvbij 43443: the mapping 𝐻 is an injective function into the range of function 𝐹. (Contributed by AV, 9-Mar-2024.) (Revised by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}    &   𝐻 = (𝑝𝑃 (𝐹𝑝))       (𝐹 Fn 𝐴𝐻:𝑃1-1→(𝐹𝐴))
 
22-Mar-2024imasetpreimafvbijlemf 43438 Lemma for imasetpreimafvbij 43443: the mapping 𝐻 is a function into the range of function 𝐹. (Contributed by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}    &   𝐻 = (𝑝𝑃 (𝐹𝑝))       (𝐹 Fn 𝐴𝐻:𝑃⟶(𝐹𝐴))
 
22-Mar-2024uniimaelsetpreimafv 43433 The union of the image of an element of the preimage of a function value is an element of the range of the function. (Contributed by AV, 5-Mar-2024.) (Revised by AV, 22-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴𝑆𝑃) → (𝐹𝑆) ∈ ran 𝐹)
 
17-Mar-2024fundcmpsurinjimaid 43448 Every function 𝐹:𝐴𝐵 can be decomposed into a surjective function onto the image (𝐹𝐴) of the domain of 𝐹 and an injective function from the image (𝐹𝐴). (Contributed by AV, 17-Mar-2024.)
𝐼 = (𝐹𝐴)    &   𝐺 = (𝑥𝐴 ↦ (𝐹𝑥))    &   𝐻 = ( I ↾ 𝐼)       (𝐹:𝐴𝐵 → (𝐺:𝐴onto𝐼𝐻:𝐼1-1𝐵𝐹 = (𝐻𝐺)))
 
13-Mar-2024fundcmpsurinjALT 43449 Alternate proof of fundcmpsurinj 43446, based on fundcmpsurinjimaid 43448: Every function 𝐹:𝐴𝐵 can be decomposed into a surjective and an injective function. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by AV, 13-Mar-2024.)
((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑝(𝑔:𝐴onto𝑝:𝑝1-1𝐵𝐹 = (𝑔)))
 
13-Mar-2024fundcmpsurinj 43446 Every function 𝐹:𝐴𝐵 can be decomposed into a surjective and an injective function. (Contributed by AV, 13-Mar-2024.)
((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑝(𝑔:𝐴onto𝑝:𝑝1-1𝐵𝐹 = (𝑔)))
 
12-Mar-2024uniimaprimaeqfv 43419 The union of the image of the preimage of a function value is the function value. (Contributed by AV, 12-Mar-2024.)
((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 “ (𝐹 “ {(𝐹𝑋)})) = (𝐹𝑋))
 
10-Mar-2024preimafvelsetpreimafv 43425 The preimage of a function value is an element of the class 𝑃 of all preimages of function values. (Contributed by AV, 10-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴𝐴𝑉𝑋𝐴) → (𝐹 “ {(𝐹𝑋)}) ∈ 𝑃)
 
10-Mar-2024elsetpreimafvb 43421 The characterization of an element of the class 𝑃 of all preimages of function values. (Contributed by AV, 10-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       (𝑆𝑉 → (𝑆𝑃 ↔ ∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)})))
 
10-Mar-2024setpreimafvex 43420 The class 𝑃 of all preimages of function values is a set. (Contributed by AV, 10-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       (𝐴𝑉𝑃 ∈ V)
 
9-Mar-2024elsetpreimafvrab 43431 An element of the preimage of a function value expressed as a restricted class abstraction. (Contributed by AV, 9-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → 𝑆 = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐹𝑋)})
 
9-Mar-2024eqfvelsetpreimafv 43430 If an element of the domain of the function has the same function value as an element of the preimage of a function value, then it is an element of the same preimage. (Contributed by AV, 9-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → ((𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋)) → 𝑌𝑆))
 
9-Mar-2024elsetpreimafvbi 43428 An element of the preimage of a function value is an element of the domain of the function with the same value as another element of the preimage. (Contributed by AV, 9-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → (𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))
 
8-Mar-2024elsetpreimafveq 43434 If two preimages of function values contain elements with identical function values, then both preimages are equal. (Contributed by AV, 8-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑅𝑃) ∧ (𝑋𝑆𝑌𝑅)) → ((𝐹𝑋) = (𝐹𝑌) → 𝑆 = 𝑅))
 
8-Mar-2024fvelsetpreimafv 43424 There is an element in a preimage 𝑆 of function values so that 𝑆 is the preimage of the function value at this element. (Contributed by AV, 8-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴𝑆𝑃) → ∃𝑥𝑆 𝑆 = (𝐹 “ {(𝐹𝑥)}))
 
8-Mar-2024elsetpreimafvssdm 43423 An element of the class 𝑃 of all preimages of function values is a subset of the domain of the function. (Contributed by AV, 8-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴𝑆𝑃) → 𝑆𝐴)
 
8-Mar-2024elsetpreimafv 43422 An element of the class 𝑃 of all preimages of function values. (Contributed by AV, 8-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       (𝑆𝑃 → ∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)}))
 
7-Mar-2024preimafvn0 43417 The preimage of a function value is not empty. (Contributed by AV, 7-Mar-2024.)
((𝐹 Fn 𝐴𝑋𝐴) → (𝐹 “ {(𝐹𝑋)}) ≠ ∅)
 
7-Mar-2024preimafvsnel 43416 The preimage of a function value at 𝑋 contains 𝑋. (Contributed by AV, 7-Mar-2024.)
((𝐹 Fn 𝐴𝑋𝐴) → 𝑋 ∈ (𝐹 “ {(𝐹𝑋)}))
 
6-Mar-20240nelsetpreimafv 43427 The empty set is not an element of the class 𝑃 of all preimages of function values. (Contributed by AV, 6-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)
 
5-Mar-2024imasetpreimafvbijlemfv1 43440 Lemma for imasetpreimafvbij 43443: for a preimage of a value of function 𝐹 there is an element of the preimage so that the value of the mapping 𝐻 at this preimage is the function value at this element. (Contributed by AV, 5-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}    &   𝐻 = (𝑝𝑃 (𝐹𝑝))       ((𝐹 Fn 𝐴𝑋𝑃) → ∃𝑦𝑋 (𝐻𝑋) = (𝐹𝑦))
 
5-Mar-2024imasetpreimafvbijlemfv 43439 Lemma for imasetpreimafvbij 43443: the value of the mapping 𝐻 at a preimage of a value of function 𝐹. (Contributed by AV, 5-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}    &   𝐻 = (𝑝𝑃 (𝐹𝑝))       ((𝐹 Fn 𝐴𝑌𝑃𝑋𝑌) → (𝐻𝑌) = (𝐹𝑋))
 
5-Mar-2024imaelsetpreimafv 43432 The image of an element of the preimage of a function value is the singleton consisting of the function value at one of its elements. (Contributed by AV, 5-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → (𝐹𝑆) = {(𝐹𝑋)})
 
5-Mar-2024elsetpreimafveqfv 43429 The elements of the preimage of a function value have the same function values. (Contributed by AV, 5-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       ((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑋𝑆𝑌𝑆)) → (𝐹𝑋) = (𝐹𝑌))
 
5-Mar-2024preimafvsspwdm 43426 The class 𝑃 of all preimages of function values is a subset of the power set of the domain of the function. (Contributed by AV, 5-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}       (𝐹 Fn 𝐴𝑃 ⊆ 𝒫 𝐴)
 
5-Mar-2024uniimafveqt 43418 The union of the image of a subset 𝑆 of the domain of a function with elements having the same function value is the function value at one of the elements of 𝑆. (Contributed by AV, 5-Mar-2024.)
((𝐹:𝐴𝐵𝑆𝐴𝑋𝑆) → (∀𝑥𝑆 (𝐹𝑥) = (𝐹𝑋) → (𝐹𝑆) = (𝐹𝑋)))
 
5-Mar-2024wl-cbvalsbi 34666 Change bounded variables in a special case. The reverse direction seems to involve ax-11 2151. My hope is that I will in some future be able to prove mo3 2641 with reversed quantifiers not using ax-11 2151. See also the remark in mo4 2643, which lead me to this effort. (Contributed by Wolf Lammen, 5-Mar-2024.)
(∀𝑥𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)
 
5-Mar-2024iuneqconst 4921 Indexed union of identical classes. (Contributed by AV, 5-Mar-2024.)
(𝑥 = 𝑋𝐵 = 𝐶)       ((𝑋𝐴 ∧ ∀𝑥𝐴 𝐵 = 𝐶) → 𝑥𝐴 𝐵 = 𝐶)
 
4-Mar-2024fundcmpsurinjlem2 43436 Lemma 2 for fundcmpsurinj 43446. (Contributed by AV, 4-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}    &   𝐺 = (𝑥𝐴 ↦ (𝐹 “ {(𝐹𝑥)}))       ((𝐹 Fn 𝐴𝐴𝑉) → 𝐺:𝐴onto𝑃)
 
4-Mar-2024fundcmpsurinjlem1 43435 Lemma 1 for fundcmpsurinj 43446. (Contributed by AV, 4-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}    &   𝐺 = (𝑥𝐴 ↦ (𝐹 “ {(𝐹𝑥)}))       ran 𝐺 = 𝑃
 
3-Mar-2024fundcmpsurinjlem3 43437 Lemma 3 for fundcmpsurinj 43446. (Contributed by AV, 3-Mar-2024.)
𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}    &   𝐻 = (𝑝𝑃 (𝐹𝑝))       ((Fun 𝐹𝑋𝑃) → (𝐻𝑋) = (𝐹𝑋))
 
3-Mar-2024mendvscafval 39668 Scalar multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 3-Mar-2024.)
𝐴 = (MEndo‘𝑀)    &    · = ( ·𝑠𝑀)    &   𝐵 = (Base‘𝐴)    &   𝑆 = (Scalar‘𝑀)    &   𝐾 = (Base‘𝑆)    &   𝐸 = (Base‘𝑀)       ( ·𝑠𝐴) = (𝑥𝐾, 𝑦𝐵 ↦ ((𝐸 × {𝑥}) ∘f · 𝑦))
 
3-Mar-2024mendmulrfval 39665 Multiplication in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 3-Mar-2024.)
𝐴 = (MEndo‘𝑀)    &   𝐵 = (Base‘𝐴)       (.r𝐴) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝑦))
 
3-Mar-2024mendplusgfval 39663 Addition in the module endomorphism algebra. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 3-Mar-2024.)
𝐴 = (MEndo‘𝑀)    &   𝐵 = (Base‘𝐴)    &    + = (+g𝑀)       (+g𝐴) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥f + 𝑦))
 
2-Mar-2024clwwlknonmpo 27795 (ClWWalksNOn‘𝐺) is an operator mapping a vertex 𝑣 and a nonnegative integer 𝑛 to the set of closed walks on 𝑣 of length 𝑛 as words over the set of vertices in a graph 𝐺. (Contributed by AV, 25-Feb-2022.) (Proof shortened by AV, 2-Mar-2024.)
(ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
 
2-Mar-2024pcofval 23541 The value of the path concatenation function on a topological space. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.) (Proof shortened by AV, 2-Mar-2024.)
(*𝑝𝐽) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑓‘(2 · 𝑥)), (𝑔‘((2 · 𝑥) − 1)))))
 
2-Mar-2024marepvfval 21102 First substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.) (Proof shortened by AV, 2-Mar-2024.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 matRepV 𝑅)    &   𝑉 = ((Base‘𝑅) ↑m 𝑁)       𝑄 = (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗)))))
 
2-Mar-2024marrepfval 21097 First substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.) (Proof shortened by AV, 2-Mar-2024.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 matRRep 𝑅)    &    0 = (0g𝑅)       𝑄 = (𝑚𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))))
 
2-Mar-2024ipffval 20720 The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &    · = (·if𝑊)        · = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦))
 
2-Mar-2024psrmulr 20092 The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)
𝑆 = (𝐼 mPwSer 𝑅)    &   𝐵 = (Base‘𝑆)    &    · = (.r𝑅)    &    = (.r𝑆)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}        = (𝑓𝐵, 𝑔𝐵 ↦ (𝑘𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦𝐷𝑦r𝑘} ↦ ((𝑓𝑥) · (𝑔‘(𝑘f𝑥)))))))
 
2-Mar-2024scaffval 19581 The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = ( ·sf𝑊)    &    · = ( ·𝑠𝑊)        = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
 
2-Mar-2024dvrfval 19363 Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 2-Mar-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)    &    / = (/r𝑅)        / = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦)))
 
2-Mar-2024oppglsm 18696 The subspace sum operation in the opposite group. (Contributed by Mario Carneiro, 19-Apr-2016.) (Proof shortened by AV, 2-Mar-2024.)
𝑂 = (oppg𝐺)    &    = (LSSum‘𝐺)       (𝑇(LSSum‘𝑂)𝑈) = (𝑈 𝑇)
 
2-Mar-2024plusffval 17846 The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 2-Mar-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (+𝑓𝐺)        = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + 𝑦))
 
2-Mar-2024xpccofval 17420 Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) (Proof shortened by AV, 2-Mar-2024.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐾 = (Hom ‘𝑇)    &    · = (comp‘𝐶)    &    = (comp‘𝐷)    &   𝑂 = (comp‘𝑇)       𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐾𝑦), 𝑓 ∈ (𝐾𝑥) ↦ ⟨((1st𝑔)(⟨(1st ‘(1st𝑥)), (1st ‘(2nd𝑥))⟩ · (1st𝑦))(1st𝑓)), ((2nd𝑔)(⟨(2nd ‘(1st𝑥)), (2nd ‘(2nd𝑥))⟩ (2nd𝑦))(2nd𝑓))⟩))
 
1-Mar-2024xpchomfval 17417 Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
𝑇 = (𝐶 ×c 𝐷)    &   𝐵 = (Base‘𝑇)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   𝐾 = (Hom ‘𝑇)       𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))
 
1-Mar-2024natfval 17204 Value of the function giving natural transformations between two categories. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
𝑁 = (𝐶 Nat 𝐷)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &    · = (comp‘𝐷)       𝑁 = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ (1st𝑓) / 𝑟(1st𝑔) / 𝑠{𝑎X𝑥𝐵 ((𝑟𝑥)𝐽(𝑠𝑥)) ∣ ∀𝑥𝐵𝑦𝐵 ∈ (𝑥𝐻𝑦)((𝑎𝑦)(⟨(𝑟𝑥), (𝑟𝑦)⟩ · (𝑠𝑦))((𝑥(2nd𝑓)𝑦)‘)) = (((𝑥(2nd𝑔)𝑦)‘)(⟨(𝑟𝑥), (𝑠𝑥)⟩ · (𝑠𝑦))(𝑎𝑥))})
 
1-Mar-2024comfffval 16956 Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
𝑂 = (compf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)       𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)𝐻𝑦), 𝑓 ∈ (𝐻𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
 
1-Mar-2024homffval 16948 Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)
𝐹 = (Homf𝐶)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)       𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥𝐻𝑦))
 
29-Feb-2024evls1pw 20417 Univariate polynomial evaluation for subrings maps the exponentiation of a polynomial to the exponentiation of the evaluated polynomial. (Contributed by SN, 29-Feb-2024.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝑈 = (𝑆s 𝑅)    &   𝑊 = (Poly1𝑈)    &   𝐺 = (mulGrp‘𝑊)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑊)    &    = (.g𝐺)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑄‘(𝑁 𝑋)) = (𝑁(.g‘(mulGrp‘(𝑆s 𝐾)))(𝑄𝑋)))
 
29-Feb-2024evlspw 20234 Polynomial evaluation for subrings maps the exponentiation of a polynomial to the exponentiation of the evaluated polynomial. (Contributed by SN, 29-Feb-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝐺 = (mulGrp‘𝑊)    &    = (.g𝐺)    &   𝑈 = (𝑆s 𝑅)    &   𝑃 = (𝑆s (𝐾m 𝐼))    &   𝐻 = (mulGrp‘𝑃)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑄‘(𝑁 𝑋)) = (𝑁(.g𝐻)(𝑄𝑋)))
 
29-Feb-2024mpllvec 20161 The polynomial ring is a vector space. (Contributed by SN, 29-Feb-2024.)
𝑃 = (𝐼 mPoly 𝑅)       ((𝐼𝑉𝑅 ∈ DivRing) → 𝑃 ∈ LVec)
 
27-Feb-2024pwmnd 18040 The power set of a class 𝐴 is a monoid under union. (Contributed by AV, 27-Feb-2024.)
(Base‘𝑀) = 𝒫 𝐴    &   (+g𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥𝑦))       𝑀 ∈ Mnd
 
27-Feb-2024pwmndid 18039 The identity of the monoid of the power set of a class 𝐴 under union is the empty set. (Contributed by AV, 27-Feb-2024.)
(Base‘𝑀) = 𝒫 𝐴    &   (+g𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥𝑦))       (0g𝑀) = ∅
 
27-Feb-2024pwmndgplus 18038 The operation of the monoid of the power set of a class 𝐴 under union. (Contributed by AV, 27-Feb-2024.)
(Base‘𝑀) = 𝒫 𝐴    &   (+g𝑀) = (𝑥 ∈ 𝒫 𝐴, 𝑦 ∈ 𝒫 𝐴 ↦ (𝑥𝑦))       ((𝑋 ∈ 𝒫 𝐴𝑌 ∈ 𝒫 𝐴) → (𝑋(+g𝑀)𝑌) = (𝑋𝑌))
 
27-Feb-2024pwuncl 7481 Power classes are closed under union. (Contributed by AV, 27-Feb-2024.)
((𝐴 ∈ 𝒫 𝑋𝐵 ∈ 𝒫 𝑋) → (𝐴𝐵) ∈ 𝒫 𝑋)
 
25-Feb-2024injsubmefmnd 43994 The set of injective endofunctions on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Feb-2024.)
𝑀 = (EndoFMnd‘𝐴)       (𝐴𝑉 → {:𝐴1-1𝐴} ∈ (SubMnd‘𝑀))
 
25-Feb-2024sursubmefmnd 43993 The set of surjective endofunctions on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Feb-2024.)
𝑀 = (EndoFMnd‘𝐴)       (𝐴𝑉 → {:𝐴onto𝐴} ∈ (SubMnd‘𝑀))
 
25-Feb-2024insubm 17971 The intersection of two submonoids is a submonoid. (Contributed by AV, 25-Feb-2024.)
((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) → (𝐴𝐵) ∈ (SubMnd‘𝑀))
 
25-Feb-2024nfsb 2558 If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. For a version requiring more disjoint variables, but fewer axioms, see nfsbv 2340. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 25-Feb-2024.)
𝑧𝜑       𝑧[𝑦 / 𝑥]𝜑
 
24-Feb-2024nfsumw 15035 Version of nfsum 15036 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 24-Feb-2024.)
𝑥𝐴    &   𝑥𝐵       𝑥Σ𝑘𝐴 𝐵
 
23-Feb-2024efmndtmd 43997 The monoid of endofunctions on a set 𝐴 is a topological monoid. Formerly part of proof for symgtgp 22637. (Contributed by AV, 23-Feb-2024.)
𝑀 = (EndoFMnd‘𝐴)       (𝐴𝑉𝑀 ∈ TopMnd)
 
22-Feb-2024selvcl 39016 Closure of the "variable selection" function. (Contributed by SN, 22-Feb-2024.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑈 = ((𝐼𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐸 = (Base‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐽𝐼)    &   (𝜑𝐹𝐵)       (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ 𝐸)
 
22-Feb-2024selvval2lem5 39015 The fifth argument passed to evalSub is in the domain (a function 𝐼𝐸). (Contributed by SN, 22-Feb-2024.)
𝑈 = ((𝐼𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐸 = (Base‘𝑇)    &   𝐹 = (𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐽𝐼)       (𝜑𝐹 ∈ (𝐸m 𝐼))
 
21-Feb-2024remulcand 39128 Commuted version of remulcan2d 39034 without ax-mulcom 10589. (Contributed by SN, 21-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵))
 
21-Feb-2024readdcan2 39120 Commuted version of readdcan 10802 without ax-mulcom 10589. (Contributed by SN, 21-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
 
21-Feb-2024evlsvarpw 20235 Polynomial evaluation for subrings maps the exponentiation of a variable to the exponentiation of the evaluated variable. (Contributed by SN, 21-Feb-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝐺 = (mulGrp‘𝑊)    &    = (.g𝐺)    &   𝑋 = ((𝐼 mVar 𝑈)‘𝑌)    &   𝑈 = (𝑆s 𝑅)    &   𝑃 = (𝑆s (𝐵m 𝐼))    &   𝐻 = (mulGrp‘𝑃)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐼𝑉)    &   (𝜑𝑌𝐼)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝑄‘(𝑁 𝑋)) = (𝑁(.g𝐻)(𝑄𝑋)))
 
21-Feb-2024sb1 2496 One direction of a simplified definition of substitution. The converse requires either a disjoint variable condition (sb5 2267) or a non-freeness hypothesis (sb5f 2531). See also sb1v 2086. (Contributed by NM, 13-May-1993.) Revise df-sb 2061. (Revised by Wolf Lammen, 21-Feb-2024.)
([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
 
21-Feb-2024sb3 2495 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Feb-2024.)
(¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))
 
21-Feb-2024sb4b 2492 Simplified definition of substitution when variables are distinct. Version of sb6 2084 with a distinctor. (Contributed by NM, 27-May-1997.) Revise df-sb 2061. (Revised by Wolf Lammen, 21-Feb-2024.)
(¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑)))
 
19-Feb-2024symgplusg 18445 The group operation of a symmetric group is the function composition. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.) (Proof shortened by AV, 19-Feb-2024.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)        + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))
 
19-Feb-2024grpsubfvalALT 18086 Shorter proof of grpsubfval 18085 using ax-rep 5181. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 19-Feb-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐼 = (invg𝐺)    &    = (-g𝐺)        = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦)))
 
19-Feb-2024grpsubfval 18085 Group subtraction (division) operation. For a shorter proof using ax-rep 5181, see grpsubfvalALT 18086. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) Remove dependency on ax-rep 5181. (Revised by Rohan Ridenour, 17-Aug-2023.) (Proof shortened by AV, 19-Feb-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐼 = (invg𝐺)    &    = (-g𝐺)        = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 + (𝐼𝑦)))
 
18-Feb-2024smndex2dlinvh 44017 The halving functions 𝐻 are left inverses of the doubling function 𝐷. (Contributed by AV, 18-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝐵 = (Base‘𝑀)    &    0 = (0g𝑀)    &   𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))    &   𝑁 ∈ ℕ0    &   𝐻 = (𝑥 ∈ ℕ0 ↦ if(2 ∥ 𝑥, (𝑥 / 2), 𝑁))       (𝐻𝐷) = 0
 
18-Feb-2024smndex2hbas 44016 The halving functions 𝐻 are endofunctions on 0. (Contributed by AV, 18-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝐵 = (Base‘𝑀)    &    0 = (0g𝑀)    &   𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))    &   𝑁 ∈ ℕ0    &   𝐻 = (𝑥 ∈ ℕ0 ↦ if(2 ∥ 𝑥, (𝑥 / 2), 𝑁))       𝐻𝐵
 
18-Feb-2024smndex2dnrinv 44015 The doubling function 𝐷 has no right inverse in the monoid of endofunctions on 0. (Contributed by AV, 18-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝐵 = (Base‘𝑀)    &    0 = (0g𝑀)    &   𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))       𝑓𝐵 (𝐷𝑓) ≠ 0
 
18-Feb-2024smndex2dbas 44014 The doubling function 𝐷 is an endofunction on 0. (Contributed by AV, 18-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝐵 = (Base‘𝑀)    &    0 = (0g𝑀)    &   𝐷 = (𝑥 ∈ ℕ0 ↦ (2 · 𝑥))       𝐷𝐵
 
18-Feb-2024symgsubmefmndALT 43996 The symmetric group on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. Alternate proof based on submefmnd 43992 and not on injsubmefmnd 43994 and sursubmefmnd 43993. (Contributed by AV, 18-Feb-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑀 = (EndoFMnd‘𝐴)    &   𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴𝑉𝐵 ∈ (SubMnd‘𝑀))
 
18-Feb-2024symgsubmefmnd 43995 The symmetric group on a set 𝐴 is a submonoid of the monoid of endofunctions on 𝐴. (Contributed by AV, 18-Feb-2024.)
𝑀 = (EndoFMnd‘𝐴)    &   𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴𝑉𝐵 ∈ (SubMnd‘𝑀))
 
18-Feb-2024efmnd2hash 43991 The monoid of endofunctions on a (proper) pair has cardinality 4. (Contributed by AV, 18-Feb-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝐴 = {𝐼, 𝐽}       ((𝐼𝑉𝐽𝑊𝐼𝐽) → (♯‘𝐵) = 4)
 
17-Feb-2024nsmndex1 44013 The base set 𝐵 of the constructed monoid 𝑆 is not a submonoid of the monoid 𝑀 of endofunctions on set 0, although 𝑀 ∈ Mnd and 𝑆 ∈ Mnd and 𝐵 ⊆ (Base‘𝑀) hold. (Contributed by AV, 17-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))    &   𝐵 = ({𝐼} ∪ 𝑛 ∈ (0..^𝑁){(𝐺𝑛)})    &   𝑆 = (𝑀s 𝐵)       𝐵 ∉ (SubMnd‘𝑀)
 
17-Feb-2024smndex1n0mnd 44012 The identity of the monoid 𝑀 of endofunctions on set 0 is not contained in the base set of the constructed monoid 𝑆. (Contributed by AV, 17-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))    &   𝐵 = ({𝐼} ∪ 𝑛 ∈ (0..^𝑁){(𝐺𝑛)})    &   𝑆 = (𝑀s 𝐵)       (0g𝑀) ∉ 𝐵
 
17-Feb-2024idresefmnd 43999 The structure with the singleton containing only the identity function restricted to a set 𝐴 as base set and the function composition as group operation, constructed by (structure) restricting the monoid of endofunctions on 𝐴 to that singleton, is a monoid whose base set is a subset of the base set of the monoid of endofunctions on 𝐴. (Contributed by AV, 17-Feb-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐸 = (𝐺s {( I ↾ 𝐴)})       (𝐴𝑉 → (𝐸 ∈ Mnd ∧ (Base‘𝐸) ⊆ (Base‘𝐺)))
 
17-Feb-2024idressubmefmnd 43998 The singleton containing only the identity function restricted to a set is a submonoid of the monoid of endofunctions on this set. (Contributed by AV, 17-Feb-2024.)
𝐺 = (EndoFMnd‘𝐴)       (𝐴𝑉 → {( I ↾ 𝐴)} ∈ (SubMnd‘𝐺))
 
17-Feb-2024submefmnd 43992 If the base set of a monoid is contained in the base set of the monoid of endofunctions on a set 𝐴, contains the identity function and has the function composition as group operation, then its base set is a submonoid of the monoid of endofunctions on set 𝐴. Analogous to pgrpsubgsymg 18466. (Contributed by AV, 17-Feb-2024.)
𝑀 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝑀)    &    0 = (0g𝑀)    &   𝐹 = (Base‘𝑆)       (𝐴𝑉 → (((𝑆 ∈ Mnd ∧ 𝐹𝐵0𝐹) ∧ (+g𝑆) = (𝑓𝐹, 𝑔𝐹 ↦ (𝑓𝑔))) → 𝐹 ∈ (SubMnd‘𝑀)))
 
17-Feb-20240subm 17970 The zero submonoid of an arbitrary monoid. (Contributed by AV, 17-Feb-2024.)
0 = (0g𝐺)       (𝐺 ∈ Mnd → { 0 } ∈ (SubMnd‘𝐺))
 
17-Feb-2024resmndismnd 17961 If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the the other monoid is contained in the base set of the monoid, then the other monoid restricted to the base set of the monoid is a monoid. Analogous to resgrpisgrp 18238. (Contributed by AV, 17-Feb-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = (Base‘𝐻)    &    0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → ((𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) → (𝐺s 𝑆) ∈ Mnd))
 
17-Feb-2024mndissubm 17960 If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the the other monoid is contained in the base set of the monoid, then the (base set of the) monoid is a submonoid of the other monoid. Analogous to grpissubg 18237. (Contributed by AV, 17-Feb-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = (Base‘𝐻)    &    0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → ((𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubMnd‘𝐺)))
 
17-Feb-2024mgmsscl 17845 If the base set of a magma is contained in the base set of another magma, and the group operation of the magma is the restriction of the group operation of the other magma to its base set, then the base set of the magma is closed under the group operation of the other magma. Formerly part of proof of grpissubg 18237. (Contributed by AV, 17-Feb-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = (Base‘𝐻)       (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋𝑆𝑌𝑆)) → (𝑋(+g𝐺)𝑌) ∈ 𝑆)
 
16-Feb-2024smndex1id 44011 The modulo function 𝐼 is the identity of the monoid of endofunctions on 0 restricted to the modulo function 𝐼 and the constant functions (𝐺𝐾). (Contributed by AV, 16-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))    &   𝐵 = ({𝐼} ∪ 𝑛 ∈ (0..^𝑁){(𝐺𝑛)})    &   𝑆 = (𝑀s 𝐵)       𝐼 = (0g𝑆)
 
16-Feb-2024smndex1mnd 44010 The monoid of endofunctions on 0 restricted to the modulo function 𝐼 and the constant functions (𝐺𝐾) is a monoid. (Contributed by AV, 16-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))    &   𝐵 = ({𝐼} ∪ 𝑛 ∈ (0..^𝑁){(𝐺𝑛)})    &   𝑆 = (𝑀s 𝐵)       𝑆 ∈ Mnd
 
16-Feb-2024smndex1mndlem 44009 Lemma for smndex1mnd 44010 and smndex1id 44011. (Contributed by AV, 16-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))    &   𝐵 = ({𝐼} ∪ 𝑛 ∈ (0..^𝑁){(𝐺𝑛)})    &   𝑆 = (𝑀s 𝐵)       (𝑋𝐵 → ((𝐼𝑋) = 𝑋 ∧ (𝑋𝐼) = 𝑋))
 
14-Feb-2024smndex1sgrp 44008 The monoid of endofunctions on 0 restricted to the modulo function 𝐼 and the constant functions (𝐺𝐾) is a semigroup. (Contributed by AV, 14-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))    &   𝐵 = ({𝐼} ∪ 𝑛 ∈ (0..^𝑁){(𝐺𝑛)})    &   𝑆 = (𝑀s 𝐵)       𝑆 ∈ Smgrp
 
14-Feb-2024smndex1mgm 44007 The monoid of endofunctions on 0 restricted to the modulo function 𝐼 and the constant functions (𝐺𝐾) is a magma. (Contributed by AV, 14-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))    &   𝐵 = ({𝐼} ∪ 𝑛 ∈ (0..^𝑁){(𝐺𝑛)})    &   𝑆 = (𝑀s 𝐵)       𝑆 ∈ Mgm
 
14-Feb-2024smndex1igid 44004 The composition of the modulo function 𝐼 and a constant function (𝐺𝐾) results in (𝐺𝐾) itself. (Contributed by AV, 14-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))       (𝐾 ∈ (0..^𝑁) → (𝐼 ∘ (𝐺𝐾)) = (𝐺𝐾))
 
14-Feb-2024smndex1gid 44003 The composition of a constant function (𝐺𝐾) with another endofunction on 0 results in (𝐺𝐾) itself. (Contributed by AV, 14-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))       ((𝐹 ∈ (Base‘𝑀) ∧ 𝐾 ∈ (0..^𝑁)) → ((𝐺𝐾) ∘ 𝐹) = (𝐺𝐾))
 
13-Feb-2024sn-ltp1 39125 ltp1 11468 without ax-mulcom 10589. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1))
 
13-Feb-2024sn-0lt1 39124 0lt1 11150 without ax-mulcom 10589. (Contributed by SN, 13-Feb-2024.)
0 < 1
 
13-Feb-2024relt0neg2 39123 Comparison of a real and its negative to zero. Compare lt0neg2 11135. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ → (0 < 𝐴 ↔ (0 − 𝐴) < 0))
 
13-Feb-2024relt0neg1 39122 Comparison of a real and its negative to zero. Compare lt0neg1 11134. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < (0 − 𝐴)))
 
13-Feb-2024sn-ltaddpos 39121 ltaddpos 11118 without ax-mulcom 10589. (Contributed by SN, 13-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴𝐵 < (𝐵 + 𝐴)))
 
13-Feb-2024renegneg 39119 A real number is equal to the negative of its negative. Compare negneg 10924. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ → (0 − (0 − 𝐴)) = 𝐴)
 
13-Feb-2024reltsubadd2 39095 'Less than' relationship between addition and subtraction. Compare ltsubadd2 11099. (Contributed by SN, 13-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 𝐵) < 𝐶𝐴 < (𝐵 + 𝐶)))
 
13-Feb-2024reltsub1 39094 Subtraction from both sides of 'less than'. Compare ltsub1 11124. (Contributed by SN, 13-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 𝐶) < (𝐵 𝐶)))
 
13-Feb-2024evlsgsummul 20233 Polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by SN, 13-Feb-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &   𝐺 = (mulGrp‘𝑊)    &    1 = (1r𝑊)    &   𝑈 = (𝑆s 𝑅)    &   𝑃 = (𝑆s (𝐾m 𝐼))    &   𝐻 = (mulGrp‘𝑃)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   ((𝜑𝑥𝑁) → 𝑌𝐵)    &   (𝜑𝑁 ⊆ ℕ0)    &   (𝜑 → (𝑥𝑁𝑌) finSupp 1 )       (𝜑 → (𝑄‘(𝐺 Σg (𝑥𝑁𝑌))) = (𝐻 Σg (𝑥𝑁 ↦ (𝑄𝑌))))
 
13-Feb-2024evlsgsumadd 20232 Polynomial evaluation maps (additive) group sums to group sums. (Contributed by SN, 13-Feb-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑊 = (𝐼 mPoly 𝑈)    &    0 = (0g𝑊)    &   𝑈 = (𝑆s 𝑅)    &   𝑃 = (𝑆s (𝐾m 𝐼))    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   ((𝜑𝑥𝑁) → 𝑌𝐵)    &   (𝜑𝑁 ⊆ ℕ0)    &   (𝜑 → (𝑥𝑁𝑌) finSupp 0 )       (𝜑 → (𝑄‘(𝑊 Σg (𝑥𝑁𝑌))) = (𝑃 Σg (𝑥𝑁 ↦ (𝑄𝑌))))
 
12-Feb-2024smndex1bas 44006 The base set of the monoid of endofunctions on 0 restricted to the modulo function 𝐼 and the constant functions (𝐺𝐾). (Contributed by AV, 12-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))    &   𝐵 = ({𝐼} ∪ 𝑛 ∈ (0..^𝑁){(𝐺𝑛)})    &   𝑆 = (𝑀s 𝐵)       (Base‘𝑆) = 𝐵
 
12-Feb-2024smndex1basss 44005 The modulo function 𝐼 and the constant functions (𝐺𝐾) are endofunctions on 0. (Contributed by AV, 12-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))    &   𝐵 = ({𝐼} ∪ 𝑛 ∈ (0..^𝑁){(𝐺𝑛)})       𝐵 ⊆ (Base‘𝑀)
 
12-Feb-2024smndex1gbas 44002 The constant functions (𝐺𝐾) are endofunctions on 0. (Contributed by AV, 12-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))    &   𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0𝑛))       (𝐾 ∈ (0..^𝑁) → (𝐺𝐾) ∈ (Base‘𝑀))
 
12-Feb-2024smndex1iidm 44001 The modulo function 𝐼 is idempotent. (Contributed by AV, 12-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))       (𝐼𝐼) = 𝐼
 
12-Feb-2024smndex1ibas 44000 The modulo function 𝐼 is an endofunction on 0. (Contributed by AV, 12-Feb-2024.)
𝑀 = (EndoFMnd‘ℕ0)    &   𝑁 ∈ ℕ    &   𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))       𝐼 ∈ (Base‘𝑀)
 
10-Feb-2024cbvexdvaw 2037 Version of cbvexdva 2422 with a disjoint variable condition, requiring fewer axioms. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Feb-2024.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
 
10-Feb-2024cbvaldvaw 2036 Version of cbvaldva 2421 with a disjoint variable condition, requiring fewer axioms. (Contributed by Gino Giotto, 10-Jan-2024.) Reduce axiom usage, along an idea of Gino Giotto. (Revised by Wolf Lammen, 10-Feb-2024.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 
5-Feb-2024remulid2 39127 Commuted version of ax-1rid 10595 and real number version of mulid2 10628 without ax-mulcom 10589. (Contributed by SN, 5-Feb-2024.)
(𝐴 ∈ ℝ → (1 · 𝐴) = 𝐴)
 
5-Feb-2024remulinvcom 39126 A left multiplicative inverse is a right multiplicative inverse. Proven without ax-mulcom 10589. (Contributed by SN, 5-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (𝐴 · 𝐵) = 1)       (𝜑 → (𝐵 · 𝐴) = 1)
 
5-Feb-2024nnmulcom 39043 Multiplication is commutative for natural numbers. (Contributed by SN, 5-Feb-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
5-Feb-2024nnmul1com 39042 Multiplication with 1 is commutative for natural numbers, without ax-mulcom 10589. Since (𝐴 · 1) is 𝐴 by ax-1rid 10595, this is equivalent to remulid2 39127 for natural numbers, but using fewer axioms (avoiding ax-resscn 10582, ax-addass 10590, ax-mulass 10591, ax-rnegex 10596, ax-pre-lttri 10599, ax-pre-lttrn 10600, ax-pre-ltadd 10601). (Contributed by SN, 5-Feb-2024.)
(𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1))
 
5-Feb-2024nnadddir 39041 Right-distributivity for natural numbers without ax-mulcom 10589. (Contributed by SN, 5-Feb-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
 
5-Feb-2024empty 1898 Two characterizations of the empty domain. (Contributed by Gérard Lang, 5-Feb-2024.)
(¬ ∃𝑥⊤ ↔ ∀𝑥⊥)
 
3-Feb-2024sbequ2 2240 An equality theorem for substitution. (Contributed by NM, 16-May-1993.) Revise df-sb 2061. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Wolf Lammen, 3-Feb-2024.)
(𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑𝜑))
 
1-Feb-2024issubmndb 17958 The submonoid predicate. Analogous to issubg 18217. (Contributed by AV, 1-Feb-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)       (𝑆 ∈ (SubMnd‘𝐺) ↔ ((𝐺 ∈ Mnd ∧ (𝐺s 𝑆) ∈ Mnd) ∧ (𝑆𝐵0𝑆)))
 
31-Jan-2024efmnd1bas 43990 The monoid of endofunctions on a singleton is consisting of the identity only. (Contributed by AV, 31-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝐴 = {𝐼}       (𝐼𝑉𝐵 = {{⟨𝐼, 𝐼⟩}})
 
31-Jan-2024efmnd0nmnd 43987 Even the monoid of endofunctions on the empty set is a actually a monoid. (Contributed by AV, 31-Jan-2024.)
(EndoFMnd‘∅) ∈ Mnd
 
31-Jan-2024efmndmnd 43986 The monoid of endofunctions on a set 𝐴 is actually a monoid. (Contributed by AV, 31-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)       (𝐴𝑉𝐺 ∈ Mnd)
 
31-Jan-2024efmndtopn 43981 The topology of the monoid of endofunctions on 𝐴. (Contributed by AV, 31-Jan-2024.)
𝐺 = (EndoFMnd‘𝑋)    &   𝐵 = (Base‘𝐺)       (𝑋𝑉 → ((∏t‘(𝑋 × {𝒫 𝑋})) ↾t 𝐵) = (TopOpen‘𝐺))
 
30-Jan-2024iotan0 6338 Representation of "the unique element such that 𝜑 " with a class expression 𝐴 which is not the empty set (that means that "the unique element such that 𝜑 " exists). (Contributed by AV, 30-Jan-2024.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝑉𝐴 ≠ ∅ ∧ 𝐴 = (℩𝑥𝜑)) → 𝜓)
 
30-Jan-2024iresn0n0 5916 The identity function restricted to a class 𝐴 is empty iff the class 𝐴 is empty. (Contributed by AV, 30-Jan-2024.)
(𝐴 = ∅ ↔ ( I ↾ 𝐴) = ∅)
 
29-Jan-2024sgrpidmnd 17904 A semigroup with an identity element which is not the empty set is a monoid. Of course there could be monoids with the empty set as identity element (see, for example, the monoid of the power set of a class under union, pwmnd 18040 and pwmndid 18039), but these cannot be proven to be monoids with this theorem. (Contributed by AV, 29-Jan-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Smgrp ∧ ∃𝑒𝐵 (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → 𝐺 ∈ Mnd)
 
29-Jan-2024ccatw2s1ccatws2 14304 The concatenation of a word with two singleton words equals the concatenation of the word with the doubleton word consisting of the symbols of the two singletons. (Contributed by Mario Carneiro/AV, 21-Oct-2018.) (Revised by AV, 29-Jan-2024.)
(𝑊 ∈ Word 𝑉 → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) = (𝑊 ++ ⟨“𝑋𝑌”⟩))
 
29-Jan-2024ccatw2s1p1 13983 Extract the symbol of the first singleton word of a word concatenated with this singleton word and another singleton word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 1-May-2020.) (Revised by AV, 1-May-2020.) (Revised by AV, 29-Jan-2024.)
((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁𝑋𝑉) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝑁) = 𝑋)
 
29-Jan-2024sbiedvw 2095 Version of sbied 2538 and sbiedv 2539 with more disjoint variable conditions, requiring fewer axioms. (Contributed by Gino Giotto, 29-Jan-2024.)
((𝜑𝑥 = 𝑦) → (𝜓𝜒))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
 
29-Jan-2024sbrimvw 2093 Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim 2304 and sbrimv 2305 based on fewer axioms, but with more disjoint variable conditions. Based on an idea of Gino Giotto. (Contributed by Wolf Lammen, 29-Jan-2024.)
([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
 
29-Jan-2024sbrimvlem 2092 Common proof template for sbrimvw 2093 and sbrimv 2305. The hypothesis is an instance of 19.21 2197. (Contributed by Wolf Lammen, 29-Jan-2024.)
(∀𝑥(𝜑 → (𝑥 = 𝑦𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦𝜓)))       ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
 
28-Jan-2024efmndsgrp 43983 The monoid of endofunctions on a class 𝐴 is a semigroup. (Contributed by AV, 28-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)       𝐺 ∈ Smgrp
 
28-Jan-2024efmndmgm 43982 The monoid of endofunctions on a class 𝐴 is a magma. (Contributed by AV, 28-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)       𝐺 ∈ Mgm
 
28-Jan-2024symggrp 18458 The symmetric group on a set 𝐴 is a group. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) (Proof shortened by AV, 28-Jan-2024.)
𝐺 = (SymGrp‘𝐴)       (𝐴𝑉𝐺 ∈ Grp)
 
28-Jan-2024symggrplem 18456 Lemma for symggrp 18458 and efmndsgrp 43983. Conditions for an operation to be associative. Formerly part of proof for symggrp 18458. (Contributed by AV, 28-Jan-2024.)
((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑥𝑦))       ((𝑋𝐵𝑌𝐵𝑍𝐵) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
 
28-Jan-2024ccat2s1fvwALT 14306 Alternate proof of ccat2s1fvw 13986 using words of length 2, see df-s2 14198. A symbol of the concatenation of a word with two single symbols corresponding to the symbol of the word. (Contributed by AV, 22-Sep-2018.) (Proof shortened by Mario Carneiro/AV, 21-Oct-2018.) (Revised by AV, 28-Jan-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑊 ∈ Word 𝑉𝐼 ∈ ℕ0𝐼 < (♯‘𝑊)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = (𝑊𝐼))
 
28-Jan-2024ccat2s1fst 13988 The first symbol of the concatenation of a word with two single symbols. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by AV, 28-Jan-2024.)
((𝑊 ∈ Word 𝑉 ∧ 0 < (♯‘𝑊)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘0) = (𝑊‘0))
 
28-Jan-2024ccat2s1fvw 13986 Extract a symbol of a word from the concatenation of the word with two single symbols. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 13-Jan-2020.) (Proof shortened by AV, 1-May-2020.) (Revised by AV, 28-Jan-2024.)
((𝑊 ∈ Word 𝑉𝐼 ∈ ℕ0𝐼 < (♯‘𝑊)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = (𝑊𝐼))
 
28-Jan-2024sbiedw 2323 Version of sbied 2538 with a disjoint variable condition, requiring fewer axioms. (Contributed by Gino Giotto, 10-Jan-2024.) Avoid ax-10 2136. (Revised by Wolf Lammen, 28-Jan-2024.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
 
28-Jan-2024sbrimv 2305 Substitution in an implication with a variable not free in the antecedent affects only the consequent. Version of sbrim 2304 not depending on ax-10 2136, but with disjoint variables. (Contributed by Wolf Lammen, 28-Jan-2024.)
𝑥𝜑       ([𝑦 / 𝑥](𝜑𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
 
27-Jan-2024efmnd1hash 43989 The monoid of endofunctions on a singleton has cardinality 1. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝐴 = {𝐼}       (𝐼𝑉 → (♯‘𝐵) = 1)
 
27-Jan-2024efmndbas0 43988 The base set of the monoid of endofunctions on the empty set is the singleton containing the empty set. (Contributed by AV, 27-Jan-2024.)
(Base‘(EndoFMnd‘∅)) = {∅}
 
27-Jan-2024ielefmnd 43984 The identity function restricted to a set 𝐴 is an element of the base set of the monoid of endofunctions on 𝐴. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)       (𝐴𝑉 → ( I ↾ 𝐴) ∈ (Base‘𝐺))
 
27-Jan-2024efmndcl 43980 The group operation of the monoid of endofunctions on 𝐴 is closed. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
 
27-Jan-2024efmndov 43979 The value of the group operation of the monoid of endofunctions on 𝐴. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑋𝑌))
 
27-Jan-2024efmndplusg 43978 The group operation of a monoid of endofunctions is the function composition. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)        + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))
 
27-Jan-2024efmndfv 43976 The function value of an endofunction. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)       ((𝐹𝐵𝑋𝐴) → (𝐹𝑋) ∈ 𝐴)
 
27-Jan-2024efmndbasfi 43975 The monoid of endofunctions on a finite set 𝐴 is finite. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴 ∈ Fin → 𝐵 ∈ Fin)
 
27-Jan-2024efmndhash 43974 The monoid of endofunctions on 𝑛 objects has cardinality 𝑛𝑛. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴 ∈ Fin → (♯‘𝐵) = ((♯‘𝐴)↑(♯‘𝐴)))
 
27-Jan-2024efmndbasf 43973 Elements in the monoid of endofunctions on 𝐴 are functions from 𝐴 into itself. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐹𝐵𝐹:𝐴𝐴)
 
27-Jan-2024elefmndbas2 43972 Two ways of saying a function is a mapping of 𝐴 to itself. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐹𝑉 → (𝐹𝐵𝐹:𝐴𝐴))
 
27-Jan-2024elefmndbas 43971 Two ways of saying a function is a mapping of 𝐴 to itself. (Contributed by AV, 27-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴𝑉 → (𝐹𝐵𝐹:𝐴𝐴))
 
27-Jan-2024mpo0v 7227 A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.) (Proof shortened by AV, 27-Jan-2024.)
(𝑥 ∈ ∅, 𝑦𝐵𝐶) = ∅
 
27-Jan-20240mpo0 7226 A mapping operation with empty domain is empty. Generalization of mpo0 7228. (Contributed by AV, 27-Jan-2024.)
((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝑥𝐴, 𝑦𝐵𝐶) = ∅)
 
26-Jan-2024nfixpw 8468 Version of nfixp 8469 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
𝑦𝐴    &   𝑦𝐵       𝑦X𝑥𝐴 𝐵
 
26-Jan-2024frsucmpt2w 8064 Version of frsucmpt2 8065 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
𝐹 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) ↾ ω)    &   (𝑦 = 𝑥𝐸 = 𝐶)    &   (𝑦 = (𝐹𝐵) → 𝐸 = 𝐷)       ((𝐵 ∈ ω ∧ 𝐷𝑉) → (𝐹‘suc 𝐵) = 𝐷)
 
26-Jan-2024elovmporab1w 7381 Version of elovmporab1 7382 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑥 / 𝑚𝑀𝜑})    &   ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 / 𝑚𝑀 ∈ V)       (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑋 / 𝑚𝑀))
 
26-Jan-2024eqoprab2bw 7213 Version of eqoprab2b 7214 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ∀𝑥𝑦𝑧(𝜑𝜓))
 
26-Jan-2024oprabidw 7176 Version of oprabid 7177 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
(⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
 
26-Jan-2024cbvriotavw 7113 Version of cbvriotav 7117 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
 
26-Jan-2024cbvriotaw 7112 Version of cbvriota 7116 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
 
26-Jan-2024nfriotadw 7111 Version of nfriotad 7114 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑𝑥𝐴)       (𝜑𝑥(𝑦𝐴 𝜓))
 
26-Jan-2024rexrnmptw 6853 Version of rexrnmpt 6855 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑦 = 𝐵 → (𝜓𝜒))       (∀𝑥𝐴 𝐵𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥𝐴 𝜒))
 
26-Jan-2024ralrnmptw 6852 Version of ralrnmpt 6854 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
𝐹 = (𝑥𝐴𝐵)    &   (𝑦 = 𝐵 → (𝜓𝜒))       (∀𝑥𝐴 𝐵𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥𝐴 𝜒))
 
26-Jan-2024elfvmptrab1w 6786 Version of elfvmptrab1 6787 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})    &   (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)       (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))
 
26-Jan-2024cbviotavw 6315 Version of cbviotav 6317 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       (℩𝑥𝜑) = (℩𝑦𝜓)
 
26-Jan-2024cbviotaw 6314 Version of cbviota 6316 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   𝑦𝜑    &   𝑥𝜓       (℩𝑥𝜑) = (℩𝑦𝜓)
 
26-Jan-2024nfiotaw 6311 Version of nfiota 6313 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
𝑥𝜑       𝑥(℩𝑦𝜑)
 
26-Jan-2024nfiotadw 6310 Version of nfiotad 6312 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥(℩𝑦𝜓))
 
26-Jan-2024eqopab2bw 5426 Version of eqopab2b 5430 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥𝑦(𝜑𝜓))
 
26-Jan-2024ssopab2bw 5425 Version of ssopab2b 5427 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥𝑦(𝜑𝜓))
 
26-Jan-2024opabidw 5403 Version of opabid 5404 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
(⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
 
26-Jan-2024copsexgw 5372 Version of copsexg 5373 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
 
26-Jan-2024disjprgw 5052 Version of disjprg 5053 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝐴𝐶 = 𝐷)    &   (𝑥 = 𝐵𝐶 = 𝐸)       ((𝐴𝑉𝐵𝑉𝐴𝐵) → (Disj 𝑥 ∈ {𝐴, 𝐵}𝐶 ↔ (𝐷𝐸) = ∅))
 
26-Jan-2024invdisjrabw 5042 Version of invdisjrab 5043 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
Disj 𝑦𝐴 {𝑥𝐵𝐶 = 𝑦}
 
26-Jan-2024nfdisjw 5034 Version of nfdisj 5035 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
𝑦𝐴    &   𝑦𝐵       𝑦Disj 𝑥𝐴 𝐵
 
26-Jan-2024sbcco3gw 4371 Version of sbcco3g 4376 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝑥 = 𝐴𝐵 = 𝐶)       (𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐶 / 𝑦]𝜑))
 
26-Jan-2024csbnestgw 4370 Version of csbnestg 4375 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝐴𝑉𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
 
26-Jan-2024sbcnestgw 4369 Version of sbcnestg 4374 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
(𝐴𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
 
26-Jan-2024csbnestgfw 4368 Version of csbnestgf 4373 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
((𝐴𝑉 ∧ ∀𝑦𝑥𝐶) → 𝐴 / 𝑥𝐵 / 𝑦𝐶 = 𝐴 / 𝑥𝐵 / 𝑦𝐶)
 
26-Jan-2024sbcnestgfw 4367 Version of sbcnestgf 4372 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
((𝐴𝑉 ∧ ∀𝑦𝑥𝜑) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦]𝜑))
 
26-Jan-2024cbvrabcsfw 3921 Version of cbvrabcsf 3925 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 26-Jan-2024.)
𝑦𝐴    &   𝑥𝐵    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑦𝐵𝜓}
 
25-Jan-2024efmndid 43985 The identity function restricted to a set 𝐴 is the identity element of the monoid of endofunctions on 𝐴. (Contributed by AV, 25-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)       (𝐴𝑉 → ( I ↾ 𝐴) = (0g𝐺))
 
25-Jan-2024efmndtset 43977 The topology of the monoid of endofunctions on 𝐴. This component is defined on a larger set than the true base - the product topology is defined on the set of all functions, not just endofunctions - but the definition of TopOpen ensures that it is trimmed down before it gets use. (Contributed by AV, 25-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)       (𝐴𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) = (TopSet‘𝐺))
 
25-Jan-2024efmndbas 43970 The base set of the monoid of endofunctions on class 𝐴. (Contributed by AV, 25-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (Base‘𝐺)       𝐵 = (𝐴m 𝐴)
 
25-Jan-2024efmnd 43969 The monoid of endofunctions on set 𝐴. (Contributed by AV, 25-Jan-2024.)
𝐺 = (EndoFMnd‘𝐴)    &   𝐵 = (𝐴m 𝐴)    &    + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))    &   𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))       (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
 
25-Jan-2024df-efmnd 43968 Define the monoid of endofunctions on set 𝑥. We represent the monoid as the set of functions from 𝑥 to itself ((𝑥m 𝑥)) under function composition, and topologize it as a function space assuming the set is discrete. Analogous to SymGrp, see df-symg 18434. (Contributed by AV, 25-Jan-2024.)
EndoFMnd = (𝑥 ∈ V ↦ (𝑥m 𝑥) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑥 × {𝒫 𝑥}))⟩})
 
23-Jan-2024resubid1 39118 Real number version of subid1 10894, without ax-mulcom 10589. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (𝐴 0) = 𝐴)
 
23-Jan-2024readdid1 39117 Real number version of addid1 10808, without ax-mulcom 10589. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴)
 
23-Jan-2024resubid 39116 Subtraction of a real number from itself (compare subid 10893). (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (𝐴 𝐴) = 0)
 
23-Jan-2024remul01 39115 Real number version of mul01 10807 proven without ax-mulcom 10589. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (𝐴 · 0) = 0)
 
23-Jan-2024sn-0ne2 39114 0ne2 11832 without ax-mulcom 10589. (Contributed by SN, 23-Jan-2024.)
0 ≠ 2
 
23-Jan-2024remul02 39113 Real number version of mul02 10806 proven without ax-mulcom 10589. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (0 · 𝐴) = 0)
 
23-Jan-2024sn-addid2 39112 addid2 10811 without ax-mulcom 10589. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
 
23-Jan-2024readdid2 39111 Real number version of addid2 10811. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (0 + 𝐴) = 𝐴)
 
23-Jan-2024re0m0e0 39110 Real number version of 0m0e0 11745 proven without ax-mulcom 10589. (Contributed by SN, 23-Jan-2024.)
(0 − 0) = 0
 
23-Jan-2024nelb 3265 A definition of ¬ 𝐴𝐵. (Contributed by Thierry Arnoux, 20-Nov-2023.) (Proof shortened by SN, 23-Jan-2024.)
𝐴𝐵 ↔ ∀𝑥𝐵 𝑥𝐴)
 
22-Jan-20243cubes 39165 Every rational number is a sum of three rational cubes. (S. Ryley, The Ladies' Diary 122 (1825), 35) (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝐴 ∈ ℚ ↔ ∃𝑎 ∈ ℚ ∃𝑏 ∈ ℚ ∃𝑐 ∈ ℚ 𝐴 = (((𝑎↑3) + (𝑏↑3)) + (𝑐↑3)))
 
22-Jan-20243cubeslem4 39164 Lemma for 3cubes 39165. This is Ryley's explicit formula for decomposing a rational 𝐴 into a sum of three rational cubes. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑𝐴 ∈ ℚ)       (𝜑𝐴 = (((((((3↑3) · (𝐴↑3)) − 1) / ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3))↑3) + ((((-((3↑3) · (𝐴↑3)) + ((3↑2) · 𝐴)) + 1) / ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3))↑3)) + (((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) / ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3))↑3)))
 
22-Jan-20243cubeslem3 39163 Lemma for 3cubes 39165. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑𝐴 ∈ ℚ)       (𝜑 → (𝐴 · (((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3)↑3)) = ((((((3↑3) · (𝐴↑3)) − 1)↑3) + (((-((3↑3) · (𝐴↑3)) + ((3↑2) · 𝐴)) + 1)↑3)) + ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴))↑3)))
 
22-Jan-20243cubeslem3r 39162 Lemma for 3cubes 39165. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑𝐴 ∈ ℚ)       (𝜑 → ((((((3↑3) · (𝐴↑3)) − 1)↑3) + (((-((3↑3) · (𝐴↑3)) + ((3↑2) · 𝐴)) + 1)↑3)) + ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴))↑3)) = (((𝐴↑7) · (3↑9)) + (((𝐴↑6) · (3↑9)) + (((𝐴↑5) · ((3↑8) + (3↑8))) + (((𝐴↑4) · (((3↑7) · 2) + (3↑6))) + (((𝐴↑3) · ((3↑6) + (3↑6))) + (((𝐴↑2) · (3↑5)) + (𝐴 · (3↑3)))))))))
 
22-Jan-20243cubeslem3l 39161 Lemma for 3cubes 39165. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑𝐴 ∈ ℚ)       (𝜑 → (𝐴 · (((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3)↑3)) = (((𝐴↑7) · (3↑9)) + (((𝐴↑6) · (3↑9)) + (((𝐴↑5) · ((3↑8) + (3↑8))) + (((𝐴↑4) · (((3↑7) · 2) + (3↑6))) + (((𝐴↑3) · ((3↑6) + (3↑6))) + (((𝐴↑2) · (3↑5)) + (𝐴 · (3↑3)))))))))
 
22-Jan-20243cubeslem2 39160 Lemma for 3cubes 39165. Used to show that the denominators in 3cubeslem4 39164 are nonzero. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑𝐴 ∈ ℚ)       (𝜑 → ¬ ((((3↑3) · (𝐴↑2)) + ((3↑2) · 𝐴)) + 3) = 0)
 
22-Jan-20243cubeslem1 39159 Lemma for 3cubes 39165. (Contributed by Igor Ieskov, 22-Jan-2024.)
(𝜑𝐴 ∈ ℚ)       (𝜑 → 0 < (((𝐴 + 1)↑2) − 𝐴))
 
21-Jan-2024rexlimdv3d 39158 An extended version of rexlimdvv 3290 to include three set variables. (Contributed by Igor Ieskov, 21-Jan-2024.)
(𝜑 → ((𝑥𝐴𝑦𝐵𝑧𝐶) → (𝜓𝜒)))       (𝜑 → (∃𝑥𝐴𝑦𝐵𝑧𝐶 𝜓𝜒))
 
21-Jan-2024negexpidd 39157 The sum of a real number to the power of N and the negative of the number to the power of N equals zero if N is a nonnegative odd integer. (Contributed by Igor Ieskov, 21-Jan-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑 → ¬ 2 ∥ 𝑁)       (𝜑 → ((𝐴𝑁) + (-𝐴𝑁)) = 0)
 
21-Jan-2024sqnegd 39156 The square of the negative of a number. (Contributed by Igor Ieskov, 21-Jan-2024.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (-𝐴↑2) = (𝐴↑2))
 
21-Jan-2024emptyex 1899 On the empty domain, any existentially quantified formula is false. (Contributed by Wolf Lammen, 21-Jan-2024.)
(¬ ∃𝑥⊤ → ¬ ∃𝑥𝜑)
 
20-Jan-2024nfiund 44705 Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) Add disjoint variable condition to avoid ax-13 2381. See nfiundg 44706 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑥𝜑    &   (𝜑𝑦𝐴)    &   (𝜑𝑦𝐵)       (𝜑𝑦 𝑥𝐴 𝐵)
 
20-Jan-2024bj-elsn0 34339 If the intersection of two classes is a set, then these classes are equal if and only if one is an element of the singleton formed on the other. Stronger form of elsng 4571 and elsn2g 4593 (which could be proved from it). (Contributed by BJ, 20-Jan-2024.)
((𝐴𝐵) ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
 
20-Jan-2024bnj1441 32011 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Add disjoint variable condition to avoid ax-13 2381. See bnj1441g 32012 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) (New usage is discouraged.)
(𝑥𝐴 → ∀𝑦 𝑥𝐴)    &   (𝜑 → ∀𝑦𝜑)       (𝑧 ∈ {𝑥𝐴𝜑} → ∀𝑦 𝑧 ∈ {𝑥𝐴𝜑})
 
20-Jan-2024cygabl 18939 A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 20-Jan-2024.)
(𝐺 ∈ CycGrp → 𝐺 ∈ Abel)
 
20-Jan-2024cycsubmcmn 18937 The set of nonnegative integer powers of an element 𝐴 of a monoid forms a commutative monoid. (Contributed by AV, 20-Jan-2024.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   𝐶 = ran 𝐹       ((𝐺 ∈ Mnd ∧ 𝐴𝐵) → (𝐺s 𝐶) ∈ CMnd)
 
20-Jan-2024cycsubmcom 18285 The operation of a monoid is commutative over the set of nonnegative integer powers of an element 𝐴 of the monoid. (Contributed by AV, 20-Jan-2024.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   𝐶 = ran 𝐹    &    + = (+g𝐺)       (((𝐺 ∈ Mnd ∧ 𝐴𝐵) ∧ (𝑋𝐶𝑌𝐶)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
 
20-Jan-2024cyccom 18284 Condition for an operation to be commutative. Lemma for cycsubmcom 18285 and cygabl 18939. Formerly part of proof for cygabl 18939. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 20-Jan-2024.)
(𝜑 → ∀𝑐𝐶𝑥𝑍 𝑐 = (𝑥 · 𝐴))    &   (𝜑 → ∀𝑚𝑍𝑛𝑍 ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴) + (𝑛 · 𝐴)))    &   (𝜑𝑋𝐶)    &   (𝜑𝑌𝐶)    &   (𝜑𝑍 ⊆ ℂ)       (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
 
20-Jan-2024mndinvmod 17929 Uniqueness of an inverse element in a monoid, if it exists. (Contributed by AV, 20-Jan-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝐵)       (𝜑 → ∃*𝑤𝐵 ((𝑤 + 𝐴) = 0 ∧ (𝐴 + 𝑤) = 0 ))
 
20-Jan-2024ccat2s1p2 13974 Extract the second of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by JJ, 20-Jan-2024.)
(𝑌𝑉 → ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘1) = 𝑌)
 
20-Jan-2024ccat2s1p1 13973 Extract the first of two concatenated singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by JJ, 20-Jan-2024.)
(𝑋𝑉 → ((⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)‘0) = 𝑋)
 
20-Jan-2024ccats1val1 13969 Value of a symbol in the left half of a word concatenated with a single symbol. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by JJ, 20-Jan-2024.)
((𝑊 ∈ Word 𝑉𝐼 ∈ (0..^(♯‘𝑊))) → ((𝑊 ++ ⟨“𝑆”⟩)‘𝐼) = (𝑊𝐼))
 
20-Jan-2024cbviinv 4957 Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) Add disjoint variable condition to avoid ax-13 2381. See cbviinvg 4959 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
(𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
 
20-Jan-2024cbviunv 4956 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 2381. See cbviunvg 4958 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
(𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
 
20-Jan-2024cbviin 4953 Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2381. See cbviing 4955 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
 
20-Jan-2024cbviun 4952 Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.) Add disjoint variable condition to avoid ax-13 2381. See cbviung 4954 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)        𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
 
20-Jan-2024nfiin 4941 Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2381. See nfiing 4943 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑦𝐴    &   𝑦𝐵       𝑦 𝑥𝐴 𝐵
 
20-Jan-2024nfiun 4940 Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2381. See nfiung 4942 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑦𝐴    &   𝑦𝐵       𝑦 𝑥𝐴 𝐵
 
20-Jan-2024nfrex 3306 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2019.) Add disjoint variable condition to avoid ax-13 2381. See nfrexg 3307 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑥𝐴    &   𝑥𝜑       𝑥𝑦𝐴 𝜑
 
20-Jan-2024nfrexd 3304 Deduction version of nfrex 3306. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2381. See nfrexdg 3305 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
 
20-Jan-2024nfaba1 2983 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2381. See nfaba1g 2984 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑥{𝑦 ∣ ∀𝑥𝜑}
 
20-Jan-2024nfab 2981 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Add disjoint variable condition to avoid ax-13 2381. See nfabg 2982 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑥𝜑       𝑥{𝑦𝜑}
 
20-Jan-2024hblem 2940 Change the free variable of a hypothesis builder. (Contributed by NM, 21-Jun-1993.) (Revised by Andrew Salmon, 11-Jul-2011.) Add disjoint variable condition to avoid ax-13 2381. See hblemg 2941 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
(𝑦𝐴 → ∀𝑥 𝑦𝐴)       (𝑧𝐴 → ∀𝑥 𝑧𝐴)
 
20-Jan-2024nfsab 2809 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Add disjoint variable condition to avoid ax-13 2381. See nfsabg 2810 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
𝑥𝜑       𝑥 𝑧 ∈ {𝑦𝜑}
 
20-Jan-2024hbab 2807 Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.) Add disjoint variable condition to avoid ax-13 2381. See hbabg 2808 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
(𝜑 → ∀𝑥𝜑)       (𝑧 ∈ {𝑦𝜑} → ∀𝑥 𝑧 ∈ {𝑦𝜑})
 
20-Jan-2024sb1v 2086 One direction of sb5 2267, provable from fewer axioms. Version of sb1 2496 with a disjoint variable condition using fewer axioms. (Contributed by Wolf Lammen, 20-Jan-2024.)
([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
 
18-Jan-2024ccatval1 13918 Value of a symbol in the left half of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 22-Sep-2015.) (Proof shortened by AV, 30-Apr-2020.) (Revised by JJ, 18-Jan-2024.)
((𝑆 ∈ Word 𝐴𝑇 ∈ Word 𝐵𝐼 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ 𝑇)‘𝐼) = (𝑆𝐼))
 
18-Jan-2024ccat0 13917 The concatenation of two words is empty iff the two words are empty. (Contributed by AV, 4-Mar-2022.) (Revised by JJ, 18-Jan-2024.)
((𝑆 ∈ Word 𝐴𝑇 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) = ∅ ↔ (𝑆 = ∅ ∧ 𝑇 = ∅)))
 
17-Jan-2024cbvmptv 5160 Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.) Add disjoint variable condition to avoid ax-13 2381. See cbvmptvg 5161 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)
(𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
 
17-Jan-2024cbvmpt 5158 Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.) Add disjoint variable condition to avoid ax-13 2381. See cbvmptg 5159 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)
𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
 
17-Jan-2024cbvmptf 5156 Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.) (Revised by Thierry Arnoux, 9-Mar-2017.) Add disjoint variable condition to avoid ax-13 2381. See cbvmptfg 5157 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝐶    &   (𝑥 = 𝑦𝐵 = 𝐶)       (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
 
17-Jan-2024cbvopab1 5130 Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2381. See cbvopab1g 5131 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)
𝑧𝜑    &   𝑥𝜓    &   (𝑥 = 𝑧 → (𝜑𝜓))       {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
 
16-Jan-2024qsidom 30884 An ideal 𝐼 in the commutative ring 𝑅 is prime if and only if the factor ring 𝑄 is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))       ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (𝑄 ∈ IDomn ↔ 𝐼 ∈ (PrmIdeal‘𝑅)))
 
16-Jan-2024qsidomlem2 30883 A quotient by a prime ideal is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))       ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ IDomn)
 
16-Jan-2024qsidomlem1 30882 If the quotient ring of a commutative ring relative to an ideal is an integral domain, that ideal must be prime. (Contributed by Thierry Arnoux, 16-Jan-2024.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))       (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (PrmIdeal‘𝑅))
 
16-Jan-2024qusxpid 30855 The Group quotient equivalence relation for the whole group is the cartesian product, i.e. all elements are in the same equivalence class. (Contributed by Thierry Arnoux, 16-Jan-2024.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Grp → (𝐺 ~QG 𝐵) = (𝐵 × 𝐵))
 
16-Jan-2024qsxpid 30854 The quotient set of a cartesian product is trivial. (Contributed by Thierry Arnoux, 16-Jan-2024.)
(𝐴 ≠ ∅ → (𝐴 / (𝐴 × 𝐴)) = {𝐴})
 
15-Jan-2024rspsnid 30864 A principal ideal contains the element that generates it. (Contributed by Thierry Arnoux, 15-Jan-2024.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐺𝐵) → 𝐺 ∈ (𝐾‘{𝐺}))
 
15-Jan-2024rspsnel 30863 Membership in a principal ideal. Analogous to lspsnel 19704. (Contributed by Thierry Arnoux, 15-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐾 = (RSpan‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (𝐼 ∈ (𝐾‘{𝑋}) ↔ ∃𝑥𝐵 𝐼 = (𝑥 · 𝑋)))
 
15-Jan-2024qustrivr 30857 Converse of qustriv 30856. (Contributed by Thierry Arnoux, 15-Jan-2024.)
𝐵 = (Base‘𝐺)    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝐻))       ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺) ∧ (Base‘𝑄) = {𝐻}) → 𝐻 = 𝐵)
 
15-Jan-2024qustriv 30856 The quotient of a group 𝐺 by itself is trivial. (Contributed by Thierry Arnoux, 15-Jan-2024.)
𝐵 = (Base‘𝐺)    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝐵))       (𝐺 ∈ Grp → (Base‘𝑄) = {𝐵})
 
15-Jan-2024eqg0el 30853 Equivalence class of a quotient group for a subgroup. (Contributed by Thierry Arnoux, 15-Jan-2024.)
= (𝐺 ~QG 𝐻)       ((𝐺 ∈ Grp ∧ 𝐻 ∈ (SubGrp‘𝐺)) → ([𝑋] = 𝐻𝑋𝐻))
 
15-Jan-2024ecxpid 30852 The equivalence class of a cartesian product is the whole set. (Contributed by Thierry Arnoux, 15-Jan-2024.)
(𝑋𝐴 → [𝑋](𝐴 × 𝐴) = 𝐴)
 
14-Jan-2024cringm4 30880 Commutative/associative law for commutative ring. (Contributed by Thierry Arnoux, 14-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ CRing ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 · 𝑌) · (𝑍 · 𝑊)) = ((𝑋 · 𝑍) · (𝑌 · 𝑊)))
 
14-Jan-2024lidlnsg 30879 An ideal is a normal subgroup. (Contributed by Thierry Arnoux, 14-Jan-2024.)
((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
 
14-Jan-2024df-prmidl 30872 Define the class of prime ideals of a ring 𝑅. A proper ideal 𝐼 of 𝑅 is prime if whenever 𝐴𝐵𝐼 for ideals 𝐴 and 𝐵, either 𝐴𝐼 or 𝐵𝐼. The more familiar definition using elements rather than ideals is equivalent provided 𝑅 is commutative; see prmidl2 30877 and isprmidlc 30881. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 14-Jan-2024.)
PrmIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑎 ∈ (LIdeal‘𝑟)∀𝑏 ∈ (LIdeal‘𝑟)(∀𝑥𝑎𝑦𝑏 (𝑥(.r𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})
 
14-Jan-2024wlklenvclwlk 27363 The number of vertices in a walk equals the length of the walk after it is "closed" (i.e. enhanced by an edge from its last vertex to its first vertex). (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 2-May-2021.) (Revised by JJ, 14-Jan-2024.)
(𝑊 ∈ Word (Vtx‘𝐺) → (⟨𝐹, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (Walks‘𝐺) → (♯‘𝐹) = (♯‘𝑊)))
 
14-Jan-2024ccat2s1len 13965 The length of the concatenation of two singleton words. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by JJ, 14-Jan-2024.)
(♯‘(⟨“𝑋”⟩ ++ ⟨“𝑌”⟩)) = 2
 
13-Jan-2024dvdemo2 5266 Demonstration of a theorem that requires the setvar variables 𝑥 and 𝑧 to be disjoint (but without any other disjointness conditions, and in particular, none on 𝑦).

That theorem bundles the theorems (𝑥(𝑥 = 𝑦𝑧𝑥) with 𝑥, 𝑦, 𝑧 disjoint), often called its "principal instance", and the two "degenerate instances" (𝑥(𝑥 = 𝑥𝑧𝑥) with 𝑥, 𝑧 disjoint) and (𝑥(𝑥 = 𝑧𝑧𝑥) with 𝑥, 𝑧 disjoint).

Compare with dvdemo1 5265, which has the same principal instance and one common degenerate instance but crucially differs in the other degenerate instance.

See https://us.metamath.org/mpeuni/mmset.html#distinct 5265 for details on the "disjoint variable" mechanism.

Note that dvdemo2 5266 is partially bundled, in that the pairs of setvar variables 𝑥, 𝑦 and 𝑦, 𝑧 need not be disjoint, and in spite of that, its proof does not require any of the auxiliary axioms ax-10 2136, ax-11 2151, ax-12 2167, ax-13 2381. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.)

𝑥(𝑥 = 𝑦𝑧𝑥)
 
13-Jan-2024dvdemo1 5265 Demonstration of a theorem that requires the setvar variables 𝑥 and 𝑦 to be disjoint (but without any other disjointness conditions, and in particular, none on 𝑧).

That theorem bundles the theorems (𝑥(𝑥 = 𝑦𝑧𝑥) with 𝑥, 𝑦, 𝑧 disjoint), often called its "principal instance", and the two "degenerate instances" (𝑥(𝑥 = 𝑦𝑥𝑥) with 𝑥, 𝑦 disjoint) and (𝑥(𝑥 = 𝑦𝑦𝑥) with 𝑥, 𝑦 disjoint).

Compare with dvdemo2 5266, which has the same principal instance and one common degenerate instance but crucially differs in the other degenerate instance.

See https://us.metamath.org/mpeuni/mmset.html#distinct 5266 for details on the "disjoint variable" mechanism. (The verb "bundle" to express this phenomenon was introduced by Raph Levien.)

Note that dvdemo1 5265 is partially bundled, in that the pairs of setvar variables 𝑥, 𝑧 and 𝑦, 𝑧 need not be disjoint, and in spite of that, its proof does not require ax-11 2151 nor ax-13 2381. (Contributed by NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.)

𝑥(𝑥 = 𝑦𝑧𝑥)
 
12-Jan-2024isprmidlc 30881 The predicate "is prime ideal" for commutative rings. Alternate definition for commutative rings. See definition in [Lang] p. 92. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ CRing → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃𝐵 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥𝑃𝑦𝑃)))))
 
12-Jan-2024prmidlidl 30878 A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ∈ (LIdeal‘𝑅))
 
12-Jan-2024prmidl2 30877 A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 35229 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (((𝑅 ∈ Ring ∧ 𝑃 ∈ (LIdeal‘𝑅)) ∧ (𝑃𝐵 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥𝑃𝑦𝑃)))) → 𝑃 ∈ (PrmIdeal‘𝑅))
 
12-Jan-2024prmidl 30876 The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) ∧ ∀𝑥𝐼𝑦𝐽 (𝑥 · 𝑦) ∈ 𝑃) → (𝐼𝑃𝐽𝑃))
 
12-Jan-2024prmidlnr 30875 A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃𝐵)
 
12-Jan-2024isprmidl 30874 The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))
 
12-Jan-2024prmidlval 30873 The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})
 
10-Jan-2024nfcsbw 3906 Version of nfcsb 3907 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑥𝐴    &   𝑥𝐵       𝑥𝐴 / 𝑦𝐵
 
10-Jan-2024csbcow 3895 Version of csbco 3896 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝐴 / 𝑦𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
 
10-Jan-2024cbvcsbw 3890 Version of cbvcsb 3891 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝐶    &   𝑥𝐷    &   (𝑥 = 𝑦𝐶 = 𝐷)       𝐴 / 𝑥𝐶 = 𝐴 / 𝑦𝐷
 
10-Jan-2024cbvsbcvw 3802 Version of cbvsbcv 3804 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)
 
10-Jan-2024cbvsbcw 3801 Version of cbvsbc 3803 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       ([𝐴 / 𝑥]𝜑[𝐴 / 𝑦]𝜓)
 
10-Jan-2024sbccow 3792 Version of sbcco 3795 with a disjoint variable condition, which requires fewer axioms. (Contributed by Gino Giotto, 10-Jan-2024.)
([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
 
10-Jan-2024nfsbcw 3791 Version of nfsbc 3794 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑥𝐴    &   𝑥𝜑       𝑥[𝐴 / 𝑦]𝜑
 
10-Jan-2024nfsbcdw 3790 Version of nfsbcd 3793 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥[𝐴 / 𝑦]𝜓)
 
10-Jan-2024euxfrw 3709 Version of euxfr 3711 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝐴 ∈ V    &   ∃!𝑦 𝑥 = 𝐴    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
 
10-Jan-2024euxfr2w 3708 Version of euxfr2 3710 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝐴 ∈ V    &   ∃*𝑦 𝑥 = 𝐴       (∃!𝑥𝑦(𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝜑)
 
10-Jan-2024cbvrabw 3487 Version of cbvrab 3488 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝐴𝜑} = {𝑦𝐴𝜓}
 
10-Jan-2024cbvrexsvw 3466 Version of cbvrexsv 3468 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
(∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 [𝑦 / 𝑥]𝜑)
 
10-Jan-2024cbvralsvw 3465 Version of cbvralsv 3467 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
(∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 [𝑦 / 𝑥]𝜑)
 
10-Jan-2024cbvral3vw 3461 Version of cbvral3v 3464 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑤 → (𝜑𝜒))    &   (𝑦 = 𝑣 → (𝜒𝜃))    &   (𝑧 = 𝑢 → (𝜃𝜓))       (∀𝑥𝐴𝑦𝐵𝑧𝐶 𝜑 ↔ ∀𝑤𝐴𝑣𝐵𝑢𝐶 𝜓)
 
10-Jan-2024cbvrex2vw 3460 Version of cbvrex2v 3463 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑧 → (𝜑𝜒))    &   (𝑦 = 𝑤 → (𝜒𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
 
10-Jan-2024cbvral2vw 3459 Version of cbvral2v 3462 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑧 → (𝜑𝜒))    &   (𝑦 = 𝑤 → (𝜒𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)
 
10-Jan-2024cbvreuvw 3449 Version of cbvreuv 3452 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
 
10-Jan-2024cbvrexvw 3448 Version of cbvrexv 3451 with a disjoint variable condition, which does not require ax-10 2136, ax-11 2151, ax-12 2167, ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
 
10-Jan-2024cbvralvw 3447 Version of cbvralv 3450 with a disjoint variable condition, which does not require ax-10 2136, ax-11 2151, ax-12 2167, ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
 
10-Jan-2024cbvrmow 3442 Version of cbvrmo 3446 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
 
10-Jan-2024cbvreuw 3441 Version of cbvreu 3445 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
 
10-Jan-2024cbvrexw 3440 Version of cbvrex 3444 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
 
10-Jan-2024cbvralw 3439 Version of cbvral 3443 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
 
10-Jan-2024cbvrexfw 3436 Version of cbvrexf 3438 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴 𝜓)
 
10-Jan-2024cbvralfw 3435 Version of cbvralf 3437 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑥𝐴    &   𝑦𝐴    &   𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
 
10-Jan-2024nfrabw 3383 Version of nfrab 3384 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑥𝜑    &   𝑥𝐴       𝑥{𝑦𝐴𝜑}
 
10-Jan-2024nfrmow 3373 Version of nfrmo 3375 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑥𝐴    &   𝑥𝜑       𝑥∃*𝑦𝐴 𝜑
 
10-Jan-2024nfreuw 3372 Version of nfreu 3374 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑥𝐴    &   𝑥𝜑       𝑥∃!𝑦𝐴 𝜑
 
10-Jan-2024ralcom2w 3360 Version of ralcom2 3361 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
(∀𝑥𝐴𝑦𝐴 𝜑 → ∀𝑦𝐴𝑥𝐴 𝜑)
 
10-Jan-2024nfra2w 3224 Version of nfra2 3225 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝑥𝐴𝑦𝐵 𝜑
 
10-Jan-2024nfralw 3222 Version of nfral 3223 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑥𝐴    &   𝑥𝜑       𝑥𝑦𝐴 𝜑
 
10-Jan-2024nfraldw 3220 Version of nfrald 3221 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   (𝜑𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝐴 𝜓)
 
10-Jan-2024nfabdw 2997 Version of nfabd 2998 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑𝑥{𝑦𝜓})
 
10-Jan-2024clelsb3fw 2978 Version of clelsb3f 2979 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑥𝐴       ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
 
10-Jan-2024cbvabw 2887 Version of cbvab 2888 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       {𝑥𝜑} = {𝑦𝜓}
 
10-Jan-2024cbveuw 2683 Version of cbveu 2684 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
 
10-Jan-2024cbvmow 2681 Version of cbvmo 2682 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
 
10-Jan-2024nfeuw 2672 Version of nfeu 2673 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑥𝜑       𝑥∃!𝑦𝜑
 
10-Jan-2024nfeudw 2670 Version of nfeud 2671 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥∃!𝑦𝜓)
 
10-Jan-2024cbval2v 2354 Version of cbval2 2423 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by BJ, 16-Jun-2019.) (Proof shortened by Gino Giotto, 10-Jan-2024.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)
 
10-Jan-2024cbvexdw 2350 Version of cbvexd 2420 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))
 
10-Jan-2024cbvaldw 2349 Version of cbvald 2419 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 
10-Jan-2024cbv2w 2348 Version of cbv2 2414 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 
10-Jan-2024hbsbw 2342 Version of hbsb 2560 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
(𝜑 → ∀𝑧𝜑)       ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
 
10-Jan-2024sbiedwOLD 2324 Obsolete version of sbiedw 2323 as of 28-Jan-2024. (Contributed by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → ([𝑦 / 𝑥]𝜓𝜒))
 
10-Jan-2024nfnaew 2144 Version of nfnae 2448 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 10-Jan-2024.)
𝑧 ¬ ∀𝑥 𝑥 = 𝑦
 
10-Jan-20242sbievw 2096 Version of 2sbiev 2540 with more disjoint variable conditions, requiring fewer axioms. (Contributed by Gino Giotto, 10-Jan-2024.)
((𝑥 = 𝑡𝑦 = 𝑢) → (𝜑𝜓))       ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑𝜓)
 
10-Jan-2024cbvex4vw 2040 Version of cbvex4v 2428 with more disjoint variable conditions, which requires fewer axioms. (Contributed by Gino Giotto, 10-Jan-2024.)
((𝑥 = 𝑣𝑦 = 𝑢) → (𝜑𝜓))    &   ((𝑧 = 𝑓𝑤 = 𝑔) → (𝜓𝜒))       (∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑣𝑢𝑓𝑔𝜒)
 
10-Jan-2024cbvex2vw 2039 Version of cbvex2vv 2427 with more disjoint variable conditions, which requires fewer axioms . (Contributed by Gino Giotto, 10-Jan-2024.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∃𝑥𝑦𝜑 ↔ ∃𝑧𝑤𝜓)
 
10-Jan-2024cbval2vw 2038 Version of cbval2vv 2426 with more disjoint variable conditions, which requires fewer axioms . (Contributed by Gino Giotto, 10-Jan-2024.)
((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)
 
7-Jan-2024bj-fvimacnv0 34456 Variant of fvimacnv 6815 where membership of 𝐴 in the domain is not needed provided the containing class 𝐵 does not contain the empty set. Note that this antecedent would not be needed with definition df-afv 43196. (Contributed by BJ, 7-Jan-2024.)
((Fun 𝐹 ∧ ¬ ∅ ∈ 𝐵) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
 
6-Jan-2024bj-rveccvec 34474 Real vector spaces are subcomplex vector spaces (elemental version). (Contributed by BJ, 6-Jan-2024.)
(𝑉 ∈ ℝ-Vec → 𝑉 ∈ ℂVec)
 
6-Jan-2024bj-rvecsscvec 34473 Real vector spaces are subcomplex vector spaces. (Contributed by BJ, 6-Jan-2024.)
ℝ-Vec ⊆ ℂVec
 
6-Jan-2024bj-rvecsscmod 34472 Real vector spaces are subcomplex modules. (Contributed by BJ, 6-Jan-2024.)
ℝ-Vec ⊆ ℂMod
 
6-Jan-2024bj-rveccmod 34471 Real vector spaces are subcomplex modules (elemental version). (Contributed by BJ, 6-Jan-2024.)
(𝑉 ∈ ℝ-Vec → 𝑉 ∈ ℂMod)
 
6-Jan-2024bj-rvecssvec 34470 Real vector spaces are vector spaces. (Contributed by BJ, 6-Jan-2024.)
ℝ-Vec ⊆ LVec
 
6-Jan-2024bj-isrvec2 34469 The predicate "is a real vector space". (Contributed by BJ, 6-Jan-2024.)
(𝜑 → (Scalar‘𝑉) = 𝐾)       (𝜑 → (𝑉 ∈ ℝ-Vec ↔ (𝑉 ∈ LVec ∧ 𝐾 = ℝfld)))
 
6-Jan-2024bj-rvecvec 34468 Real vector spaces are vector spaces (elemental version). (Contributed by BJ, 6-Jan-2024.)
(𝑉 ∈ ℝ-Vec → 𝑉 ∈ LVec)
 
6-Jan-2024bj-isrvecd 34467 The predicate "is a real vector space". (Contributed by BJ, 6-Jan-2024.)
(𝜑 → (Scalar‘𝑉) = 𝐾)       (𝜑 → (𝑉 ∈ ℝ-Vec ↔ (𝑉 ∈ LMod ∧ 𝐾 = ℝfld)))
 
6-Jan-2024bj-rvecrr 34466 The field of scalars of a real vector space is the field of real numbers. (Contributed by BJ, 6-Jan-2024.)
(𝑉 ∈ ℝ-Vec → (Scalar‘𝑉) = ℝfld)
 
6-Jan-2024bj-rvecssmod 34465 Real vector spaces are modules. (Contributed by BJ, 6-Jan-2024.)
ℝ-Vec ⊆ LMod
 
6-Jan-2024bj-rvecmod 34464 Real vector spaces are modules (elemental version). (Contributed by BJ, 6-Jan-2024.)
(𝑉 ∈ ℝ-Vec → 𝑉 ∈ LMod)
 
6-Jan-2024bj-isrvec 34463 The predicate "is a real vector space". (Contributed by BJ, 6-Jan-2024.)
(𝑉 ∈ ℝ-Vec ↔ (𝑉 ∈ LMod ∧ (Scalar‘𝑉) = ℝfld))
 
6-Jan-2024bj-isclm 34460 The predicate "is a subcomplex module". (Contributed by BJ, 6-Jan-2024.)
(𝜑𝐹 = (Scalar‘𝑊))    &   (𝜑𝐾 = (Base‘𝐹))       (𝜑 → (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld))))
 
6-Jan-2024bj-rrdrg 34459 The field of real numbers is a division ring. (Contributed by BJ, 6-Jan-2024.)
fld ∈ DivRing
 
6-Jan-2024bj-flddrng 34458 Fields are division rings. (Contributed by BJ, 6-Jan-2024.)
Field ⊆ DivRing
 
6-Jan-2024bj-isvec 34457 The predicate "is a vector space". (Contributed by BJ, 6-Jan-2024.)
(𝜑𝐾 = (Scalar‘𝑉))       (𝜑 → (𝑉 ∈ LVec ↔ (𝑉 ∈ LMod ∧ 𝐾 ∈ DivRing)))
 
5-Jan-2024bj-grpssmndel 34445 Groups are monoids (elemental version). Shorter proof of grpmnd 18048. (Contributed by BJ, 5-Jan-2024.) (Proof modification is discouraged.)
(𝐴 ∈ Grp → 𝐴 ∈ Mnd)
 
5-Jan-2024bj-grpssmnd 34444 Groups are monoids. (Contributed by BJ, 5-Jan-2024.) (Proof modification is discouraged.)
Grp ⊆ Mnd
 
4-Jan-2024cycpmco2lem7 30701 Lemma for cycpmco2 30702. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    &   (𝜑𝐾 ∈ ran 𝑊)    &   (𝜑𝐾𝐽)    &   (𝜑 → (𝑈𝐾) ∈ (0..^𝐸))       (𝜑 → ((𝑀𝑈)‘𝐾) = ((𝑀𝑊)‘𝐾))
 
4-Jan-2024cycpmco2lem6 30700 Lemma for cycpmco2 30702. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    &   (𝜑𝐾 ∈ ran 𝑊)    &   (𝜑𝐾𝐼)    &   (𝜑 → (𝑈𝐾) ∈ (𝐸..^((♯‘𝑈) − 1)))       (𝜑 → ((𝑀𝑈)‘𝐾) = ((𝑀𝑊)‘𝐾))
 
4-Jan-2024cycpmco2lem5 30699 Lemma for cycpmco2 30702. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)    &   (𝜑𝐾 ∈ ran 𝑊)    &   (𝜑 → (𝑈𝐾) = ((♯‘𝑈) − 1))       (𝜑 → ((𝑀𝑈)‘𝐾) = ((𝑀𝑊)‘𝐾))
 
4-Jan-2024cycpmco2lem4 30698 Lemma for cycpmco2 30702. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑 → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀𝑈)‘𝐼))
 
4-Jan-2024cycpmco2lem3 30697 Lemma for cycpmco2 30702. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑 → ((♯‘𝑈) − 1) = (♯‘𝑊))
 
4-Jan-2024cycpmco2lem2 30696 Lemma for cycpmco2 30702. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑 → (𝑈𝐸) = 𝐼)
 
4-Jan-2024cycpmco2lem1 30695 Lemma for cycpmco2 30702. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑 → ((𝑀𝑊)‘((𝑀‘⟨“𝐼𝐽”⟩)‘𝐼)) = ((𝑀𝑊)‘𝐽))
 
4-Jan-2024cycpmco2rn 30694 The orbit of the composition of a cyclic permutation and a well-chosen transposition is one element more than the orbit of the original permutation. (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑 → ran 𝑈 = (ran 𝑊 ∪ {𝐼}))
 
4-Jan-2024cycpmco2f1 30693 The word U used in cycpmco2 30702 is injective, so it can represent a cycle and form a cyclic permutation (𝑀𝑈). (Contributed by Thierry Arnoux, 4-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑𝑈:dom 𝑈1-1𝐷)
 
4-Jan-2024offsplitfpar 7804 Express the function operation map f by the functions defined in fsplit 7801 and fpar 7800. (Contributed by AV, 4-Jan-2024.)
𝐻 = (((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))    &   𝑆 = ((1st ↾ I ) ↾ 𝐴)       (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐹𝑉𝐺𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝐻𝑆)) = (𝐹f + 𝐺))
 
3-Jan-2024simpcntrab 43004 The center of a simple group is trivial or the group is abelian. (Contributed by SS, 3-Jan-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝑍 = (Cntr‘𝐺)    &   (𝜑𝐺 ∈ SimpGrp)       (𝜑 → (𝑍 = { 0 } ∨ 𝐺 ∈ Abel))
 
3-Jan-2024ex-fpar 28168 Formalized example provided in the comment for fpar 7800. (Contributed by AV, 3-Jan-2024.)
𝐻 = (((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))    &   𝐴 = (0[,)+∞)    &   𝐵 = ℝ    &   𝐹 = (√ ↾ 𝐴)    &   𝐺 = (sin ↾ 𝐵)       ((𝑋𝐴𝑌𝐵) → (𝑋( + ∘ 𝐻)𝑌) = ((√‘𝑋) + (sin‘𝑌)))
 
3-Jan-2024isfrgr 27966 The property of being a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.) (Revised by AV, 29-Mar-2021.) (Revised by AV, 3-Jan-2024.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑥𝑉 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝐸))
 
3-Jan-2024df-frgr 27965 Define the class of all friendship graphs: a simple graph is called a friendship graph if every pair of its vertices has exactly one common neighbor. This condition is called the friendship condition , see definition in [MertziosUnger] p. 152. (Contributed by Alexander van der Vekens and Mario Carneiro, 2-Oct-2017.) (Revised by AV, 29-Mar-2021.) (Revised by AV, 3-Jan-2024.)
FriendGraph = {𝑔 ∈ USGraph ∣ [(Vtx‘𝑔) / 𝑣][(Edg‘𝑔) / 𝑒]𝑘𝑣𝑙 ∈ (𝑣 ∖ {𝑘})∃!𝑥𝑣 {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ 𝑒}
 
3-Jan-2024fsplitfpar 7803 Merge two functions with a common argument in parallel. Combination of fsplit 7801 and fpar 7800. (Contributed by AV, 3-Jan-2024.)
𝐻 = (((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))    &   𝑆 = ((1st ↾ I ) ↾ 𝐴)       ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐻𝑆) = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩))
 
2-Jan-2024cycpmco2 30702 The composition of a cyclic permutation and a transposition of one element in the cycle and one outside the cycle results in a cyclic permutation with one more element in its orbit. (Contributed by Thierry Arnoux, 2-Jan-2024.)
𝑀 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ dom 𝑀)    &   (𝜑𝐼 ∈ (𝐷 ∖ ran 𝑊))    &   (𝜑𝐽 ∈ ran 𝑊)    &   𝐸 = ((𝑊𝐽) + 1)    &   𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)       (𝜑 → ((𝑀𝑊) ∘ (𝑀‘⟨“𝐼𝐽”⟩)) = (𝑀𝑈))
 
1-Jan-2024fzom1ne1 30450 Elementhood in a half-open interval, except the lower bound, shifted by one. (Contributed by Thierry Arnoux, 1-Jan-2024.)
((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾𝑀) → (𝐾 − 1) ∈ (𝑀..^(𝑁 − 1)))
 
1-Jan-2024fzone1 30449 Elementhood in a half-open interval, except its lower bound. (Contributed by Thierry Arnoux, 1-Jan-2024.)
((𝐾 ∈ (𝑀..^𝑁) ∧ 𝐾𝑀) → 𝐾 ∈ ((𝑀 + 1)..^𝑁))
 
1-Jan-2024fzm1ne1 30438 Elementhood of an integer and its predecessor in finite intervals of integers. (Contributed by Thierry Arnoux, 1-Jan-2024.)
((𝐾 ∈ (𝑀...𝑁) ∧ 𝐾𝑀) → (𝐾 − 1) ∈ (𝑀...(𝑁 − 1)))
 
1-Jan-2024fzne1 30437 Elementhood in a finite set of sequential integers, except its lower bound. (Contributed by Thierry Arnoux, 1-Jan-2024.)
((𝐾 ∈ (𝑀...𝑁) ∧ 𝐾𝑀) → 𝐾 ∈ ((𝑀 + 1)...𝑁))
 
1-Jan-2024ccatlen 13915 The length of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by JJ, 1-Jan-2024.)
((𝑆 ∈ Word 𝐴𝑇 ∈ Word 𝐵) → (♯‘(𝑆 ++ 𝑇)) = ((♯‘𝑆) + (♯‘𝑇)))
 
31-Dec-2023bj-elpwg 34239 If the intersection of two classes is a set, then inclusion among these classes is equivalent to membership in the powerclass. Common generalization of elpwg 4541 and elpw2g 5238 (the latter of which could be proved from it). (Contributed by BJ, 31-Dec-2023.)
((𝐴𝐵) ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
 
31-Dec-2023bj-bixor 33822 Equivalence of two ternary operations. Note the identical order and parenthesizing of the three arguments in both expressions. (Contributed by BJ, 31-Dec-2023.)
((𝜑 ↔ (𝜓𝜒)) ↔ (𝜑 ⊻ (𝜓𝜒)))
 
31-Dec-2023rinvmod 18858 Uniqueness of a right inverse element in a commutative monoid, if it exists. Corresponds to caovmo 7374. (Contributed by AV, 31-Dec-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝐵)       (𝜑 → ∃*𝑤𝐵 (𝐴 + 𝑤) = 0 )
 
31-Dec-2023fsplit 7801 A function that can be used to feed a common value to both operands of an operation. Use as the second argument of a composition with the function of fpar 7800 in order to build compound functions such as (𝑥 ∈ (0[,)+∞) ↦ ((√‘𝑥) + (sin‘𝑥))). (Contributed by NM, 17-Sep-2007.) Replace use of dfid2 5456 with df-id 5453. (Revised by BJ, 31-Dec-2023.)
(1st ↾ I ) = (𝑥 ∈ V ↦ ⟨𝑥, 𝑥⟩)
 
31-Dec-2023pwidb 4553 A class is an element of its powerclass if and only if it is a set. (Contributed by BJ, 31-Dec-2023.)
(𝐴 ∈ V ↔ 𝐴 ∈ 𝒫 𝐴)
 
31-Dec-2023elpw 4542 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.) (Proof shortened by BJ, 31-Dec-2023.)
𝐴 ∈ V       (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
 
31-Dec-2023elpwg 4541 Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 5238. (Contributed by NM, 6-Aug-2000.) (Proof shortened by BJ, 31-Dec-2023.)
(𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
 
30-Dec-2023eqwrd 13897 Two words are equal iff they have the same length and the same symbol at each position. (Contributed by AV, 13-Apr-2018.) (Revised by JJ, 30-Dec-2023.)
((𝑈 ∈ Word 𝑆𝑊 ∈ Word 𝑇) → (𝑈 = 𝑊 ↔ ((♯‘𝑈) = (♯‘𝑊) ∧ ∀𝑖 ∈ (0..^(♯‘𝑈))(𝑈𝑖) = (𝑊𝑖))))
 
30-Dec-2023pwunss 5446 The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.) (Proof shortened by BJ, 30-Dec-2023.)
(𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴𝐵)
 
29-Dec-2023mndbn0 17915 The base set of a monoid is not empty. Statement in [Lang] p. 3. (Contributed by AV, 29-Dec-2023.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Mnd → 𝐵 ≠ ∅)
 
29-Dec-2023ncanth 7101 Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 5210). Specifically, the identity function maps the universe onto its power class. Compare canth 7100 that works for sets.

This failure comes from a limitation of the collection principle (which is necessary to avoid Russell's paradox ru 3768): 𝒫 V, being a class, cannot contain proper classes, so it is no larger than V, which is why the identity function "succeeds" in being surjective onto 𝒫 V (see pwv 4827). See also the remark in ru 3768 about NF, in which Cantor's theorem fails for sets that are "too large". This theorem gives some intuition behind that failure: in NF the universal class is a set, and it equals its own power set. (Contributed by NM, 29-Jun-2004.) (Proof shortened by BJ, 29-Dec-2023.)

I :V–onto→𝒫 V
 
28-Dec-2023bj-opabssvv 34334 A variant of relopabiv 5686 (which could be proved from it, similarly to relxp 5566 from xpss 5564). (Contributed by BJ, 28-Dec-2023.)
{⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)
 
28-Dec-2023cycsubm 18283 The set of nonnegative integer powers of an element 𝐴 of a monoid forms a submonoid containing 𝐴 (see cycsubmcl 18282), called the cyclic monoid generated by the element 𝐴. This corresponds to the statement in [Lang] p. 6. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   𝐶 = ran 𝐹       ((𝐺 ∈ Mnd ∧ 𝐴𝐵) → 𝐶 ∈ (SubMnd‘𝐺))
 
28-Dec-2023cycsubmcl 18282 The set of nonnegative integer powers of an element 𝐴 contains 𝐴. Although this theorem holds for any class 𝐺, the definition of 𝐹 is only meaningful if 𝐺 is a monoid or at least a unital magma. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   𝐶 = ran 𝐹       (𝐴𝐵𝐴𝐶)
 
28-Dec-2023cycsubmel 18281 Characterization of an element of the set of nonnegative integer powers of an element 𝐴. Although this theorem holds for any class 𝐺, the definition of 𝐹 is only meaningful if 𝐺 is a monoid or at least a unital magma. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴))    &   𝐶 = ran 𝐹       (𝑋𝐶 ↔ ∃𝑖 ∈ ℕ0 𝑋 = (𝑖 · 𝐴))
 
28-Dec-2023mulgnn0gsum 18172 Group multiple (exponentiation) operation at a nonnegative integer expressed by a group sum. This corresponds to the definition in [Lang] p. 6, second formula. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋)       ((𝑁 ∈ ℕ0𝑋𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))
 
28-Dec-2023mulgnngsum 18171 Group multiple (exponentiation) operation at a positive integer expressed by a group sum. (Contributed by AV, 28-Dec-2023.)
𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋)       ((𝑁 ∈ ℕ ∧ 𝑋𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))
 
28-Dec-2023cnvrescnv 6045 Two ways to express the corestriction of a class. (Contributed by BJ, 28-Dec-2023.)
(𝑅𝐵) = (𝑅 ∩ (V × 𝐵))
 
28-Dec-2023alcomiw 2041 Weak version of alcom 2153. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Dec-2023.)
(𝑦 = 𝑧 → (𝜑𝜓))       (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
 
27-Dec-2023gsumfsupp 43966 A group sum of a family can be restricted to the support of that family without changing its value, provided that that support is finite. This corresponds to the definition of an (infinite) product in [Lang] p. 5, last two formulas. (Contributed by AV, 27-Dec-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   𝐼 = (𝐹 supp 0 )    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg (𝐹𝐼)) = (𝐺 Σg 𝐹))
 
27-Dec-2023nn0mnd 43963 The set of nonnegative integers under (complex) addition is a monoid. Example in [Lang] p. 6. Remark: 𝑀 could have also been written as (ℂflds0). (Contributed by AV, 27-Dec-2023.)
𝑀 = {⟨(Base‘ndx), ℕ0⟩, ⟨(+g‘ndx), + ⟩}       𝑀 ∈ Mnd
 
27-Dec-2023bj-idres 34344 Alternate expression for the restricted identity relation. The advantage of that expression is to expose it as a "bounded" class, being included in the Cartesian square of the restricting class. (Contributed by BJ, 27-Dec-2023.)

This is an alternate of idinxpresid 5908 (see idinxpres 5907). See also elrid 5906 and elidinxp 5904. (Proof modification is discouraged.)

( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴))
 
27-Dec-2023bj-ideqgALT 34342 Alternate proof of bj-ideqg 34341 from brabga 5412 instead of bj-opelid 34340 itself proved from bj-opelidb 34336. (Contributed by BJ, 27-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵) ∈ 𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
 
27-Dec-2023bj-inexeqex 34338 Lemma for bj-opelid 34340 (but not specific to the identity relation): if the intersection of two classes is a set and the two classes are equal, then both are sets (all three classes are equal, so they all belong to 𝑉, but it is more convenient to have V in the consequent for theorems using it). (Contributed by BJ, 27-Dec-2023.)
(((𝐴𝐵) ∈ 𝑉𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
27-Dec-2023bj-opelidb1 34337 Characterization of the ordered pair elements of the identity relation. Variant of bj-opelidb 34336 where only the sethood of the first component is expressed. (Contributed by BJ, 27-Dec-2023.)
(⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
 
27-Dec-2023bj-opelidb 34336 Characterization of the ordered pair elements of the identity relation.

Remark: in deduction-style proofs, one could save a few syntactic steps by using another antecedent than which already appears in the proof. Here for instance this could be the definition I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} but this would make the proof less easy to read. (Contributed by BJ, 27-Dec-2023.)

(⟨𝐴, 𝐵⟩ ∈ I ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
 
27-Dec-2023bj-funidres 34335 The restricted identity relation is a function. (Contributed by BJ, 27-Dec-2023.)

TODO: relabel funi 6380 to funid.

Fun ( I ↾ 𝑉)
 
27-Dec-2023bj-opelrelex 34328 The coordinates of an ordered pair that belongs to a relation are sets. TODO: Slightly shorter than brrelex12 5597, which could be proved from it. (Contributed by BJ, 27-Dec-2023.)
((Rel 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑅) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
27-Dec-2023copsex2b 34324 Biconditional form of copsex2d 34323. TODO: prove a relative version, that is, with 𝑥𝑉𝑦𝑊...(𝐴𝑉𝐵𝑊). (Contributed by BJ, 27-Dec-2023.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑦𝜒)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
 
27-Dec-2023bj-reabeq 34236 Relative form of abeq2 2942. (Contributed by BJ, 27-Dec-2023.)
((𝑉𝐴) = {𝑥𝑉𝜑} ↔ ∀𝑥𝑉 (𝑥𝐴𝜑))
 
27-Dec-2023bj-rcleq 34235 Relative version of dfcleq 2812. (Contributed by BJ, 27-Dec-2023.)
((𝑉𝐴) = (𝑉𝐵) ↔ ∀𝑥𝑉 (𝑥𝐴𝑥𝐵))
 
27-Dec-2023bj-rcleqf 34234 Relative version of cleqf 3007. (Contributed by BJ, 27-Dec-2023.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝑉       ((𝑉𝐴) = (𝑉𝐵) ↔ ∀𝑥𝑉 (𝑥𝐴𝑥𝐵))
 
27-Dec-2023gsumxp2 19029 Write a group sum over a cartesian product as a double sum in two ways. This corresponds to the first equation in [Lang] p. 6. (Contributed by AV, 27-Dec-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   (𝜑𝐹:(𝐴 × 𝐶)⟶𝐵)    &   (𝜑𝐹 finSupp 0 )       (𝜑 → (𝐺 Σg (𝑘𝐶 ↦ (𝐺 Σg (𝑗𝐴 ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗𝐴 ↦ (𝐺 Σg (𝑘𝐶 ↦ (𝑗𝐹𝑘))))))
 
27-Dec-2023mndlsmidm 18725 Subgroup sum is idempotent for monoids. This corresponds to the observation in [Lang] p. 6. (Contributed by AV, 27-Dec-2023.)
= (LSSum‘𝐺)    &   𝐵 = (Base‘𝐺)       (𝐺 ∈ Mnd → (𝐵 𝐵) = 𝐵)
 
27-Dec-2023lsmidm 18717 Subgroup sum is idempotent. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) (Proof shortened by AV, 27-Dec-2023.)
= (LSSum‘𝐺)       (𝑈 ∈ (SubGrp‘𝐺) → (𝑈 𝑈) = 𝑈)
 
27-Dec-2023smndlsmidm 18710 The direct product is idempotent for submonoids. (Contributed by AV, 27-Dec-2023.)
= (LSSum‘𝐺)       (𝑈 ∈ (SubMnd‘𝐺) → (𝑈 𝑈) = 𝑈)
 
27-Dec-2023fnresi 6469 The restricted identity relation is a function on the restricting class. (Contributed by NM, 27-Aug-2004.) (Proof shortened by BJ, 27-Dec-2023.)
( I ↾ 𝐴) Fn 𝐴
 
26-Dec-2023bj-ififc 33812 A biconditional connecting the conditional operator for propositions and the conditional operator for classes. Note that there is no sethood hypothesis on 𝑋: it is implied by either side. (Contributed by BJ, 24-Sep-2019.) Generalize statement from setvar 𝑥 to class 𝑋. (Revised by BJ, 26-Dec-2023.)
(𝑋 ∈ if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝑋𝐴, 𝑋𝐵))
 
26-Dec-2023bj-dfif 33811 Alternate definition of the conditional operator for classes, which used to be the main definition. (Contributed by BJ, 26-Dec-2023.) (Proof modification is discouraged.)
if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝜑𝑥𝐴) ∨ (¬ 𝜑𝑥𝐵))}
 
26-Dec-2023gsumreidx 18966 Re-index a finite group sum using a bijection. Corresponds to the first equation in [Lang] p. 5 with 𝑀 = 1. (Contributed by AV, 26-Dec-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐹:(𝑀...𝑁)⟶𝐵)    &   (𝜑𝐻:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))       (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹𝐻)))
 
26-Dec-2023gsumccat 17994 Homomorphic property of composites. Second formula in [Lang] p. 4. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 26-Dec-2023.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵𝑋 ∈ Word 𝐵) → (𝐺 Σg (𝑊 ++ 𝑋)) = ((𝐺 Σg 𝑊) + (𝐺 Σg 𝑋)))
 
26-Dec-2023gsumsgrpccat 17992 Homomorphic property of not empty composites of a group sum over a semigroup. Formerly part of proof for gsumccat 17994. (Contributed by AV, 26-Dec-2023.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ Smgrp ∧ (𝑊 ∈ Word 𝐵𝑋 ∈ Word 𝐵) ∧ (𝑊 ≠ ∅ ∧ 𝑋 ≠ ∅)) → (𝐺 Σg (𝑊 ++ 𝑋)) = ((𝐺 Σg 𝑊) + (𝐺 Σg 𝑋)))
 
26-Dec-2023gsumsplit1r 17885 Splitting off the rightmost summand of a group sum. This corresponds to the (inductive) definition of a (finite) product in [Lang] p. 4, first formula. (Contributed by AV, 26-Dec-2023.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   (𝜑𝐹:(𝑀...(𝑁 + 1))⟶𝐵)       (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ (𝑀...𝑁))) + (𝐹‘(𝑁 + 1))))
 
26-Dec-2023lidrididd 17868 If there is a left and right identity element for any binary operation (group operation) +, the left identity element (and therefore also the right identity element according to lidrideqd 17867) is equal to the two-sided identity element. (Contributed by AV, 26-Dec-2023.)
(𝜑𝐿𝐵)    &   (𝜑𝑅𝐵)    &   (𝜑 → ∀𝑥𝐵 (𝐿 + 𝑥) = 𝑥)    &   (𝜑 → ∀𝑥𝐵 (𝑥 + 𝑅) = 𝑥)    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)       (𝜑𝐿 = 0 )
 
26-Dec-2023lidrideqd 17867 If there is a left and right identity element for any binary operation (group operation) +, both identity elements are equal. Generalization of statement in [Lang] p. 3: it is sufficient that "e" is a left identity element and "e`" is a right identity element instead of both being (two-sided) identity elements. (Contributed by AV, 26-Dec-2023.)
(𝜑𝐿𝐵)    &   (𝜑𝑅𝐵)    &   (𝜑 → ∀𝑥𝐵 (𝐿 + 𝑥) = 𝑥)    &   (𝜑 → ∀𝑥𝐵 (𝑥 + 𝑅) = 𝑥)       (𝜑𝐿 = 𝑅)
 
26-Dec-2023rnep 5790 The range of the membership relation is the universal class minus the empty set. (Contributed by BJ, 26-Dec-2023.)
ran E = (V ∖ {∅})
 
26-Dec-2023dmep 5786 The domain of the membership relation is the universal class. (Contributed by Scott Fenton, 27-Oct-2010.) (Proof shortened by BJ, 26-Dec-2023.)
dom E = V
 
26-Dec-2023pwundif 5449 Break up the power class of a union into a union of smaller classes. (Contributed by NM, 25-Mar-2007.) (Proof shortened by Thierry Arnoux, 20-Dec-2016.) (Proof shortened by BJ, 26-Dec-2023.)
𝒫 (𝐴𝐵) = ((𝒫 (𝐴𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
 
25-Dec-2023sn-00id 39109 00id 10803 proven without ax-mulcom 10589 but using ax-1ne0 10594. (Though note that the current version of 00id 10803 can be changed to avoid ax-icn 10584, ax-addcl 10585, ax-mulcl 10587, ax-i2m1 10593, ax-cnre 10598. Most of this is by using 0cnALT3 39031 instead of 0cn 10621). (Contributed by SN, 25-Dec-2023.) (Proof modification is discouraged.)
(0 + 0) = 0
 
25-Dec-2023sn-00idlem3 39108 Lemma for sn-00id 39109. (Contributed by SN, 25-Dec-2023.)
((0 − 0) = 1 → (0 + 0) = 0)
 
25-Dec-2023sn-00idlem2 39107 Lemma for sn-00id 39109. (Contributed by SN, 25-Dec-2023.)
((0 − 0) ≠ 0 → (0 − 0) = 1)
 
25-Dec-2023sn-00idlem1 39106 Lemma for sn-00id 39109. (Contributed by SN, 25-Dec-2023.)
(𝐴 ∈ ℝ → (𝐴 · (0 − 0)) = (𝐴 𝐴))
 
25-Dec-2023re1m1e0m0 39105 Equality of two left-additive identities. See resubidaddid1 39103. Uses ax-i2m1 10593. (Contributed by SN, 25-Dec-2023.)
(1 − 1) = (0 − 0)
 
25-Dec-2023bj-ideqg1ALT 34349 Alternate proof of bj-ideqg1 using brabga 5412 instead of the "unbounded" version bj-brab2a1 34333 or brab2a 5637. (Contributed by BJ, 25-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

TODO: delete once bj-ideqg 34341 is in the main section.

((𝐴𝑉𝐵𝑊) → (𝐴 I 𝐵𝐴 = 𝐵))
 
25-Dec-2023bj-idreseq 34346 Sufficient condition for the restricted identity relation to agree with equality. Note that the instance of bj-ideqg 34341 with V substituted for 𝑉 is a direct consequence of bj-idreseq 34346. This is a strengthening of resieq 5857 which should be proved from it (note that currently, resieq 5857 relies on ideq 5716). Note that the intersection in the antecedent is not very meaningful, but is a device to prove versions with either class assumed to be a set. It could be enough to prove the version with a disjunctive antecedent: ((𝐴𝐶𝐵𝐶) → .... (Contributed by BJ, 25-Dec-2023.)
((𝐴𝐵) ∈ 𝐶 → (𝐴( I ↾ 𝐶)𝐵𝐴 = 𝐵))
 
25-Dec-2023bj-brab2a1 34333 "Unbounded" version of brab2a 5637. (Contributed by BJ, 25-Dec-2023.)
((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))    &   𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}       (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜓))
 
25-Dec-2023bj-brresdm 34330 If two classes are related by a restricted binary relation, then the first class is an element of the restricting class. See also brres 5853 and brrelex1 5598.

Remark: there are many pairs like bj-opelresdm 34329 / bj-brresdm 34330, where one uses membership of ordered pairs and the other, related classes (for instance, bj-opelresdm 34329 / brrelex12 5597 or the opelopabg 5416 / brabg 5417 family). They are straightforwardly equivalent by df-br 5058. The latter is indeed a very direct definition, introducing a "shorthand", and barely necessary, were it not for the frequency of the expression 𝐴𝑅𝐵. Therefore, in the spirit of "definitions are here to be used", most theorems, apart from the most elementary ones, should only have the "br" version, not the "opel" one. (Contributed by BJ, 25-Dec-2023.)

(𝐴(𝑅𝑋)𝐵𝐴𝑋)
 
25-Dec-2023bj-opelresdm 34329 If an ordered pair is in a restricted binary relation, then its first component is an element of the restricting class. See also opelres 5852. (Contributed by BJ, 25-Dec-2023.)
(⟨𝐴, 𝐵⟩ ∈ (𝑅𝑋) → 𝐴𝑋)
 
25-Dec-2023copsex2d 34323 Implicit substitution deduction for ordered pairs. (Contributed by BJ, 25-Dec-2023.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑦𝜒)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
 
25-Dec-2023bj-nfexd 34322 Variant of nfexd 2339. (Contributed by BJ, 25-Dec-2023.)
(𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)
 
25-Dec-2023bj-nfald 34321 Variant of nfald 2338. (Contributed by BJ, 25-Dec-2023.)
(𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥𝑦𝜓)
 
25-Dec-2023bj-exlimd 33855 A slightly more general exlimd 2208. A common usage will have 𝜑 substituted for 𝜓 and 𝜃 substituted for 𝜏, giving a form closer to exlimd 2208. (Contributed by BJ, 25-Dec-2023.)
(𝜑 → ∀𝑥𝜓)    &   (𝜑 → (∃𝑥𝜃𝜏))    &   (𝜓 → (𝜒𝜃))       (𝜑 → (∃𝑥𝜒𝜏))
 
25-Dec-2023bj-sylge 33854 Dual statement of sylg 1814 (the final "e" in the label stands for "existential (version of sylg 1814)". Variant of exlimih 2288. (Contributed by BJ, 25-Dec-2023.)
(∃𝑥𝜑𝜓)    &   (𝜒𝜑)       (∃𝑥𝜒𝜓)
 
25-Dec-2023bj-alrimd 33850 A slightly more general alrimd 2205. A common usage will have 𝜑 substituted for 𝜓 and 𝜒 substituted for 𝜃, giving a form closer to alrimd 2205. (Contributed by BJ, 25-Dec-2023.)
(𝜑 → ∀𝑥𝜓)    &   (𝜑 → (𝜒 → ∀𝑥𝜃))    &   (𝜓 → (𝜃𝜏))       (𝜑 → (𝜒 → ∀𝑥𝜏))
 
24-Dec-2023bj-imdirid 34367 Functorial property of the direct image: the direct image by the identity on a set is the identity on the powerset. (Contributed by BJ, 24-Dec-2023.)
(𝜑𝐴𝑈)       (𝜑 → ((𝐴𝒫*𝐴)‘( I ↾ 𝐴)) = ( I ↾ 𝒫 𝐴))
 
24-Dec-2023bj-ideqg1 34348 For sets, the identity relation is the same thing as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Generalize to a disjunctive antecedent. (Revised by BJ, 24-Dec-2023.)

TODO: delete once bj-ideqg 34341 is in the main section.

((𝐴𝑉𝐵𝑊) → (𝐴 I 𝐵𝐴 = 𝐵))
 
24-Dec-2023bj-idreseqb 34347 Characterization for two classes to be related under the restricted identity relation. (Contributed by BJ, 24-Dec-2023.)
(𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴𝐶𝐴 = 𝐵))
 
24-Dec-2023bj-ideqb 34343 Characterization of classes related by the identity relation. (Contributed by BJ, 24-Dec-2023.)
(𝐴 I 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
 
24-Dec-2023bj-ideqg 34341 Characterization of the classes related by the identity relation when their intersection is a set. Note that the antecedent is more general than either class being a set. (Contributed by NM, 30-Apr-2004.) Weaken the antecedent to sethood of the intersection. (Revised by BJ, 24-Dec-2023.)

TODO: replace ideqg 5715, or at least prove ideqg 5715 from it.

((𝐴𝐵) ∈ 𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
 
23-Dec-2023df-invdir 34369 Definition of the functionalized inverse image, which maps a binary relation between two given sets to its associated inverse image relation. (Contributed by BJ, 23-Dec-2023.)
𝒫* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑎𝑦𝑏) ∧ 𝑥 = (𝑟𝑦))}))
 
23-Dec-2023idfn 6468 The identity relation is a function on the universal class. See also funi 6380. (Contributed by BJ, 23-Dec-2023.)
I Fn V
 
23-Dec-2023idinxpresid 5908 The intersection of the identity relation with the cartesian square of a class is the restriction of the identity relation to that class. (Contributed by FL, 2-Aug-2009.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) (Proof shortened by BJ, 23-Dec-2023.)
( I ∩ (𝐴 × 𝐴)) = ( I ↾ 𝐴)
 
23-Dec-2023idinxpres 5907 The intersection of the identity relation with a cartesian product is the restriction of the identity relation to the intersection of the factors. (Contributed by FL, 2-Aug-2009.) (Proof shortened by Peter Mazsa, 9-Sep-2022.) Generalize statement from cartesian square (now idinxpresid 5908) to cartesian product. (Revised by BJ, 23-Dec-2023.)
( I ∩ (𝐴 × 𝐵)) = ( I ↾ (𝐴𝐵))
 
21-Dec-2023pwvabrel 5596 The powerclass of the cartesian square of the universal class is the class of all sets which are binary relations. (Contributed by BJ, 21-Dec-2023.)
𝒫 (V × V) = {𝑥 ∣ Rel 𝑥}
 
18-Dec-2023norasslem3 1523 This lemma specializes biorf 930 suitably for the proof of norass 1524. (Contributed by Wolf Lammen, 18-Dec-2023.)
𝜑 → ((𝜓𝜒) ↔ ((𝜑𝜓) → 𝜒)))
 
18-Dec-2023norasslem2 1522 This lemma specializes biimt 362 suitably for the proof of norass 1524. (Contributed by Wolf Lammen, 18-Dec-2023.)
(𝜑 → (𝜓 ↔ ((𝜑𝜒) → 𝜓)))
 
18-Dec-2023norasslem1 1521 This lemma shows the equivalence of two expressions, used in norass 1524. (Contributed by Wolf Lammen, 18-Dec-2023.)
(((𝜑𝜓) → 𝜒) ↔ ((𝜑 𝜓) ∨ 𝜒))
 
18-Dec-2023impimprbi 824 An implication and its reverse are equivalent exactly when both operands are equivalent. The right hand side resembles that of dfbi2 475, but is a weaker operator than . Note that an implication and its reverse can never be simultaneously false, because of pm2.521 177. (Contributed by Wolf Lammen, 18-Dec-2023.)
((𝜑𝜓) ↔ ((𝜑𝜓) ↔ (𝜓𝜑)))
 
17-Dec-2023brabd 34332 Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.)
(𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       (𝜑 → (𝐴𝑅𝐵𝜒))
 
17-Dec-2023brabd0 34331 Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑦𝜒)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       (𝜑 → (𝐴𝑅𝐵𝜒))
 
17-Dec-2023opelopabbv 34327 Membership of an ordered pair in a class abstraction of ordered pairs, biconditional form. (Contributed by BJ, 17-Dec-2023.)
(𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
 
17-Dec-2023opelopabb 34326 Membership of an ordered pair in a class abstraction of ordered pairs, biconditional form. (Contributed by BJ, 17-Dec-2023.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑦𝜒)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒)))
 
17-Dec-2023opelopabd 34325 Membership of an ordere pair in a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ∀𝑦𝜑)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑦𝜒)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))       (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ 𝜒))
 
17-Dec-2023subne0nn 30464 A nonnegative difference is positive if the two numbers are not equal. (Contributed by Thierry Arnoux, 17-Dec-2023.)
(𝜑𝑀 ∈ ℂ)    &   (𝜑𝑁 ∈ ℂ)    &   (𝜑 → (𝑀𝑁) ∈ ℕ0)    &   (𝜑𝑀𝑁)       (𝜑 → (𝑀𝑁) ∈ ℕ)
 
17-Dec-2023hadcoma 1590 Commutative law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
(hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜑, 𝜒))
 
17-Dec-2023falnorfal 1582 A identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
((⊥ ⊥) ↔ ⊤)
 
17-Dec-2023trunorfal 1579 A identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
((⊤ ⊥) ↔ ⊥)
 
17-Dec-2023norass 1524 A characterization of when an expression involving joint denials associates. This is identical to the case when alternative denial is associative, see nanass 1494. Remark: Like alternative denial, joint denial is also commutative, see norcom 1514. (Contributed by RP, 29-Oct-2023.) (Proof shortened by Wolf Lammen, 17-Dec-2023.)
((𝜑𝜒) ↔ (((𝜑 𝜓) 𝜒) ↔ (𝜑 (𝜓 𝜒))))
 
16-Dec-2023bj-imdirval3 34366 Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.)
(𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝑅 ⊆ (𝐴 × 𝐵))       (𝜑 → (𝑋((𝐴𝒫*𝐵)‘𝑅)𝑌 ↔ ((𝑋𝐴𝑌𝐵) ∧ (𝑅𝑋) = 𝑌)))
 
16-Dec-2023bj-imdirval2 34365 Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.)
(𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)    &   (𝜑𝑅 ⊆ (𝐴 × 𝐵))       (𝜑 → ((𝐴𝒫*𝐵)‘𝑅) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑅𝑥) = 𝑦)})
 
16-Dec-2023bj-imdirval 34364 Value of the functionalized direct image. (Contributed by BJ, 16-Dec-2023.)
(𝜑𝐴𝑈)    &   (𝜑𝐵𝑉)       (𝜑 → (𝐴𝒫*𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑟𝑥) = 𝑦)}))
 
16-Dec-2023df-imdir 34363 Definition of the functionalized direct image, which maps a binary relation between two given sets to its associated direct image relation. (Contributed by BJ, 16-Dec-2023.)
𝒫* = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑎𝑦𝑏) ∧ (𝑟𝑥) = 𝑦)}))
 
16-Dec-2023bj-pwvrelb 34111 Characterization of the elements of the powerclass of the cartesian square of the universal class: they are exactly the sets which are binary relations. (Contributed by BJ, 16-Dec-2023.)
(𝐴 ∈ 𝒫 (V × V) ↔ (𝐴 ∈ V ∧ Rel 𝐴))
 
16-Dec-2023pwvrel 5595 A set is a binary relation if and only if it belongs to the powerclass of the cartesian square of the universal class. (Contributed by Peter Mazsa, 14-Jun-2018.) (Revised by BJ, 16-Dec-2023.)
(𝐴𝑉 → (𝐴 ∈ 𝒫 (V × V) ↔ Rel 𝐴))
 
15-Dec-2023selvval2lem3 39012 The third argument passed to evalSub is in the domain. (Contributed by SN, 15-Dec-2023.)
𝑈 = (𝐼 mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ CRing)       (𝜑 → ran 𝐷 ∈ (SubRing‘𝑇))
 
15-Dec-2023selvval2lem2 39011 𝐷 is a ring homomorphism. (Contributed by SN, 15-Dec-2023.)
𝑈 = (𝐼 mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ CRing)       (𝜑𝐷 ∈ (𝑅 RingHom 𝑇))
 
15-Dec-2023selvval2lem1 39010 𝑇 is an associative algebra. For simplicity, 𝐼 stands for (𝐼𝐽) and we have 𝐽𝑊 instead of 𝐽𝐼. (Contributed by SN, 15-Dec-2023.)
𝑈 = (𝐼 mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ CRing)       (𝜑𝑇 ∈ AssAlg)
 
15-Dec-2023equs5av 2270 Version of equs5a 2472 with a disjoint variable condition, which does not require ax-13 2381. See also sb56 2268. (Contributed by Gino Giotto, 15-Dec-2023.)
(∃𝑥(𝑥 = 𝑦 ∧ ∀𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑))
 
15-Dec-2023sb4av 2234 Version of sb4a 2502 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by BJ, 15-Dec-2023.)
([𝑡 / 𝑥]∀𝑡𝜑 → ∀𝑥(𝑥 = 𝑡𝜑))
 
14-Dec-2023cu3addd 39155 Cube of sum of three numbers. (Contributed by Igor Ieskov, 14-Dec-2023.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → (((𝐴 + 𝐵) + 𝐶)↑3) = (((((𝐴↑3) + (3 · ((𝐴↑2) · 𝐵))) + ((3 · (𝐴 · (𝐵↑2))) + (𝐵↑3))) + (((3 · ((𝐴↑2) · 𝐶)) + (((3 · 2) · (𝐴 · 𝐵)) · 𝐶)) + (3 · ((𝐵↑2) · 𝐶)))) + (((3 · (𝐴 · (𝐶↑2))) + (3 · (𝐵 · (𝐶↑2)))) + (𝐶↑3))))
 
14-Dec-2023binom2d 39154 Deduction form of binom2. (Contributed by Igor Ieskov, 14-Dec-2023.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵)↑2) = (((𝐴↑2) + (2 · (𝐴 · 𝐵))) + (𝐵↑2)))
 
14-Dec-2023splfv3 30559 Symbols to the right of a splice are unaffected. (Contributed by Thierry Arnoux, 14-Dec-2023.)
(𝜑𝑆 ∈ Word 𝐴)    &   (𝜑𝐹 ∈ (0...𝑇))    &   (𝜑𝑇 ∈ (0...(♯‘𝑆)))    &   (𝜑𝑅 ∈ Word 𝐴)    &   (𝜑𝑋 ∈ (0..^((♯‘𝑆) − 𝑇)))    &   (𝜑𝐾 = (𝐹 + (♯‘𝑅)))       (𝜑 → ((𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩)‘(𝑋 + 𝐾)) = (𝑆‘(𝑋 + 𝑇)))
 
14-Dec-2023elunsn 30200 Elementhood to a union with a singleton. (Contributed by Thierry Arnoux, 14-Dec-2023.)
(𝐴𝑉 → (𝐴 ∈ (𝐵 ∪ {𝐶}) ↔ (𝐴𝐵𝐴 = 𝐶)))
 
14-Dec-2023elfzom1p1elfzo 13105 Increasing an element of a half-open range of nonnegative integers by 1 results in an element of the half-open range of nonnegative integers with an upper bound increased by 1. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Proof shortened by Thierry Arnoux, 14-Dec-2023.)
((𝑁 ∈ ℕ ∧ 𝑋 ∈ (0..^(𝑁 − 1))) → (𝑋 + 1) ∈ (0..^𝑁))
 
13-Dec-2023sn-1ne2 39036 A proof of 1ne2 11833 without using ax-mulcom 10589, ax-mulass 10591, ax-pre-mulgt0 10602. Based on mul02lem2 10805. (Contributed by SN, 13-Dec-2023.)
1 ≠ 2
 
13-Dec-2023swrdrndisj 30558 Condition for the range of two subwords of an injective word to be disjoint. (Contributed by Thierry Arnoux, 13-Dec-2023.)
(𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑀 ∈ (0...𝑁))    &   (𝜑𝑁 ∈ (0...(♯‘𝑊)))    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑𝑂 ∈ (𝑁...𝑃))    &   (𝜑𝑃 ∈ (𝑁...(♯‘𝑊)))       (𝜑 → (ran (𝑊 substr ⟨𝑀, 𝑁⟩) ∩ ran (𝑊 substr ⟨𝑂, 𝑃⟩)) = ∅)
 
13-Dec-2023swrdrn3 30556 Express the range of a subword. Stronger version of swrdrn2 30555. (Contributed by Thierry Arnoux, 13-Dec-2023.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑊 “ (𝑀..^𝑁)))
 
13-Dec-2023pfxf1 30545 Condition for a prefix to be injective. (Contributed by Thierry Arnoux, 13-Dec-2023.)
(𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝑊:dom 𝑊1-1𝑆)    &   (𝜑𝐿 ∈ (0...(♯‘𝑊)))       (𝜑 → (𝑊 prefix 𝐿):dom (𝑊 prefix 𝐿)–1-1𝑆)
 
13-Dec-2023pfxrn3 30544 Express the range of a prefix of a word. Stronger version of pfxrn2 30543. (Contributed by Thierry Arnoux, 13-Dec-2023.)
((𝑊 ∈ Word 𝑆𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) = (𝑊 “ (0..^𝐿)))
 
13-Dec-2023nelun 30201 Negated membership for a union. (Contributed by Thierry Arnoux, 13-Dec-2023.)
(𝐴 = (𝐵𝐶) → (¬ 𝑋𝐴 ↔ (¬ 𝑋𝐵 ∧ ¬ 𝑋𝐶)))
 
12-Dec-2023swrdf1 30557 Condition for a subword to be injective. (Contributed by Thierry Arnoux, 12-Dec-2023.)
(𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑀 ∈ (0...𝑁))    &   (𝜑𝑁 ∈ (0...(♯‘𝑊)))    &   (𝜑𝑊:dom 𝑊1-1𝐷)       (𝜑 → (𝑊 substr ⟨𝑀, 𝑁⟩):dom (𝑊 substr ⟨𝑀, 𝑁⟩)–1-1𝐷)
 
12-Dec-2023swrdrn2 30555 The range of a subword is a subset of the range of that word. Stronger version of swrdrn 14002. (Contributed by Thierry Arnoux, 12-Dec-2023.)
((𝑊 ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) ⊆ ran 𝑊)
 
12-Dec-2023pfxrn2 30543 The range of a prefix of a word is a subset of the range of that word. Stronger version of pfxrn 14035. (Contributed by Thierry Arnoux, 12-Dec-2023.)
((𝑊 ∈ Word 𝑆𝐿 ∈ (0...(♯‘𝑊))) → ran (𝑊 prefix 𝐿) ⊆ ran 𝑊)
 
12-Dec-2023incom 4175 Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 21-Jun-1993.) (Proof shortened by SN, 12-Dec-2023.)
(𝐴𝐵) = (𝐵𝐴)
 
12-Dec-2023rspcev 3620 Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) Drop ax-10 2136, ax-11 2151, ax-12 2167. (Revised by SN, 12-Dec-2023.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
 
12-Dec-2023rspcv 3615 Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) Drop ax-10 2136, ax-11 2151, ax-12 2167. (Revised by SN, 12-Dec-2023.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
 
11-Dec-2023ccatf1 30552 Conditions for a concatenation to be injective. (Contributed by Thierry Arnoux, 11-Dec-2023.)
(𝜑𝑆𝑉)    &   (𝜑𝐴 ∈ Word 𝑆)    &   (𝜑𝐵 ∈ Word 𝑆)    &   (𝜑𝐴:dom 𝐴1-1𝑆)    &   (𝜑𝐵:dom 𝐵1-1𝑆)    &   (𝜑 → (ran 𝐴 ∩ ran 𝐵) = ∅)       (𝜑 → (𝐴 ++ 𝐵):dom (𝐴 ++ 𝐵)–1-1𝑆)
 
11-Dec-2023s1f1 30546 Conditions for a length 1 string to be a one-to-one function. (Contributed by Thierry Arnoux, 11-Dec-2023.)
(𝜑𝐼𝐷)       (𝜑 → ⟨“𝐼”⟩:dom ⟨“𝐼”⟩–1-1𝐷)
 
9-Dec-2023bj-wnfnf 33965 When 𝜑 is substituted for 𝜓, this statement expresses nonfreeness in the weak form of nonfreeness (∃ → ∀). Note that this could also be proved from bj-nnfim 33972, bj-nnfe1 33986 and bj-nnfa1 33985. (Contributed by BJ, 9-Dec-2023.)
Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓)
 
9-Dec-2023bj-wnfenf 33951 When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the "exists" form of nonfreeness. (Contributed by BJ, 9-Dec-2023.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑𝜓))
 
9-Dec-2023bj-wnfanf 33950 When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the "forall" form of nonfreeness. (Contributed by BJ, 9-Dec-2023.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓))
 
9-Dec-2023bj-wnf2 33949 When 𝜑 is substituted for 𝜓, this is the first half of nonfreness (. → ∀) of the weak form of nonfreeness (∃ → ∀). (Contributed by BJ, 9-Dec-2023.)
(∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))
 
9-Dec-2023bj-wnf1 33948 When 𝜑 is substituted for 𝜓, this is the first half of nonfreness (. → ∀) of the weak form of nonfreeness (∃ → ∀). (Contributed by BJ, 9-Dec-2023.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓))
 
9-Dec-2023bj-eximcom 33873 A commuted form of exim 1825 which is sometimes posited as an axiom in instuitionistic modal logic. (Contributed by BJ, 9-Dec-2023.)
(∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
 
9-Dec-2023bj-aleximiALT 33872 Alternate proof of aleximi 1823 from exim 1825, which is sometimes used as an axiom in instuitionistic modal logic. (Contributed by BJ, 9-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
 
9-Dec-2023bj-eximALT 33871 Alternate proof of exim 1825 directly from alim 1802 by using df-ex 1772 (using duality of and . (Contributed by BJ, 9-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
 
9-Dec-2023bj-nfimexal 33856 A weak from of nonfreeness in either an antecedent or a consequent implies that a universally quantified implication is equivalent to the associated implication where the antecedent is existentially quantified and the consequent is universally quantified. The forward implication always holds (this is 19.38 1830) and the converse implication is the join of instances of bj-alrimg 33849 and bj-exlimg 33853 (see 19.38a 1831 and 19.38b 1832). TODO: prove a version where the antecedents use the nonfreeness quantifier. (Contributed by BJ, 9-Dec-2023.)
(((∃𝑥𝜑 → ∀𝑥𝜑) ∨ (∃𝑥𝜓 → ∀𝑥𝜓)) → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))
 
9-Dec-2023bj-exlimg 33853 The general form of the *exlim* family of theorems: if 𝜑 is substituted for 𝜓, then the antecedent expresses a form of nonfreeness of 𝑥 in 𝜑, so the theorem means that under a nonfreeness condition in a consequent, one can deduce from the universally quantified implication an implication where the antecedent is existentially quantified. Dual of bj-alrimg 33849. (Contributed by BJ, 9-Dec-2023.)
((∃𝑥𝜑𝜓) → (∀𝑥(𝜒𝜑) → (∃𝑥𝜒𝜓)))
 
9-Dec-2023bj-alrimg 33849 The general form of the *alrim* family of theorems: if 𝜑 is substituted for 𝜓, then the antecedent expresses a form of nonfreeness of 𝑥 in 𝜑, so the theorem means that under a nonfreeness condition in an antecedent, one can deduce from the universally quantified implication an implication where the consequent is universally quantified. Dual of bj-exlimg 33853. (Contributed by BJ, 9-Dec-2023.)
((𝜑 → ∀𝑥𝜓) → (∀𝑥(𝜓𝜒) → (𝜑 → ∀𝑥𝜒)))
 
8-Dec-2023noror 1519 is expressible via . (Contributed by Remi, 26-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
((𝜑𝜓) ↔ ((𝜑 𝜓) (𝜑 𝜓)))
 
8-Dec-2023noran 1517 is expressible via . (Contributed by Remi, 26-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
((𝜑𝜓) ↔ ((𝜑 𝜑) (𝜓 𝜓)))
 
8-Dec-2023nornot 1515 ¬ is expressible via . (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 8-Dec-2023.)
𝜑 ↔ (𝜑 𝜑))
 
7-Dec-2023swrdwlk 32270 Two matching subwords of a walk also represent a walk. (Contributed by BTernaryTau, 7-Dec-2023.)
((𝐹(Walks‘𝐺)𝑃𝐵 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 substr ⟨𝐵, 𝐿⟩)(Walks‘𝐺)(𝑃 substr ⟨𝐵, (𝐿 + 1)⟩))
 
7-Dec-2023trunortru 1577 A identity. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 7-Dec-2023.)
((⊤ ⊤) ↔ ⊥)
 
4-Dec-2023bj-nnclav 33781 When is substituted for 𝜓, this formula is the Clavius law with a doubly negated consequent. (Contributed by BJ, 4-Dec-2023.)
(((𝜑𝜓) → 𝜑) → ((𝜑𝜓) → 𝜓))
 
4-Dec-2023revwlkb 32269 Two words represent a walk if and only if their reverses also represent a walk. (Contributed by BTernaryTau, 4-Dec-2023.)
((𝐹 ∈ Word 𝑊𝑃 ∈ Word 𝑈) → (𝐹(Walks‘𝐺)𝑃 ↔ (reverse‘𝐹)(Walks‘𝐺)(reverse‘𝑃)))
 
3-Dec-2023swrdrevpfx 32260 A subword expressed in terms of reverses and prefixes. (Contributed by BTernaryTau, 3-Dec-2023.)
((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝑊 substr ⟨𝐹, 𝐿⟩) = (reverse‘((reverse‘(𝑊 prefix 𝐿)) prefix (𝐿𝐹))))
 
3-Dec-2023rabbidva 3476 Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 28-Nov-2003.) (Proof shortened by SN, 3-Dec-2023.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → {𝑥𝐴𝜓} = {𝑥𝐴𝜒})
 
2-Dec-2023xlimlimsupleliminf 42020 A sequence of extended reals converges if and only if its superior limit is smaller than or equal to its inferior limit. (Contributed by Glauco Siliprandi, 2-Dec-2023.)
(𝜑𝑀 ∈ ℤ)    &   𝑍 = (ℤ𝑀)    &   (𝜑𝐹:𝑍⟶ℝ*)       (𝜑 → (𝐹 ∈ dom ~~>* ↔ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))
 
2-Dec-2023bj-19.41t 34000 Closed form of 19.41 2227 from the same axioms as 19.41v 1941. The same is doable with 19.27 2219, 19.28 2220, 19.31 2226, 19.32 2225, 19.44 2229, 19.45 2230. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜓 → (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
 
2-Dec-2023bj-19.42t 33999 Closed form of 19.42 2228 from the same axioms as 19.42v 1945. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜑 → (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓)))
 
2-Dec-2023bj-19.37im 33998 One direction of 19.37 2224 from the same axioms as 19.37imv 1939. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜑 → (∃𝑥(𝜑𝜓) → (𝜑 → ∃𝑥𝜓)))
 
2-Dec-2023bj-19.36im 33997 One direction of 19.36 2222 from the same axioms as 19.36imv 1937. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜓 → (∃𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓)))
 
2-Dec-2023bj-19.23t 33996 Statement 19.23t 2200 proved from modalK (obsoleting 19.23v 1934). (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜓 → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
 
2-Dec-2023bj-19.21t 33995 Statement 19.21t 2196 proved from modalK (obsoleting 19.21v 1931). (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
 
2-Dec-2023bj-stdpc5t 33994 Alias of bj-nnf-alrim 33981 for labeling consistency (a standard predicate calculus axiom). Closed form of stdpc5 2198 proved from modalK (obsoleting stdpc5v 1930). (Contributed by BJ, 2-Dec-2023.) Use bj-nnf-alrim 33981 instead. (New usaged is discouraged.)
(Ⅎ'𝑥𝜑 → (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
 
2-Dec-2023bj-nnf-exlim 33982 Proof of the closed form of exlimi 2207 from modalK (compare exlimiv 1922). See also bj-sylget2 33852. (Contributed by BJ, 2-Dec-2023.)
(Ⅎ'𝑥𝜓 → (∀𝑥(𝜑𝜓) → (∃𝑥𝜑𝜓)))
 
2-Dec-2023bj-nnfbid 33979 Nonfreeness in both sides implies nonfreeness in the biconditional, deduction form. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
(𝜑 → Ⅎ'𝑥𝜓)    &   (𝜑 → Ⅎ'𝑥𝜒)       (𝜑 → Ⅎ'𝑥(𝜓𝜒))
 
2-Dec-2023bj-nnfbit 33978 Nonfreeness in both sides implies nonfreeness in the biconditional. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑𝜓))
 
2-Dec-2023bj-nnford 33977 Nonfreeness in both disjuncts implies nonfreeness in the disjunction, deduction form. See comments for bj-nnfor 33976 and bj-nnfand 33975. (Contributed by BJ, 2-Dec-2023.) (Proof modification is discouraged.)
(𝜑 → Ⅎ'𝑥𝜓)    &   (𝜑 → Ⅎ'𝑥𝜒)       (𝜑 → Ⅎ'𝑥(𝜓𝜒))
 
2-Dec-2023bj-nnfimd 33973 Nonfreeness in the antecedent and the consequent of an implication implies nonfreeness in the implication, deduction form. (Contributed by BJ, 2-Dec-2023.)
(𝜑 → Ⅎ'𝑥𝜓)    &   (𝜑 → Ⅎ'𝑥𝜒)       (𝜑 → Ⅎ'𝑥(𝜓𝜒))
 
2-Dec-2023pfxwlk 32267 A prefix of a walk is a walk. (Contributed by BTernaryTau, 2-Dec-2023.)
((𝐹(Walks‘𝐺)𝑃𝐿 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝐿)(Walks‘𝐺)(𝑃 prefix (𝐿 + 1)))
 
2-Dec-2023revpfxsfxrev 32259 The reverse of a prefix of a word is equal to the same-length suffix of the reverse of that word. (Contributed by BTernaryTau, 2-Dec-2023.)
((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊))) → (reverse‘(𝑊 prefix 𝐿)) = ((reverse‘𝑊) substr ⟨((♯‘𝑊) − 𝐿), (♯‘𝑊)⟩))
 
1-Dec-2023mpteq12dv 5142 An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.) Drop ax-10 2136 while shortening its proof. (Revised by Steven Nguyen and Gino Giotto, 1-Dec-2023.)
(𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 
1-Dec-2023sbcbidv 3824 Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.) Drop ax-12 2167. (Revised by Gino Giotto, 1-Dec-2023.)
(𝜑 → (𝜓𝜒))       (𝜑 → ([𝐴 / 𝑥]𝜓[𝐴 / 𝑥]𝜒))
 
1-Dec-2023rexab2 3688 Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2107. (Revised by Gino Giotto, 1-Dec-2023.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∃𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∃𝑦(𝜑𝜒))
 
1-Dec-2023ralab2 3685 Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) Drop ax-8 2107. (Revised by Gino Giotto, 1-Dec-2023.)
(𝑥 = 𝑦 → (𝜓𝜒))       (∀𝑥 ∈ {𝑦𝜑}𝜓 ↔ ∀𝑦(𝜑𝜒))
 
1-Dec-2023ceqsexgv 3644 Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.) Drop ax-10 2136 and ax-12 2167. (Revised by Gino Giotto, 1-Dec-2023.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐴𝑉 → (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
 
30-Nov-2023bj-stabpeirce 33786 Over minimal implicational calculus, Peirce's law is implied by the (classical refutation equivalent of) the double negation of the stability of any proposition. (Contributed by BJ, 30-Nov-2023.) (Proof modification is discouraged.)
((((((𝜑𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓) → (((𝜓𝜑) → 𝜓) → 𝜓))
 
30-Nov-2023bj-peircestab 33785 Over minimal implicational calculus, Peirce's law implies the double negation of the stability of any proposition (that is the interpretation when is substitued for 𝜓). (Contributed by BJ, 30-Nov-2023.) (Proof modification is discouraged.)
(((((𝜑𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓)
 
30-Nov-2023revwlk 32268 The reverse of a walk is a walk. (Contributed by BTernaryTau, 30-Nov-2023.)
(𝐹(Walks‘𝐺)𝑃 → (reverse‘𝐹)(Walks‘𝐺)(reverse‘𝑃))
 
29-Nov-2023disjdifr 30202 A class and its relative complement are disjoint. (Contributed by Thierry Arnoux, 29-Nov-2023.)
((𝐵𝐴) ∩ 𝐴) = ∅
 
27-Nov-2023nrmo 33655 "At most one" restricted existential quantifier for a statement which is never true. (Contributed by Thierry Arnoux, 27-Nov-2023.)
(𝑥𝐴 → ¬ 𝜑)       ∃*𝑥𝐴 𝜑
 
27-Nov-2023tocyccntz 30713 All elements of a (finite) set of cycles commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 27-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑍 = (Cntz‘𝑆)    &   𝑀 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑Disj 𝑥𝐴 ran 𝑥)    &   (𝜑𝐴 ⊆ dom 𝑀)       (𝜑 → (𝑀𝐴) ⊆ (𝑍‘(𝑀𝐴)))
 
27-Nov-2023cntzsnid 30623 The centralizer of the identity element is the whole base set. (Contributed by Thierry Arnoux, 27-Nov-2023.)
𝐵 = (Base‘𝑀)    &   𝑍 = (Cntz‘𝑀)    &    0 = (0g𝑀)       (𝑀 ∈ Mnd → (𝑍‘{ 0 }) = 𝐵)
 
27-Nov-2023cntzun 30622 The centralizer of a union is the intersection of the centralizers. (Contributed by Thierry Arnoux, 27-Nov-2023.)
𝐵 = (Base‘𝑀)    &   𝑍 = (Cntz‘𝑀)       ((𝑋𝐵𝑌𝐵) → (𝑍‘(𝑋𝑌)) = ((𝑍𝑋) ∩ (𝑍𝑌)))
 
27-Nov-2023disjxun0 30252 Simplify a disjoint union. (Contributed by Thierry Arnoux, 27-Nov-2023.)
((𝜑𝑥𝐵) → 𝐶 = ∅)       (𝜑 → (Disj 𝑥 ∈ (𝐴𝐵)𝐶Disj 𝑥𝐴 𝐶))
 
27-Nov-2023rmounid 30186 Case where an "at most one" restricted existential quantifier for a union is equivalent to such a quantifier for one of the sets. (Contributed by Thierry Arnoux, 27-Nov-2023.)
((𝜑𝑥𝐵) → ¬ 𝜓)       (𝜑 → (∃*𝑥 ∈ (𝐴𝐵)𝜓 ↔ ∃*𝑥𝐴 𝜓))
 
27-Nov-2023rmoun 30185 "At most one" restricted existential quantifier for a union implies the same quantifier on both sets. (Contributed by Thierry Arnoux, 27-Nov-2023.)
(∃*𝑥 ∈ (𝐴𝐵)𝜑 → (∃*𝑥𝐴 𝜑 ∧ ∃*𝑥𝐵 𝜑))
 
23-Nov-2023ich2ex 43506 Two setvar variables are always interchangeable when there are two existential quantifiers. (Contributed by SN, 23-Nov-2023.)
[𝑥𝑦]∃𝑥𝑦𝜑
 
23-Nov-2023ich2al 43505 Two setvar variables are always interchangeable when there are two universal quantifiers. (Contributed by SN, 23-Nov-2023.)
[𝑥𝑦]∀𝑥𝑦𝜑
 
23-Nov-2023ichf 43487 Setvar variables are interchangeable in a wff they are not free in. (Contributed by SN, 23-Nov-2023.)
𝑥𝜑    &   𝑦𝜑       [𝑥𝑦]𝜑
 
23-Nov-2023ichv 43486 Setvar variables are interchangeable in a wff they do not appear in. (Contributed by SN, 23-Nov-2023.)
[𝑥𝑦]𝜑
 
23-Nov-2023nf5r 2183 Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) df-nf 1776 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by Wolf Lammen, 23-Nov-2023.)
(Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
 
21-Nov-2023tocyc01 30687 Permutation cycles built from the empty set or a singleton are the identity. (Contributed by Thierry Arnoux, 21-Nov-2023.)
𝐶 = (toCyc‘𝐷)       ((𝐷𝑉𝑊 ∈ (dom 𝐶 ∩ (♯ “ {0, 1}))) → (𝐶𝑊) = ( I ↾ 𝐷))
 
21-Nov-20231cshid 30560 Cyclically shifting a single letter word keeps it unchanged. (Contributed by Thierry Arnoux, 21-Nov-2023.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ ∧ (♯‘𝑊) = 1) → (𝑊 cyclShift 𝑁) = 𝑊)
 
21-Nov-2023hashgt1 30456 Restate "set contains at least two elements" in terms of elementhood. (Contributed by Thierry Arnoux, 21-Nov-2023.)
(𝐴𝑉 → (¬ 𝐴 ∈ (♯ “ {0, 1}) ↔ 1 < (♯‘𝐴)))
 
21-Nov-2023xnn01gt 30421 An extended nonnegative integer is neither 0 nor 1 if and only if it is greater than 1. (Contributed by Thierry Arnoux, 21-Nov-2023.)
(𝑁 ∈ ℕ0* → (¬ 𝑁 ∈ {0, 1} ↔ 1 < 𝑁))
 
21-Nov-2023undifr 30211 Union of complementary parts into whole. (Contributed by Thierry Arnoux, 21-Nov-2023.)
(𝐴𝐵 ↔ ((𝐵𝐴) ∪ 𝐴) = 𝐵)
 
20-Nov-2023cycpmrn 30712 The range of the word used to build a cycle is the cycle's orbit, i.e. the set of points it moves. (Contributed by Thierry Arnoux, 20-Nov-2023.)
𝑀 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑 → 1 < (♯‘𝑊))       (𝜑 → ran 𝑊 = dom ((𝑀𝑊) ∖ I ))
 
20-Nov-2023symgcntz 30656 All elements of a (finite) set of permutations commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 20-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝑍 = (Cntz‘𝑆)    &   (𝜑𝐴𝐵)    &   (𝜑Disj 𝑥𝐴 dom (𝑥 ∖ I ))       (𝜑𝐴 ⊆ (𝑍𝐴))
 
20-Nov-2023gsumzresunsn 30618 Append an element to a finite group sum expressed as a function restriction. (Contributed by Thierry Arnoux, 20-Nov-2023.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑍 = (Cntz‘𝐺)    &   𝑌 = (𝐹𝑋)    &   (𝜑𝐹:𝐶𝐵)    &   (𝜑𝐴𝐶)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑 → ¬ 𝑋𝐴)    &   (𝜑𝑋𝐶)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝐹 “ (𝐴 ∪ {𝑋})) ⊆ (𝑍‘(𝐹 “ (𝐴 ∪ {𝑋}))))       (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ∪ {𝑋}))) = ((𝐺 Σg (𝐹𝐴)) + 𝑌))
 
20-Nov-2023eldmne0 30301 A function of nonempty domain is not empty. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝑋 ∈ dom 𝐹𝐹 ≠ ∅)
 
20-Nov-20230res 30282 Restriction of the empty function. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(∅ ↾ 𝐴) = ∅
 
20-Nov-2023iunxunpr 30247 Appending two sets to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝑥 = 𝑋𝐵 = 𝐶)    &   (𝑥 = 𝑌𝐵 = 𝐷)       ((𝑋𝑉𝑌𝑊) → 𝑥 ∈ (𝐴 ∪ {𝑋, 𝑌})𝐵 = ( 𝑥𝐴 𝐵 ∪ (𝐶𝐷)))
 
20-Nov-2023iunxunsn 30246 Appending a set to an indexed union. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝑥 = 𝑋𝐵 = 𝐶)       (𝑋𝑉 𝑥 ∈ (𝐴 ∪ {𝑋})𝐵 = ( 𝑥𝐴 𝐵𝐶))
 
20-Nov-2023neldifpr2 30221 The second element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
¬ 𝐵 ∈ (𝐶 ∖ {𝐴, 𝐵})
 
20-Nov-2023neldifpr1 30220 The first element of a pair is not an element of a difference with this pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
¬ 𝐴 ∈ (𝐶 ∖ {𝐴, 𝐵})
 
20-Nov-2023inpr0 30219 Rewrite an empty intersection with a pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
((𝐴 ∩ {𝐵, 𝐶}) = ∅ ↔ (¬ 𝐵𝐴 ∧ ¬ 𝐶𝐴))
 
20-Nov-2023nelpr 30218 A set 𝐴 not in a pair is neither element of the pair. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝐴𝑉 → (¬ 𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴𝐵𝐴𝐶)))
 
20-Nov-2023nelbOLD 30159 Obsolete version of nelb 3265 as of 23-Jan-2024. (Contributed by Thierry Arnoux, 20-Nov-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴𝐵 ↔ ∀𝑥𝐵 𝑥𝐴)
 
19-Nov-2023bj-sbft 34001 Version of sbft 2260 using Ⅎ', proved from core axioms. (Contributed by BJ, 19-Nov-2023.)
(Ⅎ'𝑥𝜑 → ([𝑡 / 𝑥]𝜑𝜑))
 
19-Nov-2023bj-nnfor 33976 Nonfreeness in both disjuncts implies nonfreeness in the disjunction. (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof using the definition of disjunction in terms of implication and negation, so using bj-nnfim 33972, bj-nnfnt 33966 and bj-nnfbi 33954, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑𝜓))
 
19-Nov-2023bj-nnfand 33975 Nonfreeness in both conjuncts implies nonfreeness in the conjunction, deduction form. Note: compared with the proof of bj-nnfan 33974, it has two more essential steps but fewer total steps (since there are fewer intermediate formulas to build) and is easier to follow and understand. This statement is of intermediate complexity: for simpler statements, closed-style proofs like that of bj-nnfan 33974 will generally be shorter than deduction-style proofs while still easy to follow, while for more complex statements, the opposite will be true (and deduction-style proofs like that of bj-nnfand 33975 will generally be easier to understand). (Contributed by BJ, 19-Nov-2023.) (Proof modification is discouraged.)
(𝜑 → Ⅎ'𝑥𝜓)    &   (𝜑 → Ⅎ'𝑥𝜒)       (𝜑 → Ⅎ'𝑥(𝜓𝜒))
 
19-Nov-2023bj-nnfan 33974 Nonfreeness in both conjuncts implies nonfreeness in the conjunction. (Contributed by BJ, 19-Nov-2023.) In classical logic, there is a proof using the definition of conjunction in terms of implication and negation, so using bj-nnfim 33972, bj-nnfnt 33966 and bj-nnfbi 33954, but we want a proof valid in intuitionistic logic. (Proof modification is discouraged.)
((Ⅎ'𝑥𝜑 ∧ Ⅎ'𝑥𝜓) → Ⅎ'𝑥(𝜑𝜓))
 
19-Nov-2023prv1n 32575 No wff encoded as a Godel-set of membership is true in a model with only one element. (Contributed by AV, 19-Nov-2023.)
((𝐼 ∈ ω ∧ 𝐽 ∈ ω ∧ 𝑋𝑉) → ¬ {𝑋}⊧(𝐼𝑔𝐽))
 
19-Nov-2023prv0 32574 Every wff encoded as 𝑈 is true in an "empty model" (𝑀 = ∅). Since is defined in terms of the interpretations making the given formula true, it is not defined on the "empty model", since there are no interpretations. In particular, the empty set on the LHS of should not be interpreted as the empty model, because 𝑥𝑥 = 𝑥 is not satisfied on the empty model. (Contributed by AV, 19-Nov-2023.)
(𝑈 ∈ (Fmla‘ω) → ∅⊧𝑈)
 
19-Nov-2023ex-sategoelel12 32571 Example of a valuation of a simplified satisfaction predicate over a proper pair (of ordinal numbers) as model for a Godel-set of membership using the properties of a successor: (𝑆‘2o) = 1o ∈ 2o = (𝑆‘2o). Remark: the indices 1o and 2o are intentionally reversed to distinguish them from elements of the model: (2o𝑔1o) should not be confused with 2o ∈ 1o, which is false. (Contributed by AV, 19-Nov-2023.)
𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 1o, 2o))       𝑆 ∈ ({1o, 2o} Sat (2o𝑔1o))
 
19-Nov-2023ex-sategoelelomsuc 32570 Example of a valuation of a simplified satisfaction predicate over the ordinal numbers as model for a Godel-set of membership using the properties of a successor: (𝑆‘2o) = 𝑍 ∈ suc 𝑍 = (𝑆‘2o). Remark: the indices 1o and 2o are intentionally reversed to distinguish them from elements of the model: (2o𝑔1o) should not be confused with 2o ∈ 1o, which is false. (Contributed by AV, 19-Nov-2023.)
𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 2o, 𝑍, suc 𝑍))       (𝑍 ∈ ω → 𝑆 ∈ (ω Sat (2o𝑔1o)))
 
19-Nov-2023fmlan0 32535 The empty set is not a Godel formula. (Contributed by AV, 19-Nov-2023.)
∅ ∉ (Fmla‘ω)
 
19-Nov-2023pm2.61i 183 Inference eliminating an antecedent. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2023.)
(𝜑𝜓)    &   𝜑𝜓)       𝜓
 
18-Nov-2023alephiso3 39796 is a strictly order-preserving mapping of On onto the class of all infinite cardinal numbers. (Contributed by RP, 18-Nov-2023.)
ℵ Isom E , ≺ (On, (ran card ∖ ω))
 
18-Nov-2023alephiso2 39795 is a strictly order-preserving mapping of On onto the class of all infinite cardinal numbers. (Contributed by RP, 18-Nov-2023.)
ℵ Isom E , ≺ (On, {𝑥 ∈ ran card ∣ ω ⊆ 𝑥})
 
18-Nov-2023aleph1min 39794 (ℵ‘1o) is the least uncountable ordinal. (Contributed by RP, 18-Nov-2023.)
(ℵ‘1o) = {𝑥 ∈ On ∣ ω ≺ 𝑥}
 
18-Nov-2023bj-elsnb 34248 Biconditional version of elsng 4571. (Contributed by BJ, 18-Nov-2023.)
(𝐴 ∈ {𝐵} ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
 
18-Nov-2023bj-elsn12g 34247 Join of elsng 4571 and elsn2g 4593. (Contributed by BJ, 18-Nov-2023.)
((𝐴𝑉𝐵𝑊) → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
 
18-Nov-2023bj-nnfbii 33956 If two formulas are equivalent for all 𝑥, then nonfreeness of 𝑥 in one of them is equivalent to nonfreeness in the other, inference form. See bj-nnfbi 33954. (Contributed by BJ, 18-Nov-2023.)
(𝜑𝜓)       (Ⅎ'𝑥𝜑 ↔ Ⅎ'𝑥𝜓)
 
18-Nov-2023rnrhmsubrg 19496 The range of a ring homomorphism is a subring. (Contributed by SN, 18-Nov-2023.)
(𝐹 ∈ (𝑀 RingHom 𝑁) → ran 𝐹 ∈ (SubRing‘𝑁))
 
18-Nov-2023notzfaus 5253 In the Separation Scheme zfauscl 5196, we require that 𝑦 not occur in 𝜑 (which can be generalized to "not be free in"). Here we show special cases of 𝐴 and 𝜑 that result in a contradiction if that requirement is not met. (Contributed by NM, 8-Feb-2006.) (Proof shortened by BJ, 18-Nov-2023.)
𝐴 = {∅}    &   (𝜑 ↔ ¬ 𝑥𝑦)        ¬ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
 
17-Nov-2023satefvfmla1 32569 The simplified satisfaction predicate at two Godel-sets of membership combined with a Godel-set for NAND. (Contributed by AV, 17-Nov-2023.)
𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿))       ((𝑀𝑉 ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (𝑀 Sat 𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼) ∈ (𝑎𝐽) ∨ ¬ (𝑎𝐾) ∈ (𝑎𝐿))})
 
17-Nov-20232goelgoanfmla1 32568 Two Godel-sets of membership combined with a Godel-set for NAND is a Godel formula of height 1. (Contributed by AV, 17-Nov-2023.)
𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿))       (((𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → 𝑋 ∈ (Fmla‘1o))
 
17-Nov-2023satfv1fvfmla1 32567 The value of the satisfaction predicate at two Godel-sets of membership combined with a Godel-set for NAND. (Contributed by AV, 17-Nov-2023.)
𝑋 = ((𝐼𝑔𝐽)⊼𝑔(𝐾𝑔𝐿))       (((𝑀𝑉𝐸𝑊) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω) ∧ (𝐾 ∈ ω ∧ 𝐿 ∈ ω)) → (((𝑀 Sat 𝐸)‘1o)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝐼)𝐸(𝑎𝐽) ∨ ¬ (𝑎𝐾)𝐸(𝑎𝐿))})
 
17-Nov-2023symgcom2 30655 Two permutations 𝑋 and 𝑌 commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 17-Nov-2023.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → (dom (𝑋 ∖ I ) ∩ dom (𝑌 ∖ I )) = ∅)       (𝜑 → (𝑋𝑌) = (𝑌𝑋))
 
17-Nov-2023nfpconfp 30305 The set of fixed points of 𝐹 is the complement of the set of points moved by 𝐹. (Contributed by Thierry Arnoux, 17-Nov-2023.)
(𝐹 Fn 𝐴 → (𝐴 ∖ dom (𝐹 ∖ I )) = dom (𝐹 ∩ I ))
 
17-Nov-2023pm2.18 128 Clavius law, or "consequentia mirabilis" ("admirable consequence"). If a formula is implied by its negation, then it is true. Can be used in proofs by contradiction. Theorem *2.18 of [WhiteheadRussell] p. 103. See also the weak Clavius law pm2.01 190. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 17-Nov-2023.)
((¬ 𝜑𝜑) → 𝜑)
 
17-Nov-2023pm2.18d 127 Deduction form of the Clavius law pm2.18 128. (Contributed by FL, 12-Jul-2009.) (Proof shortened by Andrew Salmon, 7-May-2011.) Revised to shorten pm2.18 128. (Revised by Wolf Lammen, 17-Nov-2023.)
(𝜑 → (¬ 𝜓𝜓))       (𝜑𝜓)
 
16-Nov-2023ensucne0 39773 A class equinumerous to a successor is never empty. (Contributed by RP, 11-Nov-2023.) (Proof shortened by SN, 16-Nov-2023.)
(𝐴 ≈ suc 𝐵𝐴 ≠ ∅)
 
16-Nov-2023pmtrcnelor 30662 Composing a permutation 𝐹 with a transposition which results in moving one or two less points. (Contributed by Thierry Arnoux, 16-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑇 = (pmTrsp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝐽 = (𝐹𝐼)    &   (𝜑𝐷𝑉)    &   (𝜑𝐹𝐵)    &   (𝜑𝐼 ∈ dom (𝐹 ∖ I ))    &   𝐸 = dom (𝐹 ∖ I )    &   𝐴 = dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I )       (𝜑 → (𝐴 = (𝐸 ∖ {𝐼, 𝐽}) ∨ 𝐴 = (𝐸 ∖ {𝐼})))
 
16-Nov-2023pmtrcnel2 30661 Variation on pmtrcnel 30660. (Contributed by Thierry Arnoux, 16-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑇 = (pmTrsp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝐽 = (𝐹𝐼)    &   (𝜑𝐷𝑉)    &   (𝜑𝐹𝐵)    &   (𝜑𝐼 ∈ dom (𝐹 ∖ I ))       (𝜑 → (dom (𝐹 ∖ I ) ∖ {𝐼, 𝐽}) ⊆ dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ))
 
16-Nov-2023pmtrcnel 30660 Composing a permutation 𝐹 with a transposition which results in moving at least one less point. Here the set of points moved by a permutation 𝐹 is expressed as dom (𝐹 ∖ I ). (Contributed by Thierry Arnoux, 16-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑇 = (pmTrsp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝐽 = (𝐹𝐼)    &   (𝜑𝐷𝑉)    &   (𝜑𝐹𝐵)    &   (𝜑𝐼 ∈ dom (𝐹 ∖ I ))       (𝜑 → dom (((𝑇‘{𝐼, 𝐽}) ∘ 𝐹) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∖ {𝐼}))
 
16-Nov-2023eqdif 30208 If both set differences of two sets are empty, those sets are equal. (Contributed by Thierry Arnoux, 16-Nov-2023.)
(((𝐴𝐵) = ∅ ∧ (𝐵𝐴) = ∅) → 𝐴 = 𝐵)
 
16-Nov-2023rnasclassa 20052 The scalar multiples of the unit vector form a subalgebra of the vectors. (Contributed by SN, 16-Nov-2023.)
𝐴 = (algSc‘𝑊)    &   𝑈 = (𝑊s ran 𝐴)    &   (𝜑𝑊 ∈ AssAlg)       (𝜑𝑈 ∈ AssAlg)
 
14-Nov-2023nsyl2 143 A negated syllogism inference. (Contributed by NM, 26-Jun-1994.) (Proof shortened by Wolf Lammen, 14-Nov-2023.)
(𝜑 → ¬ 𝜓)    &   𝜒𝜓)       (𝜑𝜒)
 
13-Nov-2023pnpcan 10913 Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by SN, 13-Nov-2023.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − (𝐴 + 𝐶)) = (𝐵𝐶))
 
13-Nov-2023ralrexbid 3319 Formula-building rule for restricted existential quantifier, using a restricted universal quantifier to bind the quantified variable in the antecedent. (Contributed by AV, 21-Oct-2023.) Reduce axiom usage. (Revised by SN, 13-Nov-2023.)
(𝜑 → (𝜓𝜃))       (∀𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜃))
 
11-Nov-2023ensucne0OLD 39774 A class equinumerous to a successor is never empty. (Contributed by RP, 11-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ≈ suc 𝐵𝐴 ≠ ∅)
 
11-Nov-2023dfsucon 39767 𝐴 is called a successor ordinal if it is not a limit ordinal and not the empty set. (Contributed by RP, 11-Nov-2023.)
((Ord 𝐴 ∧ ¬ Lim 𝐴𝐴 ≠ ∅) ↔ ∃𝑥 ∈ On 𝐴 = suc 𝑥)
 
10-Nov-2023adh-minimp-pm2.43 43137 Derivation of pm2.43 56 WhiteheadRussell p. 106 (also called "hilbert" or "W") from adh-minimp-ax1 43131, adh-minimp-ax2 43135, and ax-mp 5. It uses the derivation written DD22D21 in D-notation. (See head comment for an explanation.) Polish prefix notation: CCpCpqCpq . (Contributed by BJ, 31-May-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜑𝜓)) → (𝜑𝜓))
 
10-Nov-2023adh-minimp-idALT 43136 Derivation of id 22 (reflexivity of implication, PM *2.08 WhiteheadRussell p. 101) from adh-minimp-ax1 43131, adh-minimp-ax2 43135, and ax-mp 5. It uses the derivation written DD211 in D-notation. (See head comment for an explanation.) Polish prefix notation: Cpp . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜑)
 
10-Nov-2023adh-minimp-ax2 43135 Derivation of ax-2 7 from adh-minimp 43126 and ax-mp 5. Polish prefix notation: CCpCqrCCpqCpr . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
 
10-Nov-2023adh-minimp-ax2-lem4 43134 Fourth lemma for the derivation of ax-2 7 from adh-minimp 43126 and ax-mp 5. Polish prefix notation: CpCCqCprCqr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ((𝜓 → (𝜑𝜒)) → (𝜓𝜒)))
 
10-Nov-2023adh-minimp-ax2c 43133 Derivation of a commuted form of ax-2 7 from adh-minimp 43126 and ax-mp 5. Polish prefix notation: CCpqCCpCqrCpr . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒)))
 
10-Nov-2023adh-minimp-imim1 43132 Derivation of imim1 83 ("left antimonotonicity of implication", theorem *2.06 of [WhiteheadRussell] p. 100) from adh-minimp 43126 and ax-mp 5. Polish prefix notation: CCpqCCqrCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
 
10-Nov-2023adh-minimp-ax1 43131 Derivation of ax-1 6 from adh-minimp 43126 and ax-mp 5. Polish prefix notation: CpCqp . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))
 
10-Nov-2023adh-minimp-sylsimp 43130 Derivation of jarr 106 (also called "syll-simp") from minimp 1613 and ax-mp 5. Polish prefix notation: CCCpqrCqr . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) → (𝜓𝜒))
 
10-Nov-2023adh-minimp-jarr-ax2c-lem3 43129 Third lemma for the derivation of jarr 106 and a commuted form of ax-2 7, and indirectly ax-1 6 and ax-2 7 proper , from adh-minimp 43126 and ax-mp 5. Polish prefix notation: CCCCpqCCCrpCqsCpstt . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((((𝜑𝜓) → (((𝜒𝜑) → (𝜓𝜃)) → (𝜑𝜃))) → 𝜏) → 𝜏)
 
10-Nov-2023adh-minimp-jarr-lem2 43128 Second lemma for the derivation of jarr 106, and indirectly ax-1 6, a commuted form of ax-2 7, and ax-2 7 proper, from adh-minimp 43126 and ax-mp 5. Polish prefix notation: CCCpqCCCrsCCCtrCsuCruvCqv . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → (((𝜒𝜃) → (((𝜏𝜒) → (𝜃𝜂)) → (𝜒𝜂))) → 𝜁)) → (𝜓𝜁))
 
10-Nov-2023adh-minimp-jarr-imim1-ax2c-lem1 43127 First lemma for the derivation of jarr 106, imim1 83, and a commuted form of ax-2 7, and indirectly ax-1 6 and ax-2 7, from adh-minimp 43126 and ax-mp 5. Polish prefix notation: CCpqCCCrpCqsCps . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → (((𝜒𝜑) → (𝜓𝜃)) → (𝜑𝜃)))
 
10-Nov-2023adh-minimp 43126 Another single axiom for minimal implicational calculus, due to Meredith. Other single axioms of the same length are known, but it is thought to be the minimal length. Among single axioms of this length, it is the one with simplest antecedents (i.e., in the corresponding ordering of binary trees which first compares left subtrees, it is the first one). Known as "HI-2" on Dolph Edward "Ted" Ulrich's web page. In the next 4 lemmas and 5 theorems, ax-1 6 and ax-2 7 are derived from this other single axiom in 20 detachments (instances of ax-mp 5) in total. Polish prefix notation: CpCCqrCCCsqCrtCqt ; or CtCCpqCCCspCqrCpr in Carew Arthur Meredith and Arthur Norman Prior, Notes on the axiomatics of the propositional calculus, Notre Dame Journal of Formal Logic, volume IV, number 3, July 1963, pages 171--187, on page 180. (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, 10-Nov-2023.)
(𝜑 → ((𝜓𝜒) → (((𝜃𝜓) → (𝜒𝜏)) → (𝜓𝜏))))
 
10-Nov-2023adh-minim-pm2.43 43125 Derivation of pm2.43 56 WhiteheadRussell p. 106 (also called "hilbert" or "W") from adh-minim-ax1 43119, adh-minim-ax2 43123, and ax-mp 5. It uses the derivation written DD22D21 in D-notation. (See head comment for an explanation.) (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜑𝜓)) → (𝜑𝜓))
 
10-Nov-2023adh-minim-idALT 43124 Derivation of id 22 (reflexivity of implication, PM *2.08 WhiteheadRussell p. 101) from adh-minim-ax1 43119, adh-minim-ax2 43123, and ax-mp 5. It uses the derivation written DD211 in D-notation. (See head comment for an explanation.) Polish prefix notation: Cpp . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜑)
 
10-Nov-2023adh-minim-ax2 43123 Derivation of ax-2 7 from adh-minim 43114 and ax-mp 5. Carew Arthur Meredith derived ax-2 7 in A single axiom of positive logic, The Journal of Computing Systems, volume 1, issue 3, July 1953, pages 169--170. However, here we follow the shortened derivation by Ivo Thomas, On Meredith's sole positive axiom, Notre Dame Journal of Formal Logic, volume XV, number 3, July 1974, page 477. Polish prefix notation: CCpCqrCCpqCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
 
10-Nov-2023adh-minim-ax2c 43122 Derivation of a commuted form of ax-2 7 from adh-minim 43114 and ax-mp 5. Polish prefix notation: CCpqCCpCqrCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒)))
 
10-Nov-2023adh-minim-ax2-lem6 43121 Sixth lemma for the derivation of ax-2 7 from adh-minim 43114 and ax-mp 5. Polish prefix notation: CCpCCCCqrsCCrCstCrtuCpu . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → ((((𝜓𝜒) → 𝜃) → ((𝜒 → (𝜃𝜏)) → (𝜒𝜏))) → 𝜂)) → (𝜑𝜂))
 
10-Nov-2023adh-minim-ax2-lem5 43120 Fifth lemma for the derivation of ax-2 7 from adh-minim 43114 and ax-mp 5. Polish prefix notation: CpCCCqrsCCrCstCrt . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (((𝜓𝜒) → 𝜃) → ((𝜒 → (𝜃𝜏)) → (𝜒𝜏))))
 
10-Nov-2023adh-minim-ax1 43119 Derivation of ax-1 6 from adh-minim 43114 and ax-mp 5. Carew Arthur Meredith derived ax-1 6 in A single axiom of positive logic, The Journal of Computing Systems, volume 1, issue 3, July 1953, pages 169--170. However, here we follow the shortened derivation by Ivo Thomas, On Meredith's sole positive axiom, Notre Dame Journal of Formal Logic, volume XV, number 3, July 1974, page 477. Polish prefix notation: CpCqp . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜑))
 
10-Nov-2023adh-minim-ax1-ax2-lem4 43118 Fourth lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 43114 and ax-mp 5. Polish prefix notation: CCCpqrCCqCrsCqs . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) → 𝜒) → ((𝜓 → (𝜒𝜃)) → (𝜓𝜃)))
 
10-Nov-2023adh-minim-ax1-ax2-lem3 43117 Third lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 43114 and ax-mp 5. Polish prefix notation: CCpCqrCqCsCpr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜃 → (𝜑𝜒))))
 
10-Nov-2023adh-minim-ax1-ax2-lem2 43116 Second lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 43114 and ax-mp 5. Polish prefix notation: CCpCCqCCrCpsCrstCpt . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 → ((𝜓 → ((𝜒 → (𝜑𝜃)) → (𝜒𝜃))) → 𝜏)) → (𝜑𝜏))
 
10-Nov-2023adh-minim-ax1-ax2-lem1 43115 First lemma for the derivation of ax-1 6 and ax-2 7 from adh-minim 43114 and ax-mp 5. Polish prefix notation: CpCCqCCrCCsCqtCstuCqu . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ((𝜓 → ((𝜒 → ((𝜃 → (𝜓𝜏)) → (𝜃𝜏))) → 𝜂)) → (𝜓𝜂)))
 
10-Nov-2023adh-minim 43114 A single axiom for minimal implicational calculus, due to Meredith. Other single axioms of the same length are known, but it is thought to be the minimal length. This is the axiom from Carew Arthur Meredith, A single axiom of positive logic, The Journal of Computing Systems, volume 1, issue 3, July 1953, pages 169--170. A two-line review by Alonzo Church of this article can be found in The Journal of Symbolic Logic, volume 19, issue 2, June 1954, page 144, https://doi.org/10.2307/2268914. Known as "HI-1" on Dolph Edward "Ted" Ulrich's web page. In the next 6 lemmas and 3 theorems, ax-1 6 and ax-2 7 are derived from this single axiom in 16 detachments (instances of ax-mp 5) in total. Polish prefix notation: CCCpqrCsCCqCrtCqt . (Contributed by ADH, 10-Nov-2023.)
(((𝜑𝜓) → 𝜒) → (𝜃 → ((𝜓 → (𝜒𝜏)) → (𝜓𝜏))))
 
10-Nov-2023adh-jarrsc 43113 Replacement of a nested antecedent with an outer antecedent. Commuted simplificated form of elimination of a nested antecedent. Also holds intuitionistically. Polish prefix notation: CCCpqrCsCqr . (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.)
(((𝜑𝜓) → 𝜒) → (𝜃 → (𝜓𝜒)))
 
9-Nov-2023satfv1 32507 The value of the satisfaction predicate as function over wff codes of height 1. (Contributed by AV, 9-Nov-2023.)
𝑆 = (𝑀 Sat 𝐸)       ((𝑀𝑉𝐸𝑊) → (𝑆‘1o) = ((𝑆‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (¬ (𝑎𝑖)𝐸(𝑎𝑗) ∨ ¬ (𝑎𝑘)𝐸(𝑎𝑙))}) ∨ ∃𝑛 ∈ ω (𝑥 = ∀𝑔𝑛(𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝑖 = 𝑛, if-(𝑗 = 𝑛, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝑗)), if-(𝑗 = 𝑛, (𝑎𝑖)𝐸𝑧, (𝑎𝑖)𝐸(𝑎𝑗)))}))}))
 
9-Nov-2023satfv1lem 32506 Lemma for satfv1 32507. (Contributed by AV, 9-Nov-2023.)
((𝑁 ∈ ω ∧ 𝐼 ∈ ω ∧ 𝐽 ∈ ω) → {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑁, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑁}))) ∈ {𝑏 ∈ (𝑀m ω) ∣ (𝑏𝐼)𝐸(𝑏𝐽)}} = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 if-(𝐼 = 𝑁, if-(𝐽 = 𝑁, 𝑧𝐸𝑧, 𝑧𝐸(𝑎𝐽)), if-(𝐽 = 𝑁, (𝑎𝐼)𝐸𝑧, (𝑎𝐼)𝐸(𝑎𝐽)))})
 
9-Nov-20232ex2rexrot 3247 Rotate two existential quantifiers and two restricted existential quantifiers. (Contributed by AV, 9-Nov-2023.)
(∃𝑥𝑦𝑧𝐴𝑤𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵𝑥𝑦𝜑)
 
8-Nov-2023iscard5 39779 Two ways to express the property of being a cardinal number. (Contributed by RP, 8-Nov-2023.)
((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥𝐴 ¬ 𝑥𝐴))
 
8-Nov-2023iscard4 39778 Two ways to express the property of being a cardinal number. (Contributed by RP, 8-Nov-2023.)
((card‘𝐴) = 𝐴𝐴 ∈ ran card)
 
8-Nov-2023rexopabb 5406 Restricted existential quantification over an ordered-pair class abstraction. (Contributed by AV, 8-Nov-2023.)
𝑂 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    &   (𝑜 = ⟨𝑥, 𝑦⟩ → (𝜓𝜒))       (∃𝑜𝑂 𝜓 ↔ ∃𝑥𝑦(𝜑𝜒))
 
5-Nov-2023harsucnn 39781 The next cardinal after a finite ordinal is the successor ordinal. (Contributed by RP, 5-Nov-2023.)
(𝐴 ∈ ω → (har‘𝐴) = suc 𝐴)
 
5-Nov-2023nndomog 39775 Cardinal ordering agrees with ordinal number ordering when the smaller number is a natural number. Compare with nndomo 8700 when both are natural numbers. (Originally by NM, 17-Jun-1998.) (Contributed by RP, 5-Nov-2023.)
((𝐴 ∈ ω ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴𝐵))
 
5-Nov-2023selvval2lem4 39014 The fourth argument passed to evalSub is in the domain (a polynomial in (𝐼 mPoly (𝐽 mPoly ((𝐼𝐽) mPoly 𝑅)))). (Contributed by SN, 5-Nov-2023.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑈 = ((𝐼𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   𝑆 = (𝑇s ran 𝐷)    &   𝑊 = (𝐼 mPoly 𝑆)    &   𝑋 = (Base‘𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐽𝐼)    &   (𝜑𝐹𝐵)       (𝜑 → (𝐷𝐹) ∈ 𝑋)
 
5-Nov-2023selvval2lemn 39013 A lemma to illustrate the purpose of selvval2lem3 39012 and the value of 𝑄. Will be renamed in the future when this section is moved to main. (Contributed by SN, 5-Nov-2023.)
𝑈 = ((𝐼𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   𝑄 = ((𝐼 evalSub 𝑇)‘ran 𝐷)    &   𝑊 = (𝐼 mPoly 𝑆)    &   𝑆 = (𝑇s ran 𝐷)    &   𝑋 = (𝑇s (𝐵m 𝐼))    &   𝐵 = (Base‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐽𝐼)       (𝜑𝑄 ∈ (𝑊 RingHom 𝑋))
 
5-Nov-2023elnanelprv 32573 The wff (𝐴𝐵𝐵𝐴) encoded as ((𝐴𝑔𝐵) 𝑔(𝐵𝑔𝐴)) is true in any model 𝑀. This is the model theoretic proof of elnanel 9058. (Contributed by AV, 5-Nov-2023.)
((𝑀𝑉𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝑀⊧((𝐴𝑔𝐵)⊼𝑔(𝐵𝑔𝐴)))
 
5-Nov-2023prv 32572 The "proves" relation on a set. A wff encoded as 𝑈 is true in a model 𝑀 iff for every valuation 𝑠 ∈ (𝑀m ω), the interpretation of the wff using the membership relation on 𝑀 is true. (Contributed by AV, 5-Nov-2023.)
((𝑀𝑉𝑈𝑊) → (𝑀𝑈 ↔ (𝑀 Sat 𝑈) = (𝑀m ω)))
 
5-Nov-2023ex-sategoel 32566 Instance of sategoelfv 32564 for the example of a valuation of a simplified satisfaction predicate for a Godel-set of membership. (Contributed by AV, 5-Nov-2023.)
𝐸 = (𝑀 Sat (𝐴𝑔𝐵))    &   𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)))       (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → (𝑆𝐴) ∈ (𝑆𝐵))
 
5-Nov-2023ex-sategoelel 32565 Example of a valuation of a simplified satisfaction predicate for a Godel-set of membership. (Contributed by AV, 5-Nov-2023.)
𝐸 = (𝑀 Sat (𝐴𝑔𝐵))    &   𝑆 = (𝑥 ∈ ω ↦ if(𝑥 = 𝐴, 𝑍, if(𝑥 = 𝐵, 𝒫 𝑍, ∅)))       (((𝑀 ∈ WUni ∧ 𝑍𝑀) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐴𝐵)) → 𝑆𝐸)
 
5-Nov-2023sategoelfv 32564 Condition of a valuation 𝑆 of a simplified satisfaction predicate for a Godel-set of membership: The sets in model 𝑀 corresponding to the variables 𝐴 and 𝐵 under the assignment of 𝑆 are in a membership relation in 𝑀. (Contributed by AV, 5-Nov-2023.)
𝐸 = (𝑀 Sat (𝐴𝑔𝐵))       ((𝑀𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑆𝐸) → (𝑆𝐴) ∈ (𝑆𝐵))
 
5-Nov-2023sategoelfvb 32563 Characterization of a valuation 𝑆 of a simplified satisfaction predicate for a Godel-set of membership. (Contributed by AV, 5-Nov-2023.)
𝐸 = (𝑀 Sat (𝐴𝑔𝐵))       ((𝑀𝑉 ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝑆𝐸 ↔ (𝑆 ∈ (𝑀m ω) ∧ (𝑆𝐴) ∈ (𝑆𝐵))))
 
5-Nov-2023sate0 32559 The simplified satisfaction predicate for any wff code over an empty model. (Contributed by AV, 6-Oct-2023.) (Revised by AV, 5-Nov-2023.)
(𝑈𝑉 → (∅ Sat 𝑈) = (((∅ Sat ∅)‘ω)‘𝑈))
 
5-Nov-2023rnasclmulcl 20051 (Vector) multiplication is closed for scalar multiples of the unit vector. (Contributed by SN, 5-Nov-2023.)
𝐶 = (algSc‘𝑊)    &    × = (.r𝑊)    &   (𝜑𝑊 ∈ AssAlg)       ((𝜑 ∧ (𝑋 ∈ ran 𝐶𝑌 ∈ ran 𝐶)) → (𝑋 × 𝑌) ∈ ran 𝐶)
 
5-Nov-2023rnasclsubrg 20050 The scalar multiples of the unit vector form a subring of the vectors. (Contributed by SN, 5-Nov-2023.)
𝐶 = (algSc‘𝑊)    &   (𝜑𝑊 ∈ AssAlg)       (𝜑 → ran 𝐶 ∈ (SubRing‘𝑊))
 
5-Nov-2023ascldimul 20044 The algebra scalars function distributes over multiplication. (Contributed by Mario Carneiro, 8-Mar-2015.) (Proof shortened by SN, 5-Nov-2023.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    × = (.r𝑊)    &    · = (.r𝐹)       ((𝑊 ∈ AssAlg ∧ 𝑅𝐾𝑆𝐾) → (𝐴‘(𝑅 · 𝑆)) = ((𝐴𝑅) × (𝐴𝑆)))
 
5-Nov-2023elnanel 9058 Two classes are not elements of each other simultaneously. This is just a rewriting of en2lp 9057 and serves as an example in the context of Godel codes, see elnanelprv 32573. (Contributed by AV, 5-Nov-2023.) (New usage is discouraged.)
(𝐴𝐵𝐵𝐴)
 
5-Nov-2023f1iun 7634 The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by Mario Carneiro, 20-May-2013.) (Revised by Mario Carneiro, 24-Jun-2015.) (Proof shortened by AV, 5-Nov-2023.)
(𝑥 = 𝑦𝐵 = 𝐶)    &   𝐵 ∈ V       (∀𝑥𝐴 (𝐵:𝐷1-1𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵: 𝑥𝐴 𝐷1-1𝑆)
 
4-Nov-2023harval3on 39783 For any ordinal number 𝐴 let (har‘𝐴) denote the least cardinal that is greater than 𝐴; (Contributed by RP, 4-Nov-2023.)
(𝐴 ∈ On → (har‘𝐴) = {𝑥 ∈ ran card ∣ 𝐴𝑥})
 
4-Nov-2023harval3 39782 (har‘𝐴) is the least cardinal that is greater than 𝐴. (Contributed by RP, 4-Nov-2023.)
(𝐴 ∈ dom card → (har‘𝐴) = {𝑥 ∈ ran card ∣ 𝐴𝑥})
 
4-Nov-2023satefvfmla0 32562 The simplified satisfaction predicate for wff codes of height 0. (Contributed by AV, 4-Nov-2023.)
((𝑀𝑉𝑋 ∈ (Fmla‘∅)) → (𝑀 Sat 𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋))) ∈ (𝑎‘(2nd ‘(2nd𝑋)))})
 
4-Nov-2023selvval 20259 Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑈 = ((𝐼𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   (𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)    &   (𝜑𝐽𝐼)    &   (𝜑𝐹𝐵)       (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) = ((((𝐼 evalSub 𝑇)‘ran 𝐷)‘(𝐷𝐹))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))))
 
4-Nov-2023selvfval 20258 Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.)
(𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)    &   (𝜑𝐽𝐼)       (𝜑 → ((𝐼 selectVars 𝑅)‘𝐽) = (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ((𝐼𝐽) mPoly 𝑅) / 𝑢(𝐽 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝑓))‘(𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝐽) mVar 𝑅)‘𝑥)))))))
 
4-Nov-2023selvffval 20257 Value of the "variable selection" function. (Contributed by SN, 4-Nov-2023.)
(𝜑𝐼𝑉)    &   (𝜑𝑅𝑊)       (𝜑 → (𝐼 selectVars 𝑅) = (𝑗 ∈ 𝒫 𝐼 ↦ (𝑓 ∈ (Base‘(𝐼 mPoly 𝑅)) ↦ ((𝐼𝑗) mPoly 𝑅) / 𝑢(𝑗 mPoly 𝑢) / 𝑡(algSc‘𝑡) / 𝑐(𝑐 ∘ (algSc‘𝑢)) / 𝑑((((𝐼 evalSub 𝑡)‘ran 𝑑)‘(𝑑𝑓))‘(𝑥𝐼 ↦ if(𝑥𝑗, ((𝑗 mVar 𝑢)‘𝑥), (𝑐‘(((𝐼𝑗) mVar 𝑅)‘𝑥))))))))
 
4-Nov-2023fviunfun 7635 The function value of an indexed union is the value of one of the indexed functions. (Contributed by AV, 4-Nov-2023.)
𝑈 = 𝑖𝐼 (𝐹𝑖)       ((Fun 𝑈𝐽𝐼𝑋 ∈ dom (𝐹𝐽)) → (𝑈𝑋) = ((𝐹𝐽)‘𝑋))
 
3-Nov-2023spALT 40432 sp 2172 can be proven from the other classic axioms. (Contributed by Rohan Ridenour, 3-Nov-2023.) (Proof modification is discouraged.) Use sp 2172 instead. (New usage is discouraged.)
(∀𝑥𝜑𝜑)
 
3-Nov-2023csbeq12dv 3889 Formula-building inference for class substitution. (Contributed by SN, 3-Nov-2023.)
(𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑𝐴 / 𝑥𝐵 = 𝐶 / 𝑥𝐷)
 
2-Nov-2023satfv0fvfmla0 32557 The value of the satisfaction predicate as function over a wff code at . (Contributed by AV, 2-Nov-2023.)
𝑆 = (𝑀 Sat 𝐸)       ((𝑀𝑉𝐸𝑊𝑋 ∈ (Fmla‘∅)) → ((𝑆‘∅)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))})
 
1-Nov-2023bj-mpgs 33840 From a closed form theorem (the major premise) with an antecedent in the "strong necessity" modality (in the language of modal logic), deduce the inference 𝜑𝜓. Strong necessity is stronger than necessity, and equivalent to it when sp 2172 (modal T) is available. Therefore, this theorem is stronger than mpg 1789 when sp 2172 is not available. (Contributed by BJ, 1-Nov-2023.)
𝜑    &   ((𝜑 ∧ ∀𝑥𝜑) → 𝜓)       𝜓
 
1-Nov-2023evpmsubg 30716 The alternating group is a subgroup of the symmetric group. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝐴 = (pmEven‘𝐷)       (𝐷 ∈ Fin → 𝐴 ∈ (SubGrp‘𝑆))
 
1-Nov-2023cnmsgn0g 30715 The neutral element of the sign subgroup of the complex numbers. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1})       1 = (0g𝑈)
 
1-Nov-2023evpmval 30714 Value of the set of even permutations, the alternating group. (Contributed by Thierry Arnoux, 1-Nov-2023.)
𝐴 = (pmEven‘𝐷)       (𝐷𝑉𝐴 = ((pmSgn‘𝐷) “ {1}))
 
31-Oct-2023sate0fv0 32561 A simplified satisfaction predicate as function over wff codes over an empty model is an empty set. (Contributed by AV, 31-Oct-2023.)
(𝑈 ∈ (Fmla‘ω) → (𝑆 ∈ (∅ Sat 𝑈) → 𝑆 = ∅))
 
31-Oct-2023ablsimpgfindlem1 19158 Lemma for ablsimpgfind 19161. An element of an abelian finite simple group which doesn't square to the identity has finite order. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Proof shortened by Rohan Ridenour, 31-Oct-2023.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    · = (.g𝐺)    &   𝑂 = (od‘𝐺)    &   (𝜑𝐺 ∈ Abel)    &   (𝜑𝐺 ∈ SimpGrp)       (((𝜑𝑥𝐵) ∧ (2 · 𝑥) ≠ 0 ) → (𝑂𝑥) ≠ 0)
 
31-Oct-2023ablsimpgcygd 19157 An abelian simple group is cyclic. (Contributed by Rohan Ridenour, 3-Aug-2023.) (Proof shortened by Rohan Ridenour, 31-Oct-2023.)
(𝜑𝐺 ∈ Abel)    &   (𝜑𝐺 ∈ SimpGrp)       (𝜑𝐺 ∈ CycGrp)
 
30-Oct-2023satef 32560 The simplified satisfaction predicate as function over wff codes over an empty model. (Contributed by AV, 30-Oct-2023.)
((𝑀𝑉𝑈 ∈ (Fmla‘ω) ∧ 𝑆 ∈ (𝑀 Sat 𝑈)) → 𝑆:ω⟶𝑀)
 
30-Oct-2023satefv 32558 The simplified satisfaction predicate as function over wff codes in the model 𝑀 at the code 𝑈. (Contributed by AV, 30-Oct-2023.)
((𝑀𝑉𝑈𝑊) → (𝑀 Sat 𝑈) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈))
 
29-Oct-2023tr3dom 39772 An unordered triple is dominated by ordinal three. (Contributed by RP, 29-Oct-2023.)
{𝐴, 𝐵, 𝐶} ≼ 3o
 
29-Oct-2023pr2dom 39771 An unordered pair is dominated by ordinal two. (Contributed by RP, 29-Oct-2023.)
{𝐴, 𝐵} ≼ 2o
 
29-Oct-2023sn1dom 39770 A singleton is dominated by ordinal one. (Contributed by RP, 29-Oct-2023.)
{𝐴} ≼ 1o
 
29-Oct-2023satfvel 32556 An element of the value of the satisfaction predicate as function over wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 at the code 𝑈 for a wff using ∈ , ⊼ , ∀ is a valuation 𝑆:ω⟶𝑀 of the variables (v0 = (𝑆‘∅), v1 = (𝑆‘1o), etc.) so that 𝑈 is true under the assignment 𝑆. (Contributed by AV, 29-Oct-2023.)
(((𝑀𝑉𝐸𝑊) ∧ 𝑈 ∈ (Fmla‘ω) ∧ 𝑆 ∈ (((𝑀 Sat 𝐸)‘ω)‘𝑈)) → 𝑆:ω⟶𝑀)
 
29-Oct-2023satfun 32555 The satisfaction predicate as function over wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 29-Oct-2023.)
((𝑀𝑉𝐸𝑊) → ((𝑀 Sat 𝐸)‘ω):(Fmla‘ω)⟶𝒫 (𝑀m ω))
 
29-Oct-2023norassOLD 1525 Obsolete version of norass 1524 as of 17-Dec-2023. (Contributed by RP, 29-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜒) ↔ (((𝜑 𝜓) 𝜒) ↔ (𝜑 (𝜓 𝜒))))
 
28-Oct-2023satff 32554 The satisfaction predicate as function over wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 28-Oct-2023.)
((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → ((𝑀 Sat 𝐸)‘𝑁):(Fmla‘𝑁)⟶𝒫 (𝑀m ω))
 
28-Oct-2023satffun 32553 The value of the satisfaction predicate as function over wff codes at a natural number is a function. (Contributed by AV, 28-Oct-2023.)
((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁))
 
28-Oct-2023satffunlem2 32552 Lemma 2 for satffun 32553: induction step. (Contributed by AV, 28-Oct-2023.)
((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑁) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑁)))
 
28-Oct-2023satffunlem1 32551 Lemma 1 for satffun 32553: induction basis. (Contributed by AV, 28-Oct-2023.)
((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc ∅))
 
28-Oct-2023satffunlem2lem1 32548 Lemma 1 for satffunlem2 32552. (Contributed by AV, 28-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)    &   𝐴 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))    &   𝐵 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}       ((Fun (𝑆‘suc 𝑁) ∧ (𝑆𝑁) ⊆ (𝑆‘suc 𝑁)) → Fun {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴))})
 
28-Oct-2023pm2.521g 175 A general instance of Theorem *2.521 of [WhiteheadRussell] p. 107. (Contributed by BJ, 28-Oct-2023.)
(¬ (𝜑𝜓) → (𝜓𝜒))
 
28-Oct-2023conax1k 172 Weakening of conax1 171. General instance of pm2.51 173 and of pm2.52 174. (Contributed by BJ, 28-Oct-2023.)
(¬ (𝜑𝜓) → (𝜒 → ¬ 𝜓))
 
28-Oct-2023conax1 171 Contrapositive of ax-1 6. (Contributed by BJ, 28-Oct-2023.)
(¬ (𝜑𝜓) → ¬ 𝜓)
 
27-Oct-2023satffunlem2lem2 32550 Lemma 2 for satffunlem2 32552. (Contributed by AV, 27-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)    &   𝐴 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))    &   𝐵 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}       (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (dom (𝑆‘suc 𝑁) ∩ dom {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴))}) = ∅)
 
27-Oct-2023satffunlem 32545 Lemma for satffunlem1lem1 32546 and satffunlem2lem1 32548. (Contributed by AV, 27-Oct-2023.)
(((Fun 𝑍 ∧ (𝑠𝑍𝑟𝑍) ∧ (𝑢𝑍𝑣𝑍)) ∧ (𝑥 = ((1st𝑠)⊼𝑔(1st𝑟)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑠) ∩ (2nd𝑟)))) ∧ (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑤 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))) → 𝑦 = 𝑤)
 
27-Oct-2023funeldmdif 7736 Two ways of expressing membership in the difference of domains of two nested functions. (Contributed by AV, 27-Oct-2023.)
((Fun 𝐴𝐵𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) ↔ ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶))
 
27-Oct-2023funelss 7735 If the first component of an element of a function is in the domain of a subset of the function, the element is a member of this subset. (Contributed by AV, 27-Oct-2023.)
((Fun 𝐴𝐵𝐴𝑋𝐴) → ((1st𝑋) ∈ dom 𝐵𝑋𝐵))
 
26-Oct-2023releldmdifi 7733 One way of expressing membership in the difference of domains of two nested relations. (Contributed by AV, 26-Oct-2023.)
((Rel 𝐴𝐵𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴𝐵)(1st𝑥) = 𝐶))
 
26-Oct-2023nororOLD 1520 Obsolete version of noror 1519 as of 8-Dec-2023. (Contributed by Remi, 26-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) ↔ ((𝜑 𝜓) (𝜑 𝜓)))
 
26-Oct-2023noranOLD 1518 Obsolete version of noran 1517 as of 8-Dec-2023. (Contributed by Remi, 26-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓) ↔ ((𝜑 𝜑) (𝜓 𝜓)))
 
25-Oct-2023dmopab3rexdif 32549 The domain of an ordered pair class abstraction with three nested restricted existential quantifiers with differences. (Contributed by AV, 25-Oct-2023.)
((∀𝑢𝑈 (∀𝑣𝑈 𝐵𝑋 ∧ ∀𝑖𝐼 𝐷𝑊) ∧ 𝑆𝑈) → dom {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (𝑈𝑆)(∃𝑣𝑈 (𝑥 = 𝐴𝑦 = 𝐵) ∨ ∃𝑖𝐼 (𝑥 = 𝐶𝑦 = 𝐷)) ∨ ∃𝑢𝑆𝑣 ∈ (𝑈𝑆)(𝑥 = 𝐴𝑦 = 𝐵))} = {𝑥 ∣ (∃𝑢 ∈ (𝑈𝑆)(∃𝑣𝑈 𝑥 = 𝐴 ∨ ∃𝑖𝐼 𝑥 = 𝐶) ∨ ∃𝑢𝑆𝑣 ∈ (𝑈𝑆)𝑥 = 𝐴)})
 
25-Oct-2023rexdifi 4119 Restricted existential quantification over a difference. (Contributed by AV, 25-Oct-2023.)
((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 ¬ 𝜑) → ∃𝑥 ∈ (𝐴𝐵)𝜑)
 
25-Oct-2023falnorfalOLD 1583 Obsolete version of falnorfal 1582 as of 17-Dec-2023. (Contributed by Remi, 25-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊥ ⊥) ↔ ⊤)
 
25-Oct-2023falnortru 1581 A identity. (Contributed by Remi, 25-Oct-2023.)
((⊥ ⊤) ↔ ⊥)
 
25-Oct-2023trunorfalOLD 1580 Obsolete version of trunorfal 1579 as of 17-Dec-2023. (Contributed by Remi, 25-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ⊥) ↔ ⊥)
 
25-Oct-2023trunortruOLD 1578 Obsolete version of trunortru 1577 as of 7-Dec-2023. (Contributed by Remi, 25-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ⊤) ↔ ⊥)
 
25-Oct-2023nornotOLD 1516 Obsolete version of nornot 1515 as of 8-Dec-2023. (Contributed by Remi, 25-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑 ↔ (𝜑 𝜑))
 
25-Oct-2023norcom 1514 The connector is commutative. (Contributed by Remi, 25-Oct-2023.)
((𝜑 𝜓) ↔ (𝜓 𝜑))
 
25-Oct-2023df-nor 1513 Define joint denial ("not-or" or "nor"). After we define the constant true (df-tru 1531) and the constant false (df-fal 1541), we will be able to prove these truth table values: ((⊤ ⊤) ↔ ⊥) (trunortru 1577), ((⊤ ⊥) ↔ ⊥) (trunorfal 1579), ((⊥ ⊤) ↔ ⊥) (falnortru 1581), and ((⊥ ⊥) ↔ ⊤) (falnorfal 1582). Contrast with (df-an 397), (df-or 842), (wi 4), (df-nan 1476), and (df-xor 1496). (Contributed by Remi, 25-Oct-2023.)
((𝜑 𝜓) ↔ ¬ (𝜑𝜓))
 
24-Oct-2023dff15 32250 A one-to-one function in terms of different arguments never having the same function value. (Contributed by BTernaryTau, 24-Oct-2023.)
(𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ¬ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) ∧ 𝑥𝑦)))
 
23-Oct-2023satffunlem1lem2 32547 Lemma 2 for satffunlem1 32551. (Contributed by AV, 23-Oct-2023.)
((𝑀𝑉𝐸𝑊) → (dom ((𝑀 Sat 𝐸)‘∅) ∩ dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}) = ∅)
 
23-Oct-2023acycgrsubgr 32302 The subgraph of an acyclic graph is also acyclic. (Contributed by BTernaryTau, 23-Oct-2023.)
((𝐺 ∈ AcyclicGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ AcyclicGraph)
 
23-Oct-2023subgrcycl 32279 If a cycle exists in a subgraph of a graph 𝐺, then that cycle also exists in 𝐺. (Contributed by BTernaryTau, 23-Oct-2023.)
(𝑆 SubGraph 𝐺 → (𝐹(Cycles‘𝑆)𝑃𝐹(Cycles‘𝐺)𝑃))
 
23-Oct-2023dmopab2rex 5779 The domain of an ordered pair class abstraction with two nested restricted existential quantifiers. (Contributed by AV, 23-Oct-2023.)
(∀𝑢𝑈 (∀𝑣𝑉 𝐵𝑋 ∧ ∀𝑖𝐼 𝐷𝑊) → dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝑈 (∃𝑣𝑉 (𝑥 = 𝐴𝑦 = 𝐵) ∨ ∃𝑖𝐼 (𝑥 = 𝐶𝑦 = 𝐷))} = {𝑥 ∣ ∃𝑢𝑈 (∃𝑣𝑉 𝑥 = 𝐴 ∨ ∃𝑖𝐼 𝑥 = 𝐶)})
 
23-Oct-2023equsexvw 2002 Version of equsexv 2259 with a disjoint variable condition, and of equsex 2431 with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsalvw 2001. (Contributed by BJ, 31-May-2019.) (Proof shortened by Wolf Lammen, 23-Oct-2023.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 
22-Oct-2023goalr 32541 If the "Godel-set of universal quantification" applied to a class is a Godel formula, the class is also a Godel formula. Remark: The reverse is not valid for 𝐴 being of the same height as the "Godel-set of universal quantification". (Contributed by AV, 22-Oct-2023.)
((𝑁 ∈ ω ∧ ∀𝑔𝑖𝑎 ∈ (Fmla‘𝑁)) → 𝑎 ∈ (Fmla‘𝑁))
 
22-Oct-2023goalrlem 32540 Lemma for goalr 32541 (induction step). (Contributed by AV, 22-Oct-2023.)
(𝑁 ∈ ω → ((∀𝑔𝑖𝑎 ∈ (Fmla‘suc 𝑁) → 𝑎 ∈ (Fmla‘suc 𝑁)) → (∀𝑔𝑖𝑎 ∈ (Fmla‘suc suc 𝑁) → 𝑎 ∈ (Fmla‘suc suc 𝑁))))
 
22-Oct-2023goaln0 32537 The "Godel-set of universal quantification" is a Godel formula of at least height 1. (Contributed by AV, 22-Oct-2023.)
(∀𝑔𝑖𝐴 ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
 
22-Oct-2023subgrpth 32278 If a path exists in a subgraph of a graph 𝐺, then that path also exists in 𝐺. (Contributed by BTernaryTau, 22-Oct-2023.)
(𝑆 SubGraph 𝐺 → (𝐹(Paths‘𝑆)𝑃𝐹(Paths‘𝐺)𝑃))
 
22-Oct-2023subgrtrl 32277 If a trail exists in a subgraph of a graph 𝐺, then that trail also exists in 𝐺. (Contributed by BTernaryTau, 22-Oct-2023.)
(𝑆 SubGraph 𝐺 → (𝐹(Trails‘𝑆)𝑃𝐹(Trails‘𝐺)𝑃))
 
22-Oct-2023subgrwlk 32276 If a walk exists in a subgraph of a graph 𝐺, then that walk also exists in 𝐺. (Contributed by BTernaryTau, 22-Oct-2023.)
(𝑆 SubGraph 𝐺 → (𝐹(Walks‘𝑆)𝑃𝐹(Walks‘𝐺)𝑃))
 
22-Oct-2023speiv 1967 Inference from existential specialization. (Contributed by Wolf Lammen, 22-Oct-2023.)
(𝑥 = 𝑦 → (𝜓𝜑))    &   𝜓       𝑥𝜑
 
22-Oct-2023spimew 1965 Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 22-Oct-2023.)
(𝜑 → ∀𝑥𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)
 
21-Oct-2023pren2d 39793 A pair of two distinct sets is equinumerous to ordinal two. (Contributed by RP, 21-Oct-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐴𝐵)       (𝜑 → {𝐴, 𝐵} ≈ 2o)
 
21-Oct-2023pr2eldif2 39792 If an unordered pair is equinumerous to ordinal two, then a part is an element of the difference of the pair and the singleton of the other part. (Contributed by RP, 21-Oct-2023.)
({𝐴, 𝐵} ≈ 2o𝐵 ∈ ({𝐴, 𝐵} ∖ {𝐴}))
 
21-Oct-2023pr2eldif1 39791 If an unordered pair is equinumerous to ordinal two, then a part is an element of the difference of the pair and the singleton of the other part. (Contributed by RP, 21-Oct-2023.)
({𝐴, 𝐵} ≈ 2o𝐴 ∈ ({𝐴, 𝐵} ∖ {𝐵}))
 
21-Oct-2023pr2cv2 39789 If an unordered pair is equinumerous to ordinal two, then a part is a set. (Contributed by RP, 21-Oct-2023.)
({𝐴, 𝐵} ≈ 2o𝐵 ∈ V)
 
21-Oct-2023pr2el2 39788 If an unordered pair is equinumerous to ordinal two, then a part is a member. (Contributed by RP, 21-Oct-2023.)
({𝐴, 𝐵} ≈ 2o𝐵 ∈ {𝐴, 𝐵})
 
21-Oct-2023pr2cv1 39787 If an unordered pair is equinumerous to ordinal two, then a part is a set. (Contributed by RP, 21-Oct-2023.)
({𝐴, 𝐵} ≈ 2o𝐴 ∈ V)
 
21-Oct-2023pr2el1 39786 If an unordered pair is equinumerous to ordinal two, then a part is a member. (Contributed by RP, 21-Oct-2023.)
({𝐴, 𝐵} ≈ 2o𝐴 ∈ {𝐴, 𝐵})
 
21-Oct-2023gonar 32539 If the "Godel-set of NAND" applied to classes is a Godel formula, the classes are also Godel formulas. Remark: The reverse is not valid for 𝐴 or 𝐵 being of the same height as the "Godel-set of NAND". (Contributed by AV, 21-Oct-2023.)
((𝑁 ∈ ω ∧ (𝑎𝑔𝑏) ∈ (Fmla‘𝑁)) → (𝑎 ∈ (Fmla‘𝑁) ∧ 𝑏 ∈ (Fmla‘𝑁)))
 
21-Oct-2023gonarlem 32538 Lemma for gonar 32539 (induction step). (Contributed by AV, 21-Oct-2023.)
(𝑁 ∈ ω → (((𝑎𝑔𝑏) ∈ (Fmla‘suc 𝑁) → (𝑎 ∈ (Fmla‘suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc 𝑁))) → ((𝑎𝑔𝑏) ∈ (Fmla‘suc suc 𝑁) → (𝑎 ∈ (Fmla‘suc suc 𝑁) ∧ 𝑏 ∈ (Fmla‘suc suc 𝑁)))))
 
21-Oct-2023gonan0 32536 The "Godel-set of NAND" is a Godel formula of at least height 1. (Contributed by AV, 21-Oct-2023.)
((𝐴𝑔𝐵) ∈ (Fmla‘𝑁) → 𝑁 ≠ ∅)
 
21-Oct-2023fmlaomn0 32534 The empty set is not a Godel formula of any height. (Contributed by AV, 21-Oct-2023.)
(𝑁 ∈ ω → ∅ ∉ (Fmla‘𝑁))
 
21-Oct-2023satfvsucsuc 32509 The satisfaction predicate as function over wff codes of height (𝑁 + 1), expressed by the minimally necessary satisfaction predicates as function over wff codes of height 𝑁. (Contributed by AV, 21-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)    &   𝐴 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))    &   𝐵 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}       ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → (𝑆‘suc suc 𝑁) = ((𝑆‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴))}))
 
21-Oct-2023gonanegoal 32496 The Godel-set for the Sheffer stroke NAND is not equal to the Godel-set of universal quantification. (Contributed by AV, 21-Oct-2023.)
(𝑎𝑔𝑏) ≠ ∀𝑔𝑖𝑢
 
21-Oct-2023pthacycspth 32301 A path in an acyclic graph is a simple path. (Contributed by BTernaryTau, 21-Oct-2023.)
((𝐺 ∈ AcyclicGraph ∧ 𝐹(Paths‘𝐺)𝑃) → 𝐹(SPaths‘𝐺)𝑃)
 
21-Oct-2023elneeldif 3947 The elements of a set difference and the minuend are not equal. (Contributed by AV, 21-Oct-2023.)
((𝑋𝐴𝑌 ∈ (𝐵𝐴)) → 𝑋𝑌)
 
21-Oct-2023ralrexbidOLD 3320 Obsolete version of ralrexbid 3319 as of 13-Nov-2023. (Contributed by AV, 21-Oct-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜃))       (∀𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜃))
 
20-Oct-2023fmlasucdisj 32543 The valid Godel formulas of height (𝑁 + 1) is disjoint with the difference ((Fmla‘suc suc 𝑁) ∖ (Fmla‘suc 𝑁)), expressed by formulas constructed from Godel-sets for the Sheffer stroke NAND and Godel-set of universal quantification based on the valid Godel formulas of height (𝑁 + 1). (Contributed by AV, 20-Oct-2023.)
(𝑁 ∈ ω → ((Fmla‘suc 𝑁) ∩ {𝑥 ∣ (∃𝑢 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))(∃𝑣 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) ∨ ∃𝑢 ∈ (Fmla‘𝑁)∃𝑣 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑢𝑔𝑣))}) = ∅)
 
20-Oct-2023fmla0disjsuc 32542 The set of valid Godel formulas of height 0 is disjoint with the formulas constructed from Godel-sets for the Sheffer stroke NAND and Godel-set of universal quantification. (Contributed by AV, 20-Oct-2023.)
((Fmla‘∅) ∩ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) = ∅
 
20-Oct-2023fmlasssuc 32533 The Godel formulas of height 𝑁 are a subset of the Godel formulas of height 𝑁 + 1. (Contributed by AV, 20-Oct-2023.)
(𝑁 ∈ ω → (Fmla‘𝑁) ⊆ (Fmla‘suc 𝑁))
 
20-Oct-2023cusgracyclt3v 32300 A complete simple graph is acyclic if and only if it has fewer than three vertices. (Contributed by BTernaryTau, 20-Oct-2023.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ ComplUSGraph → (𝐺 ∈ AcyclicGraph ↔ (♯‘𝑉) < 3))
 
20-Oct-2023pthisspthorcycl 32272 A path is either a simple path or a cycle (or both). (Contributed by BTernaryTau, 20-Oct-2023.)
(𝐹(Paths‘𝐺)𝑃 → (𝐹(SPaths‘𝐺)𝑃𝐹(Cycles‘𝐺)𝑃))
 
20-Oct-202319.3v 1977 Version of 19.3 2192 with a disjoint variable condition, requiring fewer axioms. Any formula can be universally quantified using a variable which it does not contain. See also 19.9v 1979. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 2006. (Revised by Wolf Lammen, 4-Dec-2017.) (Proof shortened by Wolf Lammen, 20-Oct-2023.)
(∀𝑥𝜑𝜑)
 
20-Oct-2023spvw 1976 Version of sp 2172 when 𝑥 does not occur in 𝜑. Converse of ax-5 1902. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) Shorten 19.3v 1977. (Revised by Wolf Lammen, 20-Oct-2023.)
(∀𝑥𝜑𝜑)
 
20-Oct-2023exgen 1969 Rule of existential generalization, similar to universal generalization ax-gen 1787, but valid only if an individual exists. Its proof requires ax-6 1961 in our axiomatization but the equality predicate does not occur in its statement. Some fundamental theorems of predicate calculus can be proven from ax-gen 1787, ax-4 1801 and this theorem alone, not requiring ax-7 2006 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 20-Oct-2023.)
𝜑       𝑥𝜑
 
19-Oct-2023dmopabelb 5778 A set is an element of the domain of a ordered pair class abstraction iff there is a second set so that both sets fulfil the wff of the class abstraction. (Contributed by AV, 19-Oct-2023.)
(𝑥 = 𝑋 → (𝜑𝜓))       (𝑋𝑉 → (𝑋 ∈ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑦𝜓))
 
19-Oct-2023vtocl2d 3555 Implicit substitution of two classes for two setvar variables. (Contributed by Thierry Arnoux, 25-Aug-2020.) (Revised by BTernaryTau, 19-Oct-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒))    &   (𝜑𝜓)       (𝜑𝜒)
 
18-Oct-2023mo4 2643 At-most-one quantifier expressed using implicit substitution. This theorem is also a direct consequence of mo4f 2644, but this proof is based on fewer axioms.

By the way, swapping 𝑥, 𝑦 and 𝜑, 𝜓 leads to an expression for ∃*𝑦𝜓, which is equivalent to ∃*𝑥𝜑 (is a proof line), so the right hand side is a rare instance of an expression where swapping the quantifiers can be done without ax-11 2151. (Contributed by NM, 26-Jul-1995.) Reduce axiom usage. (Revised by Wolf Lammen, 18-Oct-2023.)

(𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
 
17-Oct-2023satffunlem1lem1 32546 Lemma for satffunlem1 32551. (Contributed by AV, 17-Oct-2023.)
(Fun ((𝑀 Sat 𝐸)‘𝑁) → Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑘𝑀 ({⟨𝑖, 𝑘⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})
 
17-Oct-2023upgracycusgr 32299 An acyclic pseudograph is a simple graph. (Contributed by BTernaryTau, 17-Oct-2023.)
((𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph) → 𝐺 ∈ USGraph)
 
17-Oct-2023umgracycusgr 32298 An acyclic multigraph is a simple graph. (Contributed by BTernaryTau, 17-Oct-2023.)
((𝐺 ∈ UMGraph ∧ 𝐺 ∈ AcyclicGraph) → 𝐺 ∈ USGraph)
 
17-Oct-2023umgr2cycl 32285 A multigraph with two distinct edges that connect the same vertices has a 2-cycle. (Contributed by BTernaryTau, 17-Oct-2023.)
𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ UMGraph ∧ ∃𝑗 ∈ dom 𝐼𝑘 ∈ dom 𝐼((𝐼𝑗) = (𝐼𝑘) ∧ 𝑗𝑘)) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 2))
 
17-Oct-2023umgr2cycllem 32284 Lemma for umgr2cycl 32285. (Contributed by BTernaryTau, 17-Oct-2023.)
𝐹 = ⟨“𝐽𝐾”⟩    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐺 ∈ UMGraph)    &   (𝜑𝐽 ∈ dom 𝐼)    &   (𝜑𝐽𝐾)    &   (𝜑 → (𝐼𝐽) = (𝐼𝐾))       (𝜑 → ∃𝑝 𝐹(Cycles‘𝐺)𝑝)
 
17-Oct-20231one2o 8258 Ordinal one is not ordinal two. Analogous to 1ne2 11833. (Contributed by AV, 17-Oct-2023.)
1o ≠ 2o
 
17-Oct-2023funfv1st2nd 7734 The function value for the first component of an ordered pair is the second component of the ordered pair. (Contributed by AV, 17-Oct-2023.)
((Fun 𝐹𝑋𝐹) → (𝐹‘(1st𝑋)) = (2nd𝑋))
 
17-Oct-2023omsucne 7587 A natural number is not the successor of itself. (Contributed by AV, 17-Oct-2023.)
(𝐴 ∈ ω → 𝐴 ≠ suc 𝐴)
 
17-Oct-2023relcnvtr 6113 A relation is transitive iff its converse is transitive. (Contributed by FL, 19-Sep-2011.) (Proof shortened by Peter Mazsa, 17-Oct-2023.)
(Rel 𝑅 → ((𝑅𝑅) ⊆ 𝑅 ↔ (𝑅𝑅) ⊆ 𝑅))
 
17-Oct-2023relcnvtrg 6112 General form of relcnvtr 6113. (Contributed by Peter Mazsa, 17-Oct-2023.)
((Rel 𝑅 ∧ Rel 𝑆 ∧ Rel 𝑇) → ((𝑅𝑆) ⊆ 𝑇 ↔ (𝑆𝑅) ⊆ 𝑇))
 
17-Oct-20233anidm 1096 Idempotent law for conjunction. (Contributed by Peter Mazsa, 17-Oct-2023.)
((𝜑𝜑𝜑) ↔ 𝜑)
 
16-Oct-2023gonafv 32494 The "Godel-set for the Sheffer stroke NAND" for two formulas 𝐴 and 𝐵. (Contributed by AV, 16-Oct-2023.)
((𝐴𝑉𝐵𝑊) → (𝐴𝑔𝐵) = ⟨1o, ⟨𝐴, 𝐵⟩⟩)
 
16-Oct-20232cycl2d 32283 Construction of a 2-cycle from two given edges in a graph. (Contributed by BTernaryTau, 16-Oct-2023.)
𝑃 = ⟨“𝐴𝐵𝐴”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉))    &   (𝜑𝐴𝐵)    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐴, 𝐵} ⊆ (𝐼𝐾)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐽𝐾)       (𝜑𝐹(Cycles‘𝐺)𝑃)
 
16-Oct-20232cycld 32282 Construction of a 2-cycle from two given edges in a graph. (Contributed by BTernaryTau, 16-Oct-2023.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐽𝐾)    &   (𝜑𝐴 = 𝐶)       (𝜑𝐹(Cycles‘𝐺)𝑃)
 
15-Oct-2023satfv0fun 32515 The value of the satisfaction predicate as function over wff codes at is a function. (Contributed by AV, 15-Oct-2023.)
((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
 
15-Oct-2023satfsschain 32508 The binary relation of a satisfaction predicate as function over wff codes is an increasing chain (with respect to inclusion). (Contributed by AV, 15-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)       (((𝑀𝑉𝐸𝑊) ∧ (𝐴 ∈ ω ∧ 𝐵 ∈ ω)) → (𝐵𝐴 → (𝑆𝐵) ⊆ (𝑆𝐴)))
 
15-Oct-2023upgracycumgr 32297 An acyclic pseudograph is a multigraph. (Contributed by BTernaryTau, 15-Oct-2023.)
((𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph) → 𝐺 ∈ UMGraph)
 
15-Oct-2023acycgrislfgr 32296 An acyclic hypergraph is a loop-free hypergraph. (Contributed by BTernaryTau, 15-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ AcyclicGraph ∧ 𝐺 ∈ UHGraph) → 𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})
 
15-Oct-2023lfuhgr3 32263 A hypergraph is loop-free if and only if none of its edges connect to only one vertex. (Contributed by BTernaryTau, 15-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ¬ ∃𝑎{𝑎} ∈ (Edg‘𝐺)))
 
15-Oct-2023lfuhgr2 32262 A hypergraph is loop-free if and only if every edge is not a loop. (Contributed by BTernaryTau, 15-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)(♯‘𝑥) ≠ 1))
 
15-Oct-2023lfuhgr 32261 A hypergraph is loop-free if and only if every edge connects at least two vertices. (Contributed by BTernaryTau, 15-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ UHGraph → (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ↔ ∀𝑥 ∈ (Edg‘𝐺)2 ≤ (♯‘𝑥)))
 
15-Oct-2023cyc3conja 30726 All 3-cycles are conjugate in the alternating group An for n>= 5. Property (b) of [Lang] p. 32. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐶 = (𝑀 “ (♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &    + = (+g𝑆)    &    = (-g𝑆)    &   (𝜑 → 5 ≤ 𝑁)    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑄𝐶)    &   (𝜑𝑇𝐶)       (𝜑 → ∃𝑝𝐴 𝑄 = ((𝑝 + 𝑇) 𝑝))
 
15-Oct-2023tocycfvres2 30680 A cyclic permutation is the identity outside of its orbit. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)       (𝜑 → ((𝐶𝑊) ↾ (𝐷 ∖ ran 𝑊)) = ( I ↾ (𝐷 ∖ ran 𝑊)))
 
15-Oct-2023tocycfvres1 30679 A cyclic permutation is a cyclic shift on its orbit. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)       (𝜑 → ((𝐶𝑊) ↾ ran 𝑊) = ((𝑊 cyclShift 1) ∘ 𝑊))
 
15-Oct-2023symgsubg 30658 The value of the group subtraction operation of the symmetric group. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &    = (-g𝐺)       ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋𝑌))
 
15-Oct-2023odpmco 30657 The composition of two odd permutations is even. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (pmEven‘𝐷)       ((𝐷 ∈ Fin ∧ 𝑋 ∈ (𝐵𝐴) ∧ 𝑌 ∈ (𝐵𝐴)) → (𝑋𝑌) ∈ 𝐴)
 
15-Oct-2023symgcom 30654 Two permutations 𝑋 and 𝑌 commute if their orbits are disjoint. (Contributed by Thierry Arnoux, 15-Oct-2023.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑋𝐸) = ( I ↾ 𝐸))    &   (𝜑 → (𝑌𝐹) = ( I ↾ 𝐹))    &   (𝜑 → (𝐸𝐹) = ∅)    &   (𝜑 → (𝐸𝐹) = 𝐴)       (𝜑 → (𝑋𝑌) = (𝑌𝑋))
 
14-Oct-2023isfmlasuc 32532 The characterization of a Godel formula of height at least 1. (Contributed by AV, 14-Oct-2023.)
((𝑁 ∈ ω ∧ 𝐹𝑉) → (𝐹 ∈ (Fmla‘suc 𝑁) ↔ (𝐹 ∈ (Fmla‘𝑁) ∨ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝐹 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝐹 = ∀𝑔𝑖𝑢))))
 
14-Oct-2023cycpmconjs 30725 All cycles of the same length are conjugate in the symmetric group. (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐶 = (𝑀 “ (♯ “ {𝑃}))    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &   𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &    = (-g𝑆)    &   (𝜑𝑃 ∈ (0...𝑁))    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑄𝐶)    &   (𝜑𝑇𝐶)       (𝜑 → ∃𝑝𝐵 𝑄 = ((𝑝 + 𝑇) 𝑝))
 
14-Oct-2023cycpmconjslem2 30724 Lemma for cycpmconjs 30725 (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐶 = (𝑀 “ (♯ “ {𝑃}))    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &   𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &    = (-g𝑆)    &   (𝜑𝑃 ∈ (0...𝑁))    &   (𝜑𝐷 ∈ Fin)    &   (𝜑𝑄𝐶)       (𝜑 → ∃𝑞(𝑞:(0..^𝑁)–1-1-onto𝐷 ∧ ((𝑞𝑄) ∘ 𝑞) = ((( I ↾ (0..^𝑃)) cyclShift 1) ∪ ( I ↾ (𝑃..^𝑁)))))
 
14-Oct-2023cycpmconjslem1 30723 Lemma for cycpmconjs 30725 (Contributed by Thierry Arnoux, 14-Oct-2023.)
𝐶 = (𝑀 “ (♯ “ {𝑃}))    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑 → (♯‘𝑊) = 𝑃)       (𝜑 → ((𝑊 ∘ (𝑀𝑊)) ∘ 𝑊) = (( I ↾ (0..^𝑃)) cyclShift 1))
 
13-Oct-2023satfdmfmla 32544 The domain of the satisfaction predicate as function over wff codes in any model 𝑀 and any binary relation 𝐸 on 𝑀 for a natural number 𝑁 is the set of valid Godel formulas of height 𝑁. (Contributed by AV, 13-Oct-2023.)
((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁))
 
13-Oct-2023satfrnmapom 32514 The range of the satisfaction predicate as function over wff codes in any model 𝑀 and any binary relation 𝐸 on 𝑀 for a natural number 𝑁 is a subset of the power set of all mappings from the natural numbers into the model 𝑀. (Contributed by AV, 13-Oct-2023.)
((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → ran ((𝑀 Sat 𝐸)‘𝑁) ⊆ 𝒫 (𝑀m ω))
 
13-Oct-2023satfdm 32513 The domain of the satisfaction predicate as function over wff codes does not depend on the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 13-Oct-2023.)
(((𝑀𝑉𝐸𝑊) ∧ (𝑁𝑋𝐹𝑌)) → ∀𝑛 ∈ ω dom ((𝑀 Sat 𝐸)‘𝑛) = dom ((𝑁 Sat 𝐹)‘𝑛))
 
13-Oct-2023satfbrsuc 32510 The binary relation of a satisfaction predicate as function over wff codes at a successor. (Contributed by AV, 13-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)    &   𝑃 = (𝑆𝑁)       (((𝑀𝑉𝐸𝑊) ∧ 𝑁 ∈ ω ∧ (𝐴𝑋𝐵𝑌)) → (𝐴(𝑆‘suc 𝑁)𝐵 ↔ (𝐴𝑃𝐵 ∨ ∃𝑢𝑃 (∃𝑣𝑃 (𝐴 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝐵 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀𝑔𝑖(1st𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))))
 
13-Oct-2023loop1cycl 32281 A hypergraph has a cycle of length one if and only if it has a loop. (Contributed by BTernaryTau, 13-Oct-2023.)
(𝐺 ∈ UHGraph → (∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 1 ∧ (𝑝‘0) = 𝐴) ↔ {𝐴} ∈ (Edg‘𝐺)))
 
13-Oct-2023cycpmgcl 30722 Cyclic permutations are permutations, similar to cycpmcl 30685, but where the set of cyclic permutations of length 𝑃 is expressed in terms of a preimage. (Contributed by Thierry Arnoux, 13-Oct-2023.)
𝐶 = (𝑀 “ (♯ “ {𝑃}))    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &   𝐵 = (Base‘𝑆)       ((𝐷𝑉𝑃 ∈ (0...𝑁)) → 𝐶𝐵)
 
12-Oct-2023satfdmlem 32512 Lemma for satfdm 32513. (Contributed by AV, 12-Oct-2023.)
(((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎))))
 
12-Oct-2023satfrel 32511 The value of the satisfaction predicate as function over wff codes at a natural number is a relation. (Contributed by AV, 12-Oct-2023.)
((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → Rel ((𝑀 Sat 𝐸)‘𝑁))
 
12-Oct-2023acycgr2v 32294 A simple graph with two vertices is an acyclic graph. (Contributed by BTernaryTau, 12-Oct-2023.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ (♯‘𝑉) = 2) → 𝐺 ∈ AcyclicGraph)
 
12-Oct-2023acycgr1v 32293 A multigraph with one vertex is an acyclic graph. (Contributed by BTernaryTau, 12-Oct-2023.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ UMGraph ∧ (♯‘𝑉) = 1) → 𝐺 ∈ AcyclicGraph)
 
12-Oct-2023acycgrcycl 32291 Any cycle in an acyclic graph is trivial (i.e. has one vertex and no edges). (Contributed by BTernaryTau, 12-Oct-2023.)
((𝐺 ∈ AcyclicGraph ∧ 𝐹(Cycles‘𝐺)𝑃) → 𝐹 = ∅)
 
12-Oct-2023fnunres2 30352 Restriction of a disjoint union to the domain of the second function. (Contributed by Thierry Arnoux, 12-Oct-2023.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)
 
11-Oct-2023en2pr 39784 A class is equinumerous to ordinal two iff it is a pair of distinct sets. (Contributed by RP, 11-Oct-2023.)
(𝐴 ≈ 2o ↔ ∃𝑥𝑦(𝐴 = {𝑥, 𝑦} ∧ 𝑥𝑦))
 
11-Oct-2023prclisacycgr 32295 A proper class (representing a null graph, see vtxvalprc 26757) has the property of an acyclic graph (see also acycgr0v 32292). (Contributed by BTernaryTau, 11-Oct-2023.) (New usage is discouraged.)
𝑉 = (Vtx‘𝐺)       𝐺 ∈ V → ¬ ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝𝑓 ≠ ∅))
 
11-Oct-2023acycgr0v 32292 A null graph (with no vertices) is an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
𝑉 = (Vtx‘𝐺)       ((𝐺𝑊𝑉 = ∅) → 𝐺 ∈ AcyclicGraph)
 
11-Oct-2023isacycgr1 32290 The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
(𝐺𝑊 → (𝐺 ∈ AcyclicGraph ↔ ∀𝑓𝑝(𝑓(Cycles‘𝐺)𝑝𝑓 = ∅)))
 
11-Oct-2023isacycgr 32289 The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
(𝐺𝑊 → (𝐺 ∈ AcyclicGraph ↔ ¬ ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝𝑓 ≠ ∅)))
 
11-Oct-2023dfacycgr1 32288 An alternate definition of the class of all acyclic graphs that requires all cycles to be trivial. (Contributed by BTernaryTau, 11-Oct-2023.)
AcyclicGraph = {𝑔 ∣ ∀𝑓𝑝(𝑓(Cycles‘𝑔)𝑝𝑓 = ∅)}
 
11-Oct-2023df-acycgr 32287 Define the class of all acyclic graphs. A graph is called acyclic if it has no (non-trivial) cycles. (Contributed by BTernaryTau, 11-Oct-2023.)
AcyclicGraph = {𝑔 ∣ ¬ ∃𝑓𝑝(𝑓(Cycles‘𝑔)𝑝𝑓 ≠ ∅)}
 
10-Oct-2023satfvsuc 32505 The value of the satisfaction predicate as function over wff codes at a successor. (Contributed by AV, 10-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)       ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → (𝑆‘suc 𝑁) = ((𝑆𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}))
 
9-Oct-2023cycpmconjv 30711 A formula for computing conjugacy classes of cyclic permutations. Formula in property (b) of [Lang] p. 32. (Contributed by Thierry Arnoux, 9-Oct-2023.)
𝑆 = (SymGrp‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &    + = (+g𝑆)    &    = (-g𝑆)    &   𝐵 = (Base‘𝑆)       ((𝐷𝑉𝐺𝐵𝑊 ∈ dom 𝑀) → ((𝐺 + (𝑀𝑊)) 𝐺) = (𝑀‘(𝐺𝑊)))
 
9-Oct-2023cycpmconjvlem 30710 Lemma for cycpmconjv 30711 (Contributed by Thierry Arnoux, 9-Oct-2023.)
(𝜑𝐹:𝐷1-1-onto𝐷)    &   (𝜑𝐵𝐷)       (𝜑 → ((𝐹 ↾ (𝐷𝐵)) ∘ 𝐹) = ( I ↾ (𝐷 ∖ ran (𝐹𝐵))))
 
9-Oct-2023reldisjun 30281 Split a relation into two disjoint parts based on its domain. (Contributed by Thierry Arnoux, 9-Oct-2023.)
((Rel 𝑅 ∧ dom 𝑅 = (𝐴𝐵) ∧ (𝐴𝐵) = ∅) → 𝑅 = ((𝑅𝐴) ∪ (𝑅𝐵)))
 
9-Oct-2023rabelpw 5244 A restricted class abstraction is an element of the power set of its restricting set. (Contributed by AV, 9-Oct-2023.)
(𝐴𝑉 → {𝑥𝐴𝜑} ∈ 𝒫 𝐴)
 
9-Oct-2023difelpw 5243 A difference is an element of the power set of its minuend. (Contributed by AV, 9-Oct-2023.)
(𝐴𝑉 → (𝐴𝐵) ∈ 𝒫 𝐴)
 
8-Oct-2023pren2 39790 An unordered pair is equinumerous to ordinal two iff both parts are sets not equal to each other. (Contributed by RP, 8-Oct-2023.)
({𝐴, 𝐵} ≈ 2o ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝐵))
 
8-Oct-2023pr2cv 39785 If an unordered pair is equinumerous to ordinal two, then both parts are sets. (Contributed by RP, 8-Oct-2023.)
({𝐴, 𝐵} ≈ 2o → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
8-Oct-2023snen1el 39769 A singleton is equinumerous to ordinal one if its content is an element of it. (Contributed by RP, 8-Oct-2023.)
({𝐴} ≈ 1o𝐴 ∈ {𝐴})
 
8-Oct-2023snen1g 39768 A singleton is equinumerous to ordinal one iff its content is a set. (Contributed by RP, 8-Oct-2023.)
({𝐴} ≈ 1o𝐴 ∈ V)
 
8-Oct-2023satfvsuclem2 32504 Lemma 2 for satfvsuc 32505. (Contributed by AV, 8-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)       ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))} ∈ V)
 
8-Oct-2023satfvsuclem1 32503 Lemma 1 for satfvsuc 32505. (Contributed by AV, 8-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)       ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∧ 𝑦 ∈ 𝒫 (𝑀m ω))} ∈ V)
 
8-Oct-2023satfv0 32502 The value of the satisfaction predicate as function over wff codes at . (Contributed by AV, 8-Oct-2023.)
𝑆 = (𝑀 Sat 𝐸)       ((𝑀𝑉𝐸𝑊) → (𝑆‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})})
 
8-Oct-2023goeleq12bg 32493 Two "Godel-set of membership" codes for two variables are equal iff the two corresponding variables are equal. (Contributed by AV, 8-Oct-2023.)
(((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼𝑔𝐽) = (𝑀𝑔𝑁) ↔ (𝐼 = 𝑀𝐽 = 𝑁)))
 
8-Oct-2023spthcycl 32273 A walk is a trivial path if and only if it is both a simple path and a cycle. (Contributed by BTernaryTau, 8-Oct-2023.)
((𝐹(Paths‘𝐺)𝑃𝐹 = ∅) ↔ (𝐹(SPaths‘𝐺)𝑃𝐹(Cycles‘𝐺)𝑃))
 
8-Oct-2023funen1cnv 32254 If a function is equinumerous to ordinal 1, then its converse is also a function. (Contributed by BTernaryTau, 8-Oct-2023.)
((Fun 𝐹𝐹 ≈ 1o) → Fun 𝐹)
 
6-Oct-2023satom 32500 The satisfaction predicate for wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 at omega (ω). (Contributed by AV, 6-Oct-2023.)
((𝑀𝑉𝐸𝑊) → ((𝑀 Sat 𝐸)‘ω) = 𝑛 ∈ ω ((𝑀 Sat 𝐸)‘𝑛))
 
6-Oct-2023satfn 32499 The satisfaction predicate for wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 is a function over suc ω. (Contributed by AV, 6-Oct-2023.)
((𝑀𝑉𝐸𝑊) → (𝑀 Sat 𝐸) Fn suc ω)
 
6-Oct-2023fiun 7633 The union of a chain (with respect to inclusion) of functions is a function. Analogous to f1iun 7634. (Contributed by AV, 6-Oct-2023.)
(𝑥 = 𝑦𝐵 = 𝐶)    &   𝐵 ∈ V       (∀𝑥𝐴 (𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) → 𝑥𝐴 𝐵: 𝑥𝐴 𝐷𝑆)
 
6-Oct-2023fiunlem 7632 Lemma for fiun 7633 and f1iun 7634. Formerly part of f1iun 7634. (Contributed by AV, 6-Oct-2023.)
(𝑥 = 𝑦𝐵 = 𝐶)       (((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))
 
6-Oct-2023vtoclgft 3551 Closed theorem form of vtoclgf 3563. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by JJ, 11-Aug-2021.) Avoid ax-13 2381. (Revised by Gino Giotto, 6-Oct-2023.)
(((𝑥𝐴 ∧ Ⅎ𝑥𝜓) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∀𝑥𝜑) ∧ 𝐴𝑉) → 𝜓)
 
5-Oct-2023finorwe 34545 If the Axiom of Infinity is denied, every total order is a well-order. The notion of a well-order cannot be usefully expressed without the Axiom of Infinity due to the inability to quantify over proper classes. (Contributed by ML, 5-Oct-2023.)
(¬ ω ∈ V → ( < Or 𝐴< We 𝐴))
 
5-Oct-2023usgrcyclgt2v 32275 A simple graph with a non-trivial cycle must have at least 3 vertices. (Contributed by BTernaryTau, 5-Oct-2023.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃𝐹 ≠ ∅) → 2 < (♯‘𝑉))
 
5-Oct-2023pthhashvtx 32271 A graph containing a path has at least as many vertices as there are edges in the path. (Contributed by BTernaryTau, 5-Oct-2023.)
𝑉 = (Vtx‘𝐺)       (𝐹(Paths‘𝐺)𝑃 → (♯‘𝐹) ≤ (♯‘𝑉))
 
4-Oct-2023cusgr3cyclex 32280 Every complete simple graph with more than two vertices has a 3-cycle. (Contributed by BTernaryTau, 4-Oct-2023.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ ComplUSGraph ∧ 2 < (♯‘𝑉)) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (♯‘𝑓) = 3))
 
4-Oct-2023cusgredgex2 32266 Any two distinct vertices in a complete simple graph are connected to each other by an edge. (Contributed by BTernaryTau, 4-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ ComplUSGraph → ((𝐴𝑉𝐵𝑉𝐴𝐵) → {𝐴, 𝐵} ∈ 𝐸))
 
4-Oct-2023f1resfz0f1d 32258 If a function with a sequence of nonnegative integers (starting at 0) as its domain is one-to-one when 0 is removed, and if the range of that restriction does not contain the function's value at the removed integer, then the function is itself one-to-one. (Contributed by BTernaryTau, 4-Oct-2023.)
(𝜑𝐾 ∈ ℕ0)    &   (𝜑𝐹:(0...𝐾)⟶𝑉)    &   (𝜑 → (𝐹 ↾ (1...𝐾)):(1...𝐾)–1-1𝑉)    &   (𝜑 → ((𝐹 “ {0}) ∩ (𝐹 “ (1...𝐾))) = ∅)       (𝜑𝐹:(0...𝐾)–1-1𝑉)
 
4-Oct-2023cshf1o 30563 Condition for the cyclic shift to be a bijection. (Contributed by Thierry Arnoux, 4-Oct-2023.)
((𝑊 ∈ Word 𝐷𝑊:dom 𝑊1-1𝐷𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁):dom 𝑊1-1-onto→ran 𝑊)
 
3-Oct-2023cusgredgex 32265 Any two (distinct) vertices in a complete simple graph are connected to each other by an edge. (Contributed by BTernaryTau, 3-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ ComplUSGraph → ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → {𝐴, 𝐵} ∈ 𝐸))
 
3-Oct-2023fisshasheq 32249 A finite set is equal to its subset if they are the same size. (Contributed by BTernaryTau, 3-Oct-2023.)
((𝐵 ∈ Fin ∧ 𝐴𝐵 ∧ (♯‘𝐴) = (♯‘𝐵)) → 𝐴 = 𝐵)
 
3-Oct-2023dfeumo 2612 An elementary proof showing the reverse direction of dfmoeu 2611. Here the characterizing expression of existential uniqueness (eu6 2652) is derived from that of uniqueness (df-mo 2615). (Contributed by Wolf Lammen, 3-Oct-2023.)
((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
3-Oct-20232ax6e 2486 We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. Version of 2ax6elem 2485 with a distinct variable constraint. (Contributed by Wolf Lammen, 28-Sep-2018.) (Proof shortened by Wolf Lammen, 3-Oct-2023.)
𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)
 
2-Oct-2023cplgredgex 32264 Any two (distinct) vertices in a complete graph are connected to each other by at least one edge. (Contributed by BTernaryTau, 2-Oct-2023.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ ComplGraph → ((𝐴𝑉𝐵 ∈ (𝑉 ∖ {𝐴})) → ∃𝑒𝐸 {𝐴, 𝐵} ⊆ 𝑒))
 
2-Oct-2023fzodif1 30442 Set difference of two half-open range of sequential integers sharing the same starting value. (Contributed by Thierry Arnoux, 2-Oct-2023.)
(𝐾 ∈ (𝑀...𝑁) → ((𝑀..^𝑁) ∖ (𝑀..^𝐾)) = (𝐾..^𝑁))
 
2-Oct-20232eu5 2735 An alternate definition of double existential uniqueness (see 2eu4 2734). A mistake sometimes made in the literature is to use ∃!𝑥∃!𝑦 to mean "exactly one 𝑥 and exactly one 𝑦". (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining 𝑥∃*𝑦𝜑 as an additional condition. The correct definition apparently has never been published (∃* means "exists at most one"). (Contributed by NM, 26-Oct-2003.) Avoid ax-13 2381. (Revised by Wolf Lammen, 2-Oct-2023.)
((∃!𝑥∃!𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝑥𝑦(𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))
 
2-Oct-20232eu1v 2730 Version of 2eu1 2728 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2381. (Contributed by Wolf Lammen, 2-Oct-2023.)
(∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
 
2-Oct-2023moexex 2716 "At most one" double quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.) Factor out common proof lines with moexexvw 2706. (Revised by Wolf Lammen, 2-Oct-2023.)
𝑦𝜑       ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦𝑥(𝜑𝜓))
 
2-Oct-20232exeuv 2710 Version of 2exeu 2724 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2381. (Contributed by Wolf Lammen, 2-Oct-2023.)
((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
 
2-Oct-20232euexv 2709 Version of 2euex 2719 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2381. (Contributed by Wolf Lammen, 2-Oct-2023.)
(∃!𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑)
 
2-Oct-20232moswapv 2707 Version of 2moswap 2722 with a disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 22-Aug-2023.) Factor out common proof lines with moexexvw 2706. (Revised by Wolf Lammen, 2-Oct-2023.)
(∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))
 
2-Oct-2023moexexvw 2706 Version of moexexv 2717 with an additional disjoint variable condition, which does not require ax-13 2381. (Contributed by Gino Giotto, 22-Aug-2023.) Factor out common proof lines with moexex 2716. (Revised by Wolf Lammen, 2-Oct-2023.)
((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦𝑥(𝜑𝜓))
 
2-Oct-2023moexexlem 2704 Factor out the proof skeleton of moexex 2716 and moexexvw 2706. (Contributed by Wolf Lammen, 2-Oct-2023.)
𝑦𝜑    &   𝑦∃*𝑥𝜑    &   𝑥∃*𝑦𝑥(𝜑𝜓)       ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦𝑥(𝜑𝜓))
 
2-Oct-2023nfmov 2637 Bound-variable hypothesis builder for the at-most-one quantifier. See nfmo 2639 for a version without disjoint variable conditions but requiring ax-13 2381. (Contributed by Wolf Lammen, 2-Oct-2023.)
𝑥𝜑       𝑥∃*𝑦𝜑
 
1-Oct-2023elrncard 39780 Let us define a cardinal number to be an element 𝐴 ∈ On such that 𝐴 is not equipotent with any 𝑥𝐴. (Contributed by RP, 1-Oct-2023.)
(𝐴 ∈ ran card ↔ (𝐴 ∈ On ∧ ∀𝑥𝐴 ¬ 𝑥𝐴))
 
1-Oct-2023eu0 39764 There is only one empty set. (Contributed by RP, 1-Oct-2023.)
(∀𝑥 ¬ 𝑥 ∈ ∅ ∧ ∃!𝑥𝑦 ¬ 𝑦𝑥)
 
1-Oct-2023hashf1dmcdm 32253 The size of the domain of a one-to-one set function is less than or equal to the size of its codomain, if it exists. (Contributed by BTernaryTau, 1-Oct-2023.)
((𝐹𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → (♯‘𝐴) ≤ (♯‘𝐵))
 
1-Oct-2023hashf1dmrn 32252 The size of the domain of a one-to-one set function is equal to the size of its range. (Contributed by BTernaryTau, 1-Oct-2023.)
((𝐹𝑉𝐹:𝐴1-1𝐵) → (♯‘𝐴) = (♯‘ran 𝐹))
 
1-Oct-2023euim 2694 Add unique existential quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Proof shortened by Wolf Lammen, 1-Oct-2023.)
((∃𝑥𝜑 ∧ ∀𝑥(𝜑𝜓)) → (∃!𝑥𝜓 → ∃!𝑥𝜑))
 
30-Sep-2023hashfundm 32251 The size of a set function is equal to the size of its domain. (Contributed by BTernaryTau, 30-Sep-2023.)
((𝐹𝑉 ∧ Fun 𝐹) → (♯‘𝐹) = (♯‘dom 𝐹))
 
30-Sep-20230nn0m1nnn0 32248 A number is zero if and only if it's a nonnegative integer that becomes negative after subtracting 1. (Contributed by BTernaryTau, 30-Sep-2023.)
(𝑁 = 0 ↔ (𝑁 ∈ ℕ0 ∧ ¬ (𝑁 − 1) ∈ ℕ0))
 
29-Sep-2023syl2anc2 585 Double syllogism inference combined with contraction. (Contributed by BTernaryTau, 29-Sep-2023.)
(𝜑𝜓)    &   (𝜓𝜒)    &   ((𝜓𝜒) → 𝜃)       (𝜑𝜃)
 
28-Sep-2023f1resrcmplf1d 32257 If a function's restriction to a subclass of its domain and its restriction to the relative complement of that subclass are both one-to-one, and if the ranges of those two restrictions are disjoint, then the function is itself one-to-one. (Contributed by BTernaryTau, 28-Sep-2023.)
(𝜑𝐶𝐴)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → (𝐹𝐶):𝐶1-1𝐵)    &   (𝜑 → (𝐹 ↾ (𝐴𝐶)):(𝐴𝐶)–1-1𝐵)    &   (𝜑 → ((𝐹𝐶) ∩ (𝐹 “ (𝐴𝐶))) = ∅)       (𝜑𝐹:𝐴1-1𝐵)
 
27-Sep-2023infordmin 39777 ω is the smallest infinite ordinal. (Contributed by RP, 27-Sep-2023.)
𝑥 ∈ (On ∖ Fin)ω ⊆ 𝑥
 
27-Sep-2023dfom6 39776 Let ω be defined to be the union of the set of all finite ordinals. (Contributed by RP, 27-Sep-2023.)
ω = (On ∩ Fin)
 
27-Sep-2023ontric3g 39766 For all 𝑥, 𝑦 ∈ On, one and only one of the following hold: 𝑥𝑦, 𝑦 = 𝑥, or 𝑦𝑥. This is a transparent strict trichotomy. (Contributed by RP, 27-Sep-2023.)
𝑥 ∈ On ∀𝑦 ∈ On ((𝑥𝑦 ↔ ¬ (𝑦 = 𝑥𝑦𝑥)) ∧ (𝑦 = 𝑥 ↔ ¬ (𝑥𝑦𝑦𝑥)) ∧ (𝑦𝑥 ↔ ¬ (𝑥𝑦𝑦 = 𝑥)))
 
27-Sep-2023epelon2 39765 Over the ordinal numbers, one may define the relation 𝐴 E 𝐵 iff 𝐴𝐵 and one finds that, under this ordering, On is a well-ordered class, see epweon 7486. This is a weak form of epelg 5459 which only requires that we know 𝐵 to be a set. (Contributed by RP, 27-Sep-2023.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 E 𝐵𝐴𝐵))
 
27-Sep-2023f1resrcmplf1dlem 32256 Lemma for f1resrcmplf1d 32257. (Contributed by BTernaryTau, 27-Sep-2023.)
(𝜑𝐶𝐴)    &   (𝜑𝐷𝐴)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑 → ((𝐹𝐶) ∩ (𝐹𝐷)) = ∅)       (𝜑 → ((𝑋𝐶𝑌𝐷) → ((𝐹𝑋) = (𝐹𝑌) → 𝑋 = 𝑌)))
 
27-Sep-2023f1resveqaeq 32255 If a function restricted to a class is one-to-one, then for any two elements of the class, the values of the function at those elements are equal only if the two elements are the same element. (Contributed by BTernaryTau, 27-Sep-2023.)
(((𝐹𝐴):𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))
 
27-Sep-2023prsrcmpltd 32244 If a statement is true for all pairs of elements of a class, all pairs of elements of its complement relative to a second class, and all pairs with one element in each, then it is true for all pairs of elements of the second class. (Contributed by BTernaryTau, 27-Sep-2023.)
(𝜑 → ((𝐶𝐴𝐷𝐴) → 𝜓))    &   (𝜑 → ((𝐶𝐴𝐷 ∈ (𝐵𝐴)) → 𝜓))    &   (𝜑 → ((𝐶 ∈ (𝐵𝐴) ∧ 𝐷𝐴) → 𝜓))    &   (𝜑 → ((𝐶 ∈ (𝐵𝐴) ∧ 𝐷 ∈ (𝐵𝐴)) → 𝜓))       (𝜑 → ((𝐶𝐵𝐷𝐵) → 𝜓))
 
27-Sep-2023srcmpltd 32243 If a statement is true for every element of a class and for every element of its complement relative to a second class, then it is true for every element in the second class. (Contributed by BTernaryTau, 27-Sep-2023.)
(𝜑 → (𝐶𝐴𝜓))    &   (𝜑 → (𝐶 ∈ (𝐵𝐴) → 𝜓))       (𝜑 → (𝐶𝐵𝜓))
 
27-Sep-2023cyc3genpm 30721 The alternating group 𝐴 is generated by 3-cycles. Property (a) of [Lang] p. 32 . (Contributed by Thierry Arnoux, 27-Sep-2023.)
𝐶 = (𝑀 “ (♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)       (𝐷 ∈ Fin → (𝑄𝐴 ↔ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤)))
 
27-Sep-2023s3clhash 30551 Closure of the words of length 3 in a preimage using the hash function. (Contributed by Thierry Arnoux, 27-Sep-2023.)
⟨“𝐼𝐽𝐾”⟩ ∈ (♯ “ {3})
 
27-Sep-2023fnpr2ob 16819 Biconditional version of fnpr2o 16818. (Contributed by Jim Kingdon, 27-Sep-2023.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
 
27-Sep-2023elunant 4151 A statement is true for every element of the union of a pair of classes if and only if it is true for every element of the first class and for every element of the second class. (Contributed by BTernaryTau, 27-Sep-2023.)
((𝐶 ∈ (𝐴𝐵) → 𝜑) ↔ ((𝐶𝐴𝜑) ∧ (𝐶𝐵𝜑)))
 
27-Sep-2023eubii 2663 Introduce unique existential quantifier to both sides of an equivalence. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) Avoid ax-5 1902. (Revised by Wolf Lammen, 27-Sep-2023.)
(𝜑𝜓)       (∃!𝑥𝜑 ↔ ∃!𝑥𝜓)
 
26-Sep-2023wrdt2ind 30554 Perform an induction over the structure of a word of even length. (Contributed by Thierry Arnoux, 26-Sep-2023.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 ++ ⟨“𝑖𝑗”⟩) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   ((𝑦 ∈ Word 𝐵𝑖𝐵𝑗𝐵) → (𝜒𝜃))       ((𝐴 ∈ Word 𝐵 ∧ 2 ∥ (♯‘𝐴)) → 𝜏)
 
26-Sep-2023pfxlsw2ccat 30553 Reconstruct a word from its prefix and its last two symbols. (Contributed by Thierry Arnoux, 26-Sep-2023.)
𝑁 = (♯‘𝑊)       ((𝑊 ∈ Word 𝑉 ∧ 2 ≤ 𝑁) → 𝑊 = ((𝑊 prefix (𝑁 − 2)) ++ ⟨“(𝑊‘(𝑁 − 2))(𝑊‘(𝑁 − 1))”⟩))
 
25-Sep-2023trsp2cyc 30692 Exhibit the word a transposition corresponds to, as a cycle. (Contributed by Thierry Arnoux, 25-Sep-2023.)
𝑇 = ran (pmTrsp‘𝐷)    &   𝐶 = (toCyc‘𝐷)       ((𝐷𝑉𝑃𝑇) → ∃𝑖𝐷𝑗𝐷 (𝑖𝑗𝑃 = (𝐶‘⟨“𝑖𝑗”⟩)))
 
25-Sep-2023tocycf 30686 The permutation cycle builder as a function. (Contributed by Thierry Arnoux, 25-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝑆)       (𝐷𝑉𝐶:{𝑤 ∈ Word 𝐷𝑤:dom 𝑤1-1𝐷}⟶𝐵)
 
25-Sep-2023mhplss 20270 Homogeneous polynomials form a linear subspace of the polynomials. (Contributed by SN, 25-Sep-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐻𝑁) ∈ (LSubSp‘𝑃))
 
25-Sep-2023mhpvscacl 20269 Homogeneous polynomials are closed under scalar multiplication. (Contributed by SN, 25-Sep-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &    · = ( ·𝑠𝑃)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋𝐾)    &   (𝜑𝐹 ∈ (𝐻𝑁))       (𝜑 → (𝑋 · 𝐹) ∈ (𝐻𝑁))
 
25-Sep-2023mhpsubg 20268 Homogeneous polynomials form a subgroup of the polynomials. (Contributed by SN, 25-Sep-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Grp)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐻𝑁) ∈ (SubGrp‘𝑃))
 
25-Sep-2023xpsrnbas 16832 The indexed structure product that appears in xpsval 16831 has the same base as the target of the function 𝐹. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   𝐹 = (𝑥𝑋, 𝑦𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    &   𝐺 = (Scalar‘𝑅)    &   𝑈 = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})       (𝜑 → ran 𝐹 = (Base‘𝑈))
 
25-Sep-2023xpsval 16831 Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
𝑇 = (𝑅 ×s 𝑆)    &   𝑋 = (Base‘𝑅)    &   𝑌 = (Base‘𝑆)    &   (𝜑𝑅𝑉)    &   (𝜑𝑆𝑊)    &   𝐹 = (𝑥𝑋, 𝑦𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})    &   𝐺 = (Scalar‘𝑅)    &   𝑈 = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})       (𝜑𝑇 = (𝐹s 𝑈))
 
25-Sep-2023fvpr1o 16821 The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.)
(𝐵𝑉 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘1o) = 𝐵)
 
25-Sep-2023fvpr0o 16820 The value of a function with a domain of (at most) two elements. (Contributed by Jim Kingdon, 25-Sep-2023.)
(𝐴𝑉 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘∅) = 𝐴)
 
25-Sep-2023fnpr2o 16818 Function with a domain of 2o. (Contributed by Jim Kingdon, 25-Sep-2023.)
((𝐴𝑉𝐵𝑊) → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o)
 
25-Sep-2023df-xps 16771 Define a binary product on structures. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) “s ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩})))
 
25-Sep-2023axsepgfromrep 5192 A more general version axsepg 5195 of the axiom scheme of separation ax-sep 5194 derived from the axiom scheme of replacement ax-rep 5181 (and first-order logic). The extra generality consists in the absence of a disjoint variable condition on 𝑧, 𝜑 (that is, variable 𝑧 may occur in formula 𝜑). See linked statements for more information. (Contributed by NM, 11-Sep-2006.) Remove dependencies on ax-9 2115 to ax-13 2381. (Revised by SN, 25-Sep-2023.) Use ax-sep 5194 instead (or axsepg 5195 if the extra generality is needed). (New usage is discouraged.)
𝑦𝑥(𝑥𝑦 ↔ (𝑥𝑧𝜑))
 
24-Sep-2023usgrgt2cycl 32274 A non-trivial cycle in a simple graph has a length greater than 2. (Contributed by BTernaryTau, 24-Sep-2023.)
((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃𝐹 ≠ ∅) → 2 < (♯‘𝐹))
 
24-Sep-2023nn0ltp1ne 32247 Nonnegative integer ordering relation. (Contributed by BTernaryTau, 24-Sep-2023.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0) → ((𝐴 + 1) < 𝐵 ↔ (𝐴 < 𝐵𝐵 ≠ (𝐴 + 1))))
 
24-Sep-2023nnltp1ne 32246 Positive integer ordering relation. (Contributed by BTernaryTau, 24-Sep-2023.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 + 1) < 𝐵 ↔ (𝐴 < 𝐵𝐵 ≠ (𝐴 + 1))))
 
24-Sep-2023zltp1ne 32245 Integer ordering relation. (Contributed by BTernaryTau, 24-Sep-2023.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 + 1) < 𝐵 ↔ (𝐴 < 𝐵𝐵 ≠ (𝐴 + 1))))
 
24-Sep-2023cyc3genpmlem 30720 Lemma for cyc3genpm 30721. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (𝑀 “ (♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   𝑁 = (♯‘𝐷)    &   𝑀 = (toCyc‘𝐷)    &    · = (+g𝑆)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐿𝐷)    &   (𝜑𝐸 = (𝑀‘⟨“𝐼𝐽”⟩))    &   (𝜑𝐹 = (𝑀‘⟨“𝐾𝐿”⟩))    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐽)    &   (𝜑𝐾𝐿)       (𝜑 → ∃𝑐 ∈ Word 𝐶(𝐸 · 𝐹) = (𝑆 Σg 𝑐))
 
24-Sep-2023cyc3evpm 30719 3-Cycles are even permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = ((toCyc‘𝐷) “ (♯ “ {3}))    &   𝐴 = (pmEven‘𝐷)       (𝐷 ∈ Fin → 𝐶𝐴)
 
24-Sep-2023cyc2fv2 30691 Function value of a 2-cycle at the second point. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐼𝐽)    &   𝑆 = (SymGrp‘𝐷)       (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐽) = 𝐼)
 
24-Sep-2023cyc2fv1 30690 Function value of a 2-cycle at the first point. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐼𝐽)    &   𝑆 = (SymGrp‘𝐷)       (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐼) = 𝐽)
 
24-Sep-2023cycpm2cl 30689 Closure for the 2-cycles. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐼𝐽)    &   𝑆 = (SymGrp‘𝐷)       (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩) ∈ (Base‘𝑆))
 
24-Sep-2023cycpm2tr 30688 A cyclic permutation of 2 elements is a transposition. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐼𝐽)    &   𝑇 = (pmTrsp‘𝐷)       (𝜑 → (𝐶‘⟨“𝐼𝐽”⟩) = (𝑇‘{𝐼, 𝐽}))
 
24-Sep-2023cycpmcl 30685 Cyclic permutations are permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   𝑆 = (SymGrp‘𝐷)       (𝜑 → (𝐶𝑊) ∈ (Base‘𝑆))
 
24-Sep-2023coprprop 30361 Composition of two pairs of ordered pairs with matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)    &   (𝜑𝐴𝐶)    &   (𝜑𝐸𝑋)    &   (𝜑𝐹𝑋)    &   (𝜑𝐸𝐹)       (𝜑 → ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∘ {⟨𝐸, 𝐴⟩, ⟨𝐹, 𝐶⟩}) = {⟨𝐸, 𝐵⟩, ⟨𝐹, 𝐷⟩})
 
24-Sep-2023mptprop 30360 Rewrite pairs of ordered pairs as mapping to functions. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)    &   (𝜑𝐴𝐶)       (𝜑 → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
 
24-Sep-2023brprop 30359 Binary relation for a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)       (𝜑 → (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}𝑌 ↔ ((𝑋 = 𝐴𝑌 = 𝐵) ∨ (𝑋 = 𝐶𝑌 = 𝐷))))
 
24-Sep-2023cnvprop 30358 Converse of a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑉𝐷𝑊)) → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {⟨𝐵, 𝐴⟩, ⟨𝐷, 𝐶⟩})
 
24-Sep-2023cosnop 30357 Composition of two ordered pair singletons with matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)       (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐴⟩}) = {⟨𝐶, 𝐵⟩})
 
24-Sep-2023cosnopne 30356 Composition of two ordered pair singletons with non-matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)    &   (𝜑𝐴𝐷)       (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐷⟩}) = ∅)
 
24-Sep-2023mobii 2624 Formula-building rule for the at-most-one quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.) Avoid ax-5 1902. (Revised by Wolf Lammen, 24-Sep-2023.)
(𝜓𝜒)       (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)
 
23-Sep-2023ichnfb 43502 If 𝑥 and 𝑦 are interchangeable in 𝜑, they are both free or both not free in 𝜑. (Contributed by Wolf Lammen, 6-Aug-2023.) (Revised by AV, 23-Sep-2023.)
([𝑥𝑦]𝜑 → (∀𝑥𝑦𝜑 ↔ ∀𝑦𝑥𝜑))
 
23-Sep-2023ichnfim 43501 If in an interchangeability context 𝑥 is not free in 𝜑, the same holds for 𝑦. (Contributed by Wolf Lammen, 6-Aug-2023.) (Revised by AV, 23-Sep-2023.)
((∀𝑦𝑥𝜑 ∧ [𝑥𝑦]𝜑) → ∀𝑥𝑦𝜑)
 
23-Sep-2023currysetALT 34158 Alternate proof of curryset 34154, or more precisely alternate exposal of the same proof. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.) (New usage is discouraged.)
¬ {𝑥 ∣ (𝑥𝑥𝜑)} ∈ 𝑉
 
23-Sep-2023currysetlem3 34157 Lemma for currysetALT 34158. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
𝑋 = {𝑥 ∣ (𝑥𝑥𝜑)}        ¬ 𝑋𝑉
 
23-Sep-2023currysetlem2 34156 Lemma for currysetALT 34158. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
𝑋 = {𝑥 ∣ (𝑥𝑥𝜑)}       (𝑋𝑉 → (𝑋𝑋𝜑))
 
23-Sep-2023currysetlem1 34155 Lemma for currysetALT 34158. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
𝑋 = {𝑥 ∣ (𝑥𝑥𝜑)}       (𝑋𝑉 → (𝑋𝑋 ↔ (𝑋𝑋𝜑)))
 
23-Sep-2023curryset 34154 Curry's paradox in set theory. This can be seen as a generalization of Russell's paradox, which corresponds to the case where 𝜑 is . See alternate exposal of basically the same proof currysetALT 34158. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
¬ {𝑥 ∣ (𝑥𝑥𝜑)} ∈ 𝑉
 
23-Sep-2023currysetlem 34153 Lemma for currysetlem 34153, where it is used with (𝑥𝑥𝜑) substituted for 𝜓. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
({𝑥𝜓} ∈ 𝑉 → ({𝑥𝜓} ∈ {𝑥 ∣ (𝑥𝑥𝜑)} ↔ ({𝑥𝜓} ∈ {𝑥𝜓} → 𝜑)))
 
23-Sep-2023bj-currypara 33792 Curry's paradox. Note that the proof is intuitionistic (use ax-3 8 comes from the unusual definition of the biconditional in set.mm). The paradox comes from the case where 𝜑 is the self-referential sentence "If this sentence is true, then 𝜓", so that one can prove everything. Therefore, a consistent system cannot allow the formation of such self-referential sentences. This has lead to the study of logics rejecting contraction pm2.43 56, such as affine logic and linear logic. (Contributed by BJ, 23-Sep-2023.)
((𝜑 ↔ (𝜑𝜓)) → 𝜓)
 
23-Sep-2023bj-animbi 33791 Conjunction in terms of implication and biconditional. Note that the proof is intuitionistic (use of ax-3 8 comes from the unusual definition of the biconditional in set.mm). (Contributed by BJ, 23-Sep-2023.)
((𝜑𝜓) ↔ (𝜑 ↔ (𝜑𝜓)))
 
23-Sep-2023brsnop 30355 Binary relation for an ordered pair singleton. (Contributed by Thierry Arnoux, 23-Sep-2023.)
((𝐴𝑉𝐵𝑊) → (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ (𝑋 = 𝐴𝑌 = 𝐵)))
 
23-Sep-2023sbal2 2566 Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.) Remove a distinct variable constraint. (Revised by Wolf Lammen, 24-Dec-2022.) (Proof shortened by Wolf Lammen, 23-Sep-2023.)
(¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑))
 
22-Sep-2023sn-axprlem3 38987 axprlem3 5316 using only Tarski's FOL axiom schemes and ax-rep 5181. (Contributed by SN, 22-Sep-2023.)
𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 if-(𝜑, 𝑧 = 𝑎, 𝑧 = 𝑏))
 
22-Sep-2023satfvsucom 32501 The satisfaction predicate as function over wff codes at a successor of ω. (Contributed by AV, 22-Sep-2023.)
𝑆 = (𝑀 Sat 𝐸)       ((𝑀𝑉𝐸𝑊𝑁 ∈ suc ω) → (𝑆𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝑓 (∃𝑣𝑓 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})})‘𝑁))
 
22-Sep-2023satfsucom 32498 The satisfaction predicate for wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 at an element of the successor of ω. (Contributed by AV, 22-Sep-2023.)
((𝑀𝑉𝐸𝑊𝑁 ∈ suc ω) → ((𝑀 Sat 𝐸)‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝑓 (∃𝑣𝑓 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})})‘𝑁))
 
22-Sep-2023cycpmfv3 30684 Values outside of the orbit are unchanged by a cycle. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑𝑋𝐷)    &   (𝜑 → ¬ 𝑋 ∈ ran 𝑊)       (𝜑 → ((𝐶𝑊)‘𝑋) = 𝑋)
 
22-Sep-2023cycpmfv2 30683 Value of a cycle function for the last element of the orbit. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑 → 0 < (♯‘𝑊))    &   (𝜑𝑁 = ((♯‘𝑊) − 1))       (𝜑 → ((𝐶𝑊)‘(𝑊𝑁)) = (𝑊‘0))
 
22-Sep-2023cycpmfv1 30682 Value of a cycle function for any element but the last. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑𝑁 ∈ (0..^((♯‘𝑊) − 1)))       (𝜑 → ((𝐶𝑊)‘(𝑊𝑁)) = (𝑊‘(𝑁 + 1)))
 
22-Sep-2023cycpmfvlem 30681 Lemma for cycpmfv1 30682 and cycpmfv2 30683. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)    &   (𝜑𝑁 ∈ (0..^(♯‘𝑊)))       (𝜑 → ((𝐶𝑊)‘(𝑊𝑁)) = (((𝑊 cyclShift 1) ∘ 𝑊)‘(𝑊𝑁)))
 
22-Sep-2023tocycval 30677 Value of the cycle builder. (Contributed by Thierry Arnoux, 22-Sep-2023.)
𝐶 = (toCyc‘𝐷)       (𝐷𝑉𝐶 = (𝑤 ∈ {𝑢 ∈ Word 𝐷𝑢:dom 𝑢1-1𝐷} ↦ (( I ↾ (𝐷 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ 𝑤))))
 
21-Sep-2023sn-dtru 38989 dtru 5262 without ax-8 2107 or ax-12 2167. (Contributed by SN, 21-Sep-2023.)
¬ ∀𝑥 𝑥 = 𝑦
 
21-Sep-2023sn-axrep5v 38986 A condensed form of axrep5 5187. (Contributed by SN, 21-Sep-2023.)
(∀𝑤𝑥 ∃*𝑧𝜑 → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 𝜑))
 
21-Sep-2023prmsimpcyc 30783 A group of prime order is cyclic if and only if it is simple. This is the first family of finite simple groups. (Contributed by Thierry Arnoux, 21-Sep-2023.)
𝐵 = (Base‘𝐺)       ((♯‘𝐵) ∈ ℙ → (𝐺 ∈ SimpGrp ↔ 𝐺 ∈ CycGrp))
 
21-Sep-2023hashgt23el 13773 A set with more than two elements has at least three different elements. (Contributed by BTernaryTau, 21-Sep-2023.)
((𝑉𝑊 ∧ 2 < (♯‘𝑉)) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑎𝑏𝑎𝑐𝑏𝑐))
 
21-Sep-2023eu6 2652 Alternate definition of the unique existential quantifier df-eu 2647 not using the at-most-one quantifier. (Contributed by NM, 12-Aug-1993.) This used to be the definition of the unique existential quantifier, while df-eu 2647 was then proved as dfeu 2674. (Revised by BJ, 30-Sep-2022.) (Proof shortened by Wolf Lammen, 3-Jan-2023.) Remove use of ax-11 2151. (Revised by SN, 21-Sep-2023.)
(∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
 
20-Sep-2023fmla1 32531 The valid Godel formulas of height 1 is the set of all formulas of the form (𝑎𝑔𝑏) and 𝑔𝑘𝑎 with atoms 𝑎, 𝑏 of the form 𝑥𝑦. (Contributed by AV, 20-Sep-2023.)
(Fmla‘1o) = (({∅} × (ω × ω)) ∪ {𝑥 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ∃𝑘 ∈ ω (∃𝑙 ∈ ω 𝑥 = ((𝑖𝑔𝑗)⊼𝑔(𝑘𝑔𝑙)) ∨ 𝑥 = ∀𝑔𝑘(𝑖𝑔𝑗))})
 
20-Sep-2023fmlasuc 32530 The valid Godel formulas of height (𝑁 + 1), expressed by the valid Godel formulas of height 𝑁. (Contributed by AV, 20-Sep-2023.)
(𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘𝑁)(∃𝑣 ∈ (Fmla‘𝑁)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}))
 
20-Sep-2023fmla 32525 The set of all valid Godel formulas. (Contributed by AV, 20-Sep-2023.)
(Fmla‘ω) = 𝑛 ∈ ω (Fmla‘𝑛)
 
20-Sep-2023ineq1 4178 Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.) (Proof shortened by SN, 20-Sep-2023.)
(𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
 
20-Sep-2023nfsab1 2805 Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove use of ax-12 2167. (Revised by SN, 20-Sep-2023.)
𝑥 𝑦 ∈ {𝑥𝜑}
 
19-Sep-2023fmlafvel 32529 A class is a valid Godel formula of height 𝑁 iff it is the first component of a member of the value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation at 𝑁. (Contributed by AV, 19-Sep-2023.)
(𝑁 ∈ ω → (𝐹 ∈ (Fmla‘𝑁) ↔ ⟨𝐹, ∅⟩ ∈ ((∅ Sat ∅)‘𝑁)))
 
19-Sep-2023satf0n0 32522 The value of the satisfaction predicate as function over wff codes in the empty model and the empty binary relation does not contain the empty set. (Contributed by AV, 19-Sep-2023.)
(𝑁 ∈ ω → ∅ ∉ ((∅ Sat ∅)‘𝑁))
 
19-Sep-2023satf0op 32521 An element of a value of the satisfaction predicate as function over wff codes in the empty model and the empty binary relation expressed as ordered pair. (Contributed by AV, 19-Sep-2023.)
𝑆 = (∅ Sat ∅)       (𝑁 ∈ ω → (𝑋 ∈ (𝑆𝑁) ↔ ∃𝑥(𝑋 = ⟨𝑥, ∅⟩ ∧ ⟨𝑥, ∅⟩ ∈ (𝑆𝑁))))
 
19-Sep-2023satf0suc 32520 The value of the satisfaction predicate as function over wff codes in the empty model and the empty binary relation at a successor. (Contributed by AV, 19-Sep-2023.)
𝑆 = (∅ Sat ∅)       (𝑁 ∈ ω → (𝑆‘suc 𝑁) = ((𝑆𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))}))
 
19-Sep-2023satf0suclem 32519 Lemma for satf0suc 32520, sat1el2xp 32523 and fmlasuc0 32528. (Contributed by AV, 19-Sep-2023.)
((𝑋𝑈𝑌𝑉𝑍𝑊) → {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢𝑋 (∃𝑣𝑌 𝑥 = 𝐵 ∨ ∃𝑤𝑍 𝑥 = 𝐶))} ∈ V)
 
19-Sep-2023cyc3co2 30709 Represent a 3-cycle as a composition of two 2-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)    &    · = (+g𝑆)       (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) = ((𝐶‘⟨“𝐼𝐾”⟩) · (𝐶‘⟨“𝐼𝐽”⟩)))
 
19-Sep-2023cyc3fv3 30708 Function value of a 3-cycle at the third point. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐾) = 𝐼)
 
19-Sep-2023cyc3fv2 30707 Function value of a 3-cycle at the second point. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐽) = 𝐾)
 
19-Sep-2023cyc3fv1 30706 Function value of a 3-cycle at the first point. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ((𝐶‘⟨“𝐼𝐽𝐾”⟩)‘𝐼) = 𝐽)
 
19-Sep-2023cycpm3cl2 30705 Closure of the 3-cycles in the class of 3-cycles. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (𝐶 “ (♯ “ {3})))
 
19-Sep-2023cycpm3cl 30704 Closure of the 3-cycles in the permutations. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → (𝐶‘⟨“𝐼𝐽𝐾”⟩) ∈ (Base‘𝑆))
 
19-Sep-2023cyc2fvx 30703 Function value of a 2-cycle outside of its orbit. (Contributed by Thierry Arnoux, 19-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   𝑆 = (SymGrp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ((𝐶‘⟨“𝐼𝐽”⟩)‘𝐾) = 𝐾)
 
19-Sep-2023df-tocyc 30676 Define a convenience permutation cycle builder. Given a list of elements to be cycled, in the form of a word, this function produces the corresponding permutation cycle. See definition in [Lang] p. 30. (Contributed by Thierry Arnoux, 19-Sep-2023.)
toCyc = (𝑑 ∈ V ↦ (𝑤 ∈ {𝑢 ∈ Word 𝑑𝑢:dom 𝑢1-1𝑑} ↦ (( I ↾ (𝑑 ∖ ran 𝑤)) ∪ ((𝑤 cyclShift 1) ∘ 𝑤))))
 
19-Sep-2023cshwrnid 30562 Cyclically shifting a word preserves its range. (Contributed by Thierry Arnoux, 19-Sep-2023.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → ran (𝑊 cyclShift 𝑁) = ran 𝑊)
 
19-Sep-2023cshw1s2 30561 Cyclically shifting a length 2 word swaps its symbols. (Contributed by Thierry Arnoux, 19-Sep-2023.)
((𝐴𝑉𝐵𝑉) → (⟨“𝐴𝐵”⟩ cyclShift 1) = ⟨“𝐵𝐴”⟩)
 
19-Sep-2023s3f1 30550 Conditions for a length 3 string to be a one-to-one function. (Contributed by Thierry Arnoux, 19-Sep-2023.)
(𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)    &   (𝜑𝐼𝐽)    &   (𝜑𝐽𝐾)    &   (𝜑𝐾𝐼)       (𝜑 → ⟨“𝐼𝐽𝐾”⟩:dom ⟨“𝐼𝐽𝐾”⟩–1-1𝐷)
 
19-Sep-2023s3rn 30549 Range of a length 3 string. (Contributed by Thierry Arnoux, 19-Sep-2023.)
(𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐾𝐷)       (𝜑 → ran ⟨“𝐼𝐽𝐾”⟩ = {𝐼, 𝐽, 𝐾})
 
19-Sep-2023s2f1 30548 Conditions for a length 2 string to be a one-to-one function. (Contributed by Thierry Arnoux, 19-Sep-2023.)
(𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)    &   (𝜑𝐼𝐽)       (𝜑 → ⟨“𝐼𝐽”⟩:dom ⟨“𝐼𝐽”⟩–1-1𝐷)
 
19-Sep-2023s2rn 30547 Range of a length 2 string. (Contributed by Thierry Arnoux, 19-Sep-2023.)
(𝜑𝐼𝐷)    &   (𝜑𝐽𝐷)       (𝜑 → ran ⟨“𝐼𝐽”⟩ = {𝐼, 𝐽})
 
19-Sep-2023pfx1s2 30542 The prefix of length 1 of a length 2 word. (Contributed by Thierry Arnoux, 19-Sep-2023.)
((𝐴𝑉𝐵𝑉) → (⟨“𝐴𝐵”⟩ prefix 1) = ⟨“𝐴”⟩)
 
19-Sep-2023fnimatp 30351 The image of a triplet under a function. (Contributed by Thierry Arnoux, 19-Sep-2023.)
(𝜑𝐹 Fn 𝐷)    &   (𝜑𝐴𝐷)    &   (𝜑𝐵𝐷)    &   (𝜑𝐶𝐷)       (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹𝐴), (𝐹𝐵), (𝐹𝐶)})
 
18-Sep-2023sn-el 38988 A version of el 5261 with an inner existential quantifier on 𝑥, which avoids ax-7 2006 and ax-8 2107. (Contributed by SN, 18-Sep-2023.)
𝑦𝑥 𝑥𝑦
 
18-Sep-2023fmlasuc0 32528 The valid Godel formulas of height (𝑁 + 1). (Contributed by AV, 18-Sep-2023.)
(𝑁 ∈ ω → (Fmla‘suc 𝑁) = ((Fmla‘𝑁) ∪ {𝑥 ∣ ∃𝑢 ∈ ((∅ Sat ∅)‘𝑁)(∃𝑣 ∈ ((∅ Sat ∅)‘𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))}))
 
18-Sep-2023goaleq12d 32495 Equality of the "Godel-set of universal quantification". (Contributed by AV, 18-Sep-2023.)
(𝜑𝑀 = 𝑁)    &   (𝜑𝐴 = 𝐵)       (𝜑 → ∀𝑔𝑀𝐴 = ∀𝑔𝑁𝐵)
 
18-Sep-2023freshmansdream 30786 For a prime number 𝑃, if 𝑋 and 𝑌 are members of a commutative ring 𝑅 of characteristic 𝑃, then ((𝑋 + 𝑌)↑𝑃) = ((𝑋𝑃) + (𝑌𝑃)). This theorem is sometimes referred to as "the freshman's dream" . (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.g‘(mulGrp‘𝑅))    &   𝑃 = (chr‘𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑃 (𝑋 + 𝑌)) = ((𝑃 𝑋) + (𝑃 𝑌)))
 
18-Sep-2023altgnsg 30718 The alternating group (pmEven‘𝐷) is a normal subgroup of the symmetric group. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝑆 = (SymGrp‘𝐷)       (𝐷 ∈ Fin → (pmEven‘𝐷) ∈ (NrmSGrp‘𝑆))
 
18-Sep-2023evpmid 30717 The identity is an even permutation. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝑆 = (SymGrp‘𝐷)       (𝐷 ∈ Fin → ( I ↾ 𝐷) ∈ (pmEven‘𝐷))
 
18-Sep-2023tocycfv 30678 Function value of a permutation cycle built from a word. (Contributed by Thierry Arnoux, 18-Sep-2023.)
𝐶 = (toCyc‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝐷)    &   (𝜑𝑊:dom 𝑊1-1𝐷)       (𝜑 → (𝐶𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ 𝑊)))
 
18-Sep-2023axrep6 5188 A condensed form of ax-rep 5181. (Contributed by SN, 18-Sep-2023.)
(∀𝑤∃*𝑧𝜑 → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤𝑥 𝜑))
 
17-Sep-2023sat1el2xp 32523 The first component of an element of the value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation is a member of a doubled Cartesian product. (Contributed by AV, 17-Sep-2023.)
(𝑁 ∈ ω → ∀𝑤 ∈ ((∅ Sat ∅)‘𝑁)∃𝑎𝑏(1st𝑤) ∈ (ω × (𝑎 × 𝑏)))
 
17-Sep-2023prmdvdsbc 30458 Condition for a prime number to divide a binomial coefficient. (Contributed by Thierry Arnoux, 17-Sep-2023.)
((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∥ (𝑃C𝑁))
 
17-Sep-2023dvdszzq 30457 Divisibility for an integer quotient. (Contributed by Thierry Arnoux, 17-Sep-2023.)
𝑁 = (𝐴 / 𝐵)    &   (𝜑𝑃 ∈ ℙ)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝑃𝐴)    &   (𝜑 → ¬ 𝑃𝐵)       (𝜑𝑃𝑁)
 
17-Sep-2023opabex3rd 7656 Existence of an ordered pair abstraction if the second components are elements of a set. (Contributed by AV, 17-Sep-2023.)
(𝜑𝐴 ∈ V)    &   ((𝜑𝑦𝐴) → {𝑥𝜓} ∈ V)       (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝜓)} ∈ V)
 
16-Sep-2023goelel3xp 32492 A "Godel-set of membership" is a member of a doubled Cartesian product. (Contributed by AV, 16-Sep-2023.)
((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) ∈ (ω × (ω × ω)))
 
16-Sep-2023equvini 2469 A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require 𝑧 to be distinct from 𝑥 and 𝑦. See equvinv 2027 for a shorter proof requiring fewer axioms when 𝑧 is required to be distinct from 𝑥 and 𝑦. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 16-Sep-2023.)
(𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑧 = 𝑦))
 
15-Sep-2023fmla0xp 32527 The valid Godel formulas of height 0 is the set of all formulas of the form vi vj ("Godel-set of membership") coded as ⟨∅, ⟨𝑖, 𝑗⟩⟩. (Contributed by AV, 15-Sep-2023.)
(Fmla‘∅) = ({∅} × (ω × ω))
 
15-Sep-2023fmlafv 32524 The valid Godel formulas of height 𝑁 is the domain of the value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation at 𝑁. (Contributed by AV, 15-Sep-2023.)
(𝑁 ∈ suc ω → (Fmla‘𝑁) = dom ((∅ Sat ∅)‘𝑁))
 
15-Sep-2023goel 32491 A "Godel-set of membership". The variables are identified by their indices (which are natural numbers), and the membership vi vj is coded as ⟨∅, ⟨𝑖, 𝑗⟩⟩. (Contributed by AV, 15-Sep-2023.)
((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
 
15-Sep-2023exdifsn 32242 There exists an element in a class excluding a singleton if and only if there exists an element in the original class not equal to the singleton element. (Contributed by BTernaryTau, 15-Sep-2023.)
(∃𝑥 𝑥 ∈ (𝐴 ∖ {𝐵}) ↔ ∃𝑥𝐴 𝑥𝐵)
 
15-Sep-2023omsucelsucb 8083 Membership is inherited by successors for natural numbers. (Contributed by AV, 15-Sep-2023.)
(𝑁 ∈ ω ↔ suc 𝑁 ∈ suc ω)
 
15-Sep-2023suppcofnd 7860 The support of the composition of two functions. (Contributed by SN, 15-Sep-2023.)
(𝜑𝑍𝑈)    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐴𝑉)    &   (𝜑𝐺 Fn 𝐵)    &   (𝜑𝐵𝑊)       (𝜑 → ((𝐹𝐺) supp 𝑍) = {𝑥𝐵 ∣ ((𝐺𝑥) ∈ 𝐴 ∧ (𝐹‘(𝐺𝑥)) ≠ 𝑍)})
 
15-Sep-2023suppco 7859 The support of the composition of two functions is the inverse image by the inner function of the support of the outer function. (Contributed by AV, 30-May-2019.) Extract this statement from the proof of supp0cosupp0 7861. (Revised by SN, 15-Sep-2023.)
((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))
 
14-Sep-2023fmla0 32526 The valid Godel formulas of height 0 is the set of all formulas of the form vi vj ("Godel-set of membership") coded as ⟨∅, ⟨𝑖, 𝑗⟩⟩. (Contributed by AV, 14-Sep-2023.)
(Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)}
 
14-Sep-2023satf00 32518 The value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation at . (Contributed by AV, 14-Sep-2023.)
((∅ Sat ∅)‘∅) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}
 
14-Sep-2023satf0sucom 32517 The satisfaction predicate as function over wff codes in the empty model with an empty binary relation at a successor of ω. (Contributed by AV, 14-Sep-2023.)
(𝑁 ∈ suc ω → ((∅ Sat ∅)‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))})‘𝑁))
 
14-Sep-2023satf0 32516 The satisfaction predicate as function over wff codes in the empty model with an empty binary relation. (Contributed by AV, 14-Sep-2023.)
(∅ Sat ∅) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢𝑓 (∃𝑣𝑓 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗))}) ↾ suc ω)
 
14-Sep-2023satf 32497 The satisfaction predicate as function over wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 14-Sep-2023.)
((𝑀𝑉𝐸𝑊) → (𝑀 Sat 𝐸) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝑓 (∃𝑣𝑓 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})}) ↾ suc ω))
 
12-Sep-2023mhpinvcl 20267 Homogeneous polynomials are closed under taking the opposite. (Contributed by SN, 12-Sep-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝑀 = (invg𝑃)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Grp)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋 ∈ (𝐻𝑁))       (𝜑 → (𝑀𝑋) ∈ (𝐻𝑁))
 
12-Sep-2023mhp0cl 20265 The zero polynomial is homogeneous. (Contributed by SN, 12-Sep-2023.)
𝐻 = (𝐼 mHomP 𝑅)    &    0 = (0g𝑅)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Grp)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐷 × { 0 }) ∈ (𝐻𝑁))
 
11-Sep-2023frr3 33043 Law of general founded recursion, part three. Finally, we show that 𝐹 is unique. We do this by showing that any function 𝐻 with the same properties we proved of 𝐹 in frr1 33041 and frr2 33042 is identical to 𝐹. (Contributed by Scott Fenton, 11-Sep-2023.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)       (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐻𝑧) = (𝑧𝐺(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝐹 = 𝐻)
 
11-Sep-2023frr2 33042 Law of general founded recursion, part two. Now we establish the value of 𝐹 within 𝐴. (Contributed by Scott Fenton, 11-Sep-2023.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)       (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ 𝑋𝐴) → (𝐹𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
 
11-Sep-2023frr1 33041 Law of general founded recursion, part one. This may look like a restatement of the founded partial recursion theorems dropping the partial ordering requirement, but that change mandates that we use the Axiom of Infinity. (Contributed by Scott Fenton, 11-Sep-2023.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)       ((𝑅 Fr 𝐴𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
 
11-Sep-2023frrlem16 33040 Lemma for general founded recursion. Establish a subset relationship. (Contributed by Scott Fenton, 11-Sep-2023.)
(((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ 𝑧𝐴) → ∀𝑤 ∈ TrPred (𝑅, 𝐴, 𝑧)Pred(𝑅, 𝐴, 𝑤) ⊆ TrPred(𝑅, 𝐴, 𝑧))
 
11-Sep-2023frrlem15 33039 Lemma for general founded recursion. Two acceptable functions are compatible. (Contributed by Scott Fenton, 11-Sep-2023.)
𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}    &   𝐹 = frecs(𝑅, 𝐴, 𝐺)       (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
 
11-Sep-2023fpr3 33038 Law of founded partial recursion, part three. Finally, we show that 𝐹 is unique. We do this by showing that any function 𝐻 with the same properties we proved of 𝐹 in fpr1 33036 and fpr2 33037 is identical to 𝐹. (Contributed by Scott Fenton, 11-Sep-2023.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)       (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐻𝑧) = (𝑧𝐺(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝐹 = 𝐻)
 
11-Sep-2023fpr2 33037 Law of founded partial recursion, part two. Now we establish the value of 𝐹 within 𝐴. (Contributed by Scott Fenton, 11-Sep-2023.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)       (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑋𝐴) → (𝐹𝑋) = (𝑋𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
 
11-Sep-2023fpr1 33036 Law of founded partial recursion, part one. This development mostly follows the well-founded recursion development. Note that by requiring a partial ordering we can avoid using the Axiom of Infinity. (Contributed by Scott Fenton, 11-Sep-2023.)
𝐹 = frecs(𝑅, 𝐴, 𝐺)       ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
 
11-Sep-2023fprlem2 33035 Lemma for founded partial recursion. Establish a subset relationship. (Contributed by Scott Fenton, 11-Sep-2023.)
(((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑧𝐴) → ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)Pred(𝑅, 𝐴, 𝑤) ⊆ Pred(𝑅, 𝐴, 𝑧))
 
11-Sep-2023fprlem1 33034 Lemma for founded partial recursion. Two acceptable functions are compatible. (Contributed by Scott Fenton, 11-Sep-2023.)
𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}    &   𝐹 = frecs(𝑅, 𝐴, 𝐺)       (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝑔𝐵𝐵)) → ((𝑥𝑔𝑢𝑥𝑣) → 𝑢 = 𝑣))
 
11-Sep-2023cbval2 2423 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-Sep-2023.)
𝑧𝜑    &   𝑤𝜑    &   𝑥𝜓    &   𝑦𝜓    &   ((𝑥 = 𝑧𝑦 = 𝑤) → (𝜑𝜓))       (∀𝑥𝑦𝜑 ↔ ∀𝑧𝑤𝜓)
 
11-Sep-2023cbvexv 2410 Rule used to change bound variables, using implicit substitution. See cbvexvw 2035 for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-10 2136, shorten. (Revised by Wolf Lammen, 11-Sep-2023.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)
 
11-Sep-2023cbvalv 2409 Rule used to change bound variables, using implicit substitution. See cbvalvw 2034 for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-10 2136, shorten. (Revised by Wolf Lammen, 11-Sep-2023.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)
 
10-Sep-2023hashunsnggt 13743 The size of a set is greater than a nonnegative integer N if and only if the size of the union of that set with a disjoint singleton is greater than N + 1. (Contributed by BTernaryTau, 10-Sep-2023.)
(((𝐴𝑉𝐵𝑊𝑁 ∈ ℕ0) ∧ ¬ 𝐵𝐴) → (𝑁 < (♯‘𝐴) ↔ (𝑁 + 1) < (♯‘(𝐴 ∪ {𝐵}))))
 
10-Sep-2023cbv2 2414 Rule used to change bound variables, using implicit substitution. See cbv2w 2348 with disjoint variable conditions, not depending on ax-13 2381. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) Format hypotheses to common style, avoid ax-10 2136. (Revised by Wolf Lammen, 10-Sep-2023.)
𝑥𝜑    &   𝑦𝜑    &   (𝜑 → Ⅎ𝑦𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
 
9-Sep-2023mathbox 30146 (This theorem is a dummy placeholder for these guidelines. The label of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.)

A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of set.mm.

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2. If at all possible, please use only nullary class constants for new definitions, for example as in df-div 11286.

3. Each $p and $a statement must be immediately preceded with the comment that will be shown on its web page description. The Metamath program "MM> WRITE SOURCE set.mm / REWRAP" command will take care of indentation conventions and line wrapping.

4. All mathbox content will be on public display and should hopefully reflect the overall quality of the website.

5. Mathboxes must be independent from one another (checked by "verify markup *"). If you need a theorem from another mathbox, typically it is moved to the main part of set.mm. New users should consult with more experienced users before doing this.

6. If a contributor is no longer active, we will continue the usual maintenance edits. As time goes on, often theorems will be moved to main or removed in favor of similar replacements. But we are also willing to maintain mathboxes in place, as work by others from years ago may form the foundation of future work; you could even argue that all of mathematics is like that.

7. For theorems of importance (for example, a Metamath 100 theorem or a dependency of one), we prefer to eventually move them out of mathboxes (although a mathbox is perfectly appropriate as proofs are being developed and refined). (Contributed by NM, 20-Feb-2007.) (Revised by the Metamath team, 9-Sep-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

𝜑       𝜑
 
9-Sep-2023hashunsngx 13742 The size of the union of a set with a disjoint singleton is the extended real addition of the size of the set and 1, analogous to hashunsng 13741. (Contributed by BTernaryTau, 9-Sep-2023.)
((𝐴𝑉𝐵𝑊) → (¬ 𝐵𝐴 → (♯‘(𝐴 ∪ {𝐵})) = ((♯‘𝐴) +𝑒 1)))
 
7-Sep-2023onadju 9607 The cardinal and ordinal sums are always equinumerous. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Jim Kingdon, 7-Sep-2023.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ≈ (𝐴𝐵))
 
6-Sep-2023frlmvscadiccat 39023 Scalar multiplication distributes over concatenation. (Contributed by SN, 6-Sep-2023.)
𝑊 = (𝐾 freeLMod (0..^𝐿))    &   𝑋 = (𝐾 freeLMod (0..^𝑀))    &   𝑌 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝐶 = (Base‘𝑋)    &   𝐷 = (Base‘𝑌)    &   (𝜑𝐾 ∈ Ring)    &   (𝜑 → (𝑀 + 𝑁) = 𝐿)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑈𝐶)    &   (𝜑𝑉𝐷)    &   𝑂 = ( ·𝑠𝑊)    &    = ( ·𝑠𝑋)    &    · = ( ·𝑠𝑌)    &   𝑆 = (Base‘𝐾)    &   (𝜑𝐴𝑆)       (𝜑 → (𝐴𝑂(𝑈 ++ 𝑉)) = ((𝐴 𝑈) ++ (𝐴 · 𝑉)))
 
6-Sep-2023ccatcan2d 39005 Cancellation law for concatenation. (Contributed by SN, 6-Sep-2023.)
(𝜑𝐴 ∈ Word 𝑉)    &   (𝜑𝐵 ∈ Word 𝑉)    &   (𝜑𝐶 ∈ Word 𝑉)       (𝜑 → ((𝐴 ++ 𝐶) = (𝐵 ++ 𝐶) ↔ 𝐴 = 𝐵))
 
6-Sep-2023djuexb 9326 The disjoint union of two classes is a set iff both classes are sets. (Contributed by Jim Kingdon, 6-Sep-2023.)
((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
 
5-Sep-2023dtru 5262 At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Note that we may not substitute the same variable for both 𝑥 and 𝑦 (as indicated by the distinct variable requirement), for otherwise we would contradict stdpc6 2026.

This theorem is proved directly from set theory axioms (no set theory definitions) and does not use ax-ext 2790 or ax-sep 5194. See dtruALT 5279 for a shorter proof using these axioms.

The proof makes use of dummy variables 𝑧 and 𝑤 which do not appear in the final theorem. They must be distinct from each other and from 𝑥 and 𝑦. In other words, if we were to substitute 𝑥 for 𝑧 throughout the proof, the proof would fail. (Contributed by NM, 7-Nov-2006.) Avoid ax-13 2381. (Revised by Gino Giotto, 5-Sep-2023.)

¬ ∀𝑥 𝑥 = 𝑦
 
4-Sep-2023sbbib 2371 Reversal of substitution. (Contributed by AV, 6-Aug-2023.) (Proof shortened by Wolf Lammen, 4-Sep-2023.)
𝑦𝜑    &   𝑥𝜓       (∀𝑦([𝑦 / 𝑥]𝜑𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓))
 
4-Sep-2023sb5 2267 Alternate definition of substitution when variables are disjoint. Similar to Theorem 6.1 of [Quine] p. 40. The implication "to the right" is sb1v 2086 and even needs no disjoint variable condition, see sb1 2496. Theorem sb5f 2531 replaces the disjoint variable condition with a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) Shorten sb56 2268. (Revised by Wolf Lammen, 4-Sep-2023.)
([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑))
 
3-Sep-2023reppncan 39101 Cancellation law for mixed addition and real subtraction. Compare ppncan 10916. (Contributed by SN, 3-Sep-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) + (𝐵 𝐶)) = (𝐴 + 𝐵))
 
3-Sep-2023dfsb7 2276 An alternate definition of proper substitution df-sb 2061. By introducing a dummy variable 𝑦 in the definiens, we are able to eliminate any distinct variable restrictions among the variables 𝑡, 𝑥, and 𝜑 of the definiendum. No distinct variable conflicts arise because 𝑦 effectively insulates 𝑡 from 𝑥. To achieve this, we use a chain of two substitutions in the form of sb5 2267, first 𝑦 for 𝑥 then 𝑡 for 𝑦. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2797. Theorem sb7h 2562 provides a version where 𝜑 and 𝑦 don't have to be distinct. (Contributed by NM, 28-Jan-2004.) Revise df-sb 2061. (Revised by BJ, 25-Dec-2020.) (Proof shortened by Wolf Lammen, 3-Sep-2023.)
([𝑡 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
 
3-Sep-2023exlimdd 2210 Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof shortened by Wolf Lammen, 3-Sep-2023.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → ∃𝑥𝜓)    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)
 
3-Sep-2023exlimimdd 2209 Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020.) Shorten exlimdd 2210. (Revised by Wolf Lammen, 3-Sep-2023.)
𝑥𝜑    &   𝑥𝜒    &   (𝜑 → ∃𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑𝜒)
 
2-Sep-2023vextru 2803 Every setvar is a member of {𝑥 ∣ ⊤}, which is therefore "a" universal class. Once class extensionality dfcleq 2812 is available, we can say "the" universal class (see df-v 3494). This is sbtru 2063 expressed using class abstractions. (Contributed by BJ, 2-Sep-2023.)
𝑦 ∈ {𝑥 ∣ ⊤}
 
2-Sep-2023equsb3r 2101 Substitution applied to the atomic wff with equality. Variant of equsb3 2100. (Contributed by AV, 29-Jul-2023.) (Proof shortened by Wolf Lammen, 2-Sep-2023.)
([𝑦 / 𝑥]𝑧 = 𝑥𝑧 = 𝑦)
 
2-Sep-2023sbtru 2063 The result of substituting in the truth constant "true" is true. (Contributed by BJ, 2-Sep-2023.)
[𝑦 / 𝑥]⊤
 
1-Sep-2023frlmfzowrdb 39021 The vectors of a module with indices 0 to 𝑁 − 1 are the length- 𝑁 words over the scalars of the module. (Contributed by SN, 1-Sep-2023.)
𝑊 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       ((𝐾𝑉𝑁 ∈ ℕ0) → (𝑋𝐵 ↔ (𝑋 ∈ Word 𝑆 ∧ (♯‘𝑋) = 𝑁)))
 
1-Sep-2023frlmfzolen 39020 The dimension of a vector of a module with indices from 0 to 𝑁 − 1. (Contributed by SN, 1-Sep-2023.)
𝑊 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       ((𝑁 ∈ ℕ0𝑋𝐵) → (♯‘𝑋) = 𝑁)
 
1-Sep-2023djudom1 9596 Ordering law for cardinal addition. Exercise 4.56(f) of [Mendelson] p. 258. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 1-Sep-2023.)
((𝐴𝐵𝐶𝑉) → (𝐴𝐶) ≼ (𝐵𝐶))
 
31-Aug-2023frlmfzoccat 39022 The concatenation of two vectors of dimension 𝑁 and 𝑀 forms a vector of dimension 𝑁 + 𝑀. (Contributed by SN, 31-Aug-2023.)
𝑊 = (𝐾 freeLMod (0..^𝐿))    &   𝑋 = (𝐾 freeLMod (0..^𝑀))    &   𝑌 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝐶 = (Base‘𝑋)    &   𝐷 = (Base‘𝑌)    &   (𝜑𝐾 ∈ Ring)    &   (𝜑 → (𝑀 + 𝑁) = 𝐿)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑈𝐶)    &   (𝜑𝑉𝐷)       (𝜑 → (𝑈 ++ 𝑉) ∈ 𝐵)
 
31-Aug-2023frlmfzowrd 39019 A vector of a module with indices from 0 to 𝑁 − 1 is a word over the scalars of the module. (Contributed by SN, 31-Aug-2023.)
𝑊 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       (𝑋𝐵𝑋 ∈ Word 𝑆)
 
31-Aug-2023frlmfzwrd 39018 A vector of a module with indices from 0 to 𝑁 is a word over the scalars of the module. (Contributed by SN, 31-Aug-2023.)
𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       (𝑋𝐵𝑋 ∈ Word 𝑆)
 
31-Aug-2023frlmfielbas 39017 The vectors of a finite free module are the functions from 𝐼 to 𝑁. (Contributed by SN, 31-Aug-2023.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑁 = (Base‘𝑅)    &   𝐵 = (Base‘𝐹)       ((𝑅𝑉𝐼 ∈ Fin) → (𝑋𝐵𝑋:𝐼𝑁))
 
30-Aug-2023ichal 43504 Move a universal quantifier inside interchangability. (Contributed by SN, 30-Aug-2023.)
(∀𝑥[𝑎𝑏]𝜑 → [𝑎𝑏]∀𝑥𝜑)
 
30-Aug-2023ichn 43503 Negation does not affect interchangability. (Contributed by SN, 30-Aug-2023.)
([𝑎𝑏]𝜑 ↔ [𝑎𝑏] ¬ 𝜑)
 
28-Aug-2023ichreuopeq 43512 If the setvar variables are interchangeable in a wff, and there is a unique ordered pair fulfilling the wff, then both setvar variables must be equal. (Contributed by AV, 28-Aug-2023.)
([𝑎𝑏]𝜑 → (∃!𝑝 ∈ (𝑋 × 𝑋)∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃𝑎𝑏(𝑎 = 𝑏𝜑)))
 
28-Aug-2023suppofssd 7856 Condition for the support of a function operation to be a subset of the union of the supports of the left and right function terms. (Contributed by Steven Nguyen, 28-Aug-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑 → (𝑍𝑋𝑍) = 𝑍)       (𝜑 → ((𝐹f 𝑋𝐺) supp 𝑍) ⊆ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))
 
28-Aug-2023vex 3495 All setvar variables are sets (see isset 3504). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 26-May-1993.) Remove use of ax-12 2167. (Revised by SN, 28-Aug-2023.)
𝑥 ∈ V
 
27-Aug-2023ichnreuop 43511 If the setvar variables are interchangeable in a wff, there is never a unique ordered pair with different components fulfilling the wff (because if 𝑎, 𝑏 fulfils the wff, then also 𝑏, 𝑎 fulfils the wff). (Contributed by AV, 27-Aug-2023.)
([𝑎𝑏]𝜑 → ¬ ∃!𝑝 ∈ (𝑋 × 𝑋)∃𝑎𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑎𝑏𝜑))

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 http://us2.metamath.org:88/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: http://us2.metamath.org:88/ocat/mmj2/mmj2.zip http://us2.metamath.org:88/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 us2.metamath.org:8888 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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