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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 25-Oct-2024 at 5:19 AM ET.
Recent Additions to the Metamath Proof Explorer   Notes (last updated 7-Dec-2020 )
DateLabelDescription
Theorem
 
24-Oct-2024eqtr3 2760 A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Wolf Lammen, 24-Oct-2024.)
((𝐴 = 𝐶 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐵)
 
24-Oct-2024eqtr2 2758 A transitive law for class equality. (Contributed by NM, 20-May-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 24-Oct-2024.)
((𝐴 = 𝐵 ∧ 𝐴 = 𝐶) → 𝐵 = 𝐶)
 
23-Oct-2024sticksstones19 39798 Extend sticks and stones to finite sets, bijective builder. (Contributed by metakunt, 23-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}    &   (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆)    &   𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))    &   𝐺 = (𝑏 ∈ 𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))))    ⇒   (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
 
23-Oct-2024sticksstones18 39797 Extend sticks and stones to finite sets, bijective builder. (Contributed by metakunt, 23-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}    &   (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆)    &   𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))))    ⇒   (𝜑 → 𝐹:𝐴⟶𝐵)
 
23-Oct-2024sticksstones17 39796 Extend sticks and stones to finite sets, bijective builder. (Contributed by metakunt, 23-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧ Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}    &   (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆)    &   𝐺 = (𝑏 ∈ 𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))))    ⇒   (𝜑 → 𝐺:𝐵⟶𝐴)
 
23-Oct-2024eqeq12 2751 Equality relationship among four classes. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Wolf Lammen, 23-Oct-2024.)
((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
 
23-Oct-2024eqeq12d 2750 A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Oct-2024.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → 𝐶 = 𝐷)    ⇒   (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
 
23-Oct-2024eqeqan12d 2748 A useful inference for substituting definitions into an equality. See also eqeqan12dALT 2756. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) Shorten other proofs. (Revised by Wolf Lammen, 23-Oct-2024.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜓 → 𝐶 = 𝐷)    ⇒   ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
 
20-Oct-2024sticksstones16 39795 Sticks and stones with collapsed definitions for positive integers. (Contributed by metakunt, 20-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ)    &   𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧ Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}    ⇒   (𝜑 → (♯‘𝐴) = ((𝑁 + (𝐾 − 1))C(𝐾 − 1)))
 
16-Oct-2024thincciso 45957 Two thin categories are isomorphic iff the induced preorders are order-isomorphic. Example 3.26(2) of [Adamek] p. 33. (Contributed by Zhi Wang, 16-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   𝑅 = (Base‘𝑋)    &   𝑆 = (Base‘𝑌)    &   𝐻 = (Hom ‘𝑋)    &   𝐽 = (Hom ‘𝑌)    &   (𝜑 → 𝑈 ∈ 𝑉)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑋 ∈ ThinCat)    &   (𝜑 → 𝑌 ∈ ThinCat)    ⇒   (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑓(∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)))
 
16-Oct-2024bj-elabd2ALT 34807 Alternate proof of elabd2 3572 bypassing elab6g 3571 (and using sbiedvw 2100 instead of the ∀𝑥(𝑥 = 𝑦 → 𝜓) idiom). (Contributed by BJ, 16-Oct-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 = {𝑥 ∣ 𝜓})    &   ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (𝐴 ∈ 𝐵 ↔ 𝜒))
 
16-Oct-2024omsinds 7654 Strong (or "total") induction principle over the finite ordinals. (Contributed by Scott Fenton, 17-Jul-2015.) (Proof shortened by BJ, 16-Oct-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   (𝑥 ∈ ω → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑))    ⇒   (𝐴 ∈ ω → 𝜒)
 
16-Oct-2024predon 7558 The predecessor of an ordinal under E and On is itself. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof shortened by BJ, 16-Oct-2024.)
(𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)
 
16-Oct-2024elpred 6165 Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011.) (Proof shortened by BJ, 16-Oct-2024.)
𝑌 ∈ V    ⇒   (𝑋 ∈ 𝐷 → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))
 
16-Oct-2024elpredim 6164 Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.) (Proof shortened by BJ, 16-Oct-2024.)
𝑋 ∈ V    ⇒   (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)
 
16-Oct-2024elpredimg 6163 Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.) Generalize to closed form. (Revised by BJ, 16-Oct-2024.)
((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)) → 𝑌𝑅𝑋)
 
16-Oct-2024elpredg 6162 Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.) (Proof shortened by BJ, 16-Oct-2024.)
((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐴) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑌𝑅𝑋))
 
16-Oct-2024elpredgg 6161 Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.) Generalize to closed form. (Revised by BJ, 16-Oct-2024.)
((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))
 
16-Oct-2024epin 5952 Any set is equal to its preimage under the converse membership relation. (Contributed by Mario Carneiro, 9-Mar-2013.) Put in closed form. (Revised by BJ, 16-Oct-2024.)
(𝐴 ∈ 𝑉 → (◡ E “ {𝐴}) = 𝐴)
 
16-Oct-2024elinisegg 5950 Membership in the inverse image of a singleton. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Put in closed form and shorten. (Revised by BJ, 16-Oct-2024.)
((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
 
16-Oct-2024elimasn 5946 Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by BJ, 16-Oct-2024.) TODO: replace existing usages by usages of elimasn1 5944, remove, and relabel elimasn1 5944 to "elimasn".
𝐵 ∈ V    &   𝐶 ∈ V    ⇒   (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
 
16-Oct-2024elimasn1 5944 Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Use df-br 5044 and shorten. (Revised by BJ, 16-Oct-2024.)
𝐵 ∈ V    &   𝐶 ∈ V    ⇒   (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶)
 
16-Oct-2024elimasng1 5943 Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) Revise to use df-br 5044 and to prove elimasn1 5944 from it. (Revised by BJ, 16-Oct-2024.)
((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
 
16-Oct-2024elabd2 3572 Membership in a class abstraction, using implicit substitution. Deduction version of elab 3580. (Contributed by Gino Giotto, 12-Oct-2024.) (Revised by BJ, 16-Oct-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 = {𝑥 ∣ 𝜓})    &   ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (𝐴 ∈ 𝐵 ↔ 𝜒))
 
15-Oct-2024eloprabga 7307 The law of concretion for operation class abstraction. Compare elopab 5397. (Contributed by NM, 14-Sep-1999.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.) Avoid ax-10 2141, ax-11 2158. (Revised by Wolf Lammen, 15-Oct-2024.)
((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓))
 
15-Oct-2024cbvopabv 5115 Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.) Reduce axiom usage. (Revised by Gino Giotto, 15-Oct-2024.)
((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))    ⇒   {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}
 
15-Oct-2024notabw 4208 A class abstraction defined by a negation. Version of notab 4209 using implicit substitution, which does not require ax-10 2141, ax-12 2175. (Contributed by Gino Giotto, 15-Oct-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑦 ∣ 𝜓})
 
15-Oct-2024unabw 4202 Union of two class abstractions. Version of unab 4203 using implicit substitution, which does not require ax-8 2112, ax-10 2141, ax-12 2175. (Contributed by Gino Giotto, 15-Oct-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))    ⇒   ({𝑥 ∣ 𝜑} ∪ {𝑥 ∣ 𝜓}) = {𝑦 ∣ (𝜒 ∨ 𝜃)}
 
15-Oct-2024csbied 3840 Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.) Reduce axiom usage. (Revised by Gino Giotto, 15-Oct-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    ⇒   (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)
 
15-Oct-2024csbie 3838 Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019.) Reduce axiom usage. (Revised by Gino Giotto, 15-Oct-2024.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⦋𝐴 / 𝑥⦌𝐵 = 𝐶
 
15-Oct-2024csbconstg 3821 Substitution doesn't affect a constant 𝐵 (in which 𝑥 does not occur). csbconstgf 3820 with distinct variable requirement. (Contributed by Alan Sare, 22-Jul-2012.) Avoid ax-12 2175. (Revised by Gino Giotto, 15-Oct-2024.)
(𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐵)
 
14-Oct-2024catcxpccl 17686 The category of categories for a weak universe is closed under the product category operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   𝑇 = (𝑋 ×c 𝑌)    &   (𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → 𝑇 ∈ 𝐵)
 
14-Oct-2024catcfuccl 17597 The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   𝑄 = (𝑋 FuncCat 𝑌)    &   (𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → 𝑄 ∈ 𝐵)
 
14-Oct-2024catcccocl 17594 The composition operation of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 17595. (Contributed by AV, 14-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (comp‘𝑋) ∈ 𝑈)
 
14-Oct-2024catchomcl 17593 The Hom-set of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 17595. (Contributed by AV, 14-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
 
14-Oct-2024catcbaselcl 17592 The base set of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 17595. (Contributed by AV, 14-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (Base‘𝑋) ∈ 𝑈)
 
14-Oct-2024catcslotelcl 17591 A slot entry of an element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 17595. (Contributed by AV, 14-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   𝐸 = Slot (𝐸‘ndx)    ⇒   (𝜑 → (𝐸‘𝑋) ∈ 𝑈)
 
14-Oct-2024catcbascl 17590 An element of the base set of the category of categories for a weak universe belongs to the weak universe. Formerly part of the proof for catcoppccl 17595. (Contributed by AV, 14-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → 𝑋 ∈ 𝑈)
 
14-Oct-2024fuchom 17441 The morphisms in the functor category are natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)
𝑄 = (𝐶 FuncCat 𝐷)    &   𝑁 = (𝐶 Nat 𝐷)    ⇒   𝑁 = (Hom ‘𝑄)
 
14-Oct-2024oppchomfval 17189 Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)
𝐻 = (Hom ‘𝐶)    &   𝑂 = (oppCat‘𝐶)    ⇒   tpos 𝐻 = (Hom ‘𝑂)
 
13-Oct-2024edgfndxnn 27055 The index value of the edge function extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 13-Oct-2024.)
(.ef‘ndx) ∈ ℕ
 
13-Oct-2024edgfndx 27054 Index value of the df-edgf 27052 slot. (Contributed by AV, 13-Oct-2024.) (New usage is discouraged.)
(.ef‘ndx) = 18
 
13-Oct-20240pos 17800 Technical lemma to simplify the statement of ipopos 18014. The empty set is (rather pathologically) a poset under our definitions, since it has an empty base set (str0 16735) and any relation partially orders an empty set. (Contributed by Stefan O'Rear, 30-Jan-2015.) (Proof shortened by AV, 13-Oct-2024.)
∅ ∈ Poset
 
13-Oct-2024catcoppccl 17595 The category of categories for a weak universe is closed under taking opposites. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.)
𝐶 = (CatCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   𝑂 = (oppCat‘𝑋)    &   (𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → ω ∈ 𝑈)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → 𝑂 ∈ 𝐵)
 
13-Oct-2024wunnat 17435 A weak universe is closed under the natural transformation operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.)
(𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → 𝐶 ∈ 𝑈)    &   (𝜑 → 𝐷 ∈ 𝑈)    ⇒   (𝜑 → (𝐶 Nat 𝐷) ∈ 𝑈)
 
13-Oct-2024wunfunc 17377 A weak universe is closed under the functor set operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.)
(𝜑 → 𝑈 ∈ WUni)    &   (𝜑 → 𝐶 ∈ 𝑈)    &   (𝜑 → 𝐷 ∈ 𝑈)    ⇒   (𝜑 → (𝐶 Func 𝐷) ∈ 𝑈)
 
13-Oct-2024rescco 17309 Composition in the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 13-Oct-2024.)
𝐷 = (𝐶 ↾cat 𝐻)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜑 → 𝐻 Fn (𝑆 × 𝑆))    &   (𝜑 → 𝑆 ⊆ 𝐵)    &    · = (comp‘𝐶)    ⇒   (𝜑 → · = (comp‘𝐷))
 
13-Oct-2024prdsvallem 16931 Lemma for prdsval 16932. (Contributed by Stefan O'Rear, 3-Jan-2015.) Extracted from the former proof of prdsval 16932, dependency on df-hom 16791 removed. (Revised by AV, 13-Oct-2024.)
(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) ∈ V
 
13-Oct-2024basendxnn 16749 The index value of the base set extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 13-Oct-2024.)
(Base‘ndx) ∈ ℕ
 
13-Oct-2024pcxnn0cl 16394 Extended nonnegative integer closure of the general prime count function. (Contributed by Jim Kingdon, 13-Oct-2024.)
((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (𝑃 pCnt 𝑁) ∈ ℕ0*)
 
12-Oct-2024trpred 6178 The class of predecessors of an element of a transitive class for the membership relation is that element. (Contributed by BJ, 12-Oct-2024.)
((Tr 𝐴 ∧ 𝑋 ∈ 𝐴) → Pred( E , 𝐴, 𝑋) = 𝑋)
 
12-Oct-2024sbc2ie 3769 Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Gino Giotto, 12-Oct-2024.)
𝐴 ∈ V    &   𝐵 ∈ V    &   ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)
 
12-Oct-2024sbcg 3765 Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 3763. (Contributed by Alan Sare, 10-Nov-2012.) Reduce axiom usage. (Revised by Gino Giotto, 12-Oct-2024.)
(𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
 
12-Oct-2024sbcimdv 3760 Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1818). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) Reduce axiom usage. (Revised by Gino Giotto, 12-Oct-2024.)
(𝜑 → (𝜓 → 𝜒))    ⇒   (𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))
 
12-Oct-2024sbcied 3732 Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) Avoid ax-10 2141, ax-12 2175. (Revised by Gino Giotto, 12-Oct-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
 
12-Oct-2024sbcieg 3727 Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) Avoid ax-10 2141, ax-12 2175. (Revised by Gino Giotto, 12-Oct-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
 
12-Oct-2024elabgt 3574 Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 3578.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) Reduce axiom usage. (Revised by Gino Giotto, 12-Oct-2024.)
((𝐴 ∈ 𝐵 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
 
12-Oct-2024elabd3 3573 Membership in a class abstraction, using implicit substitution. Deduction version of elab 3580. (Contributed by Gino Giotto, 12-Oct-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒))
 
12-Oct-2024ceqsexv 3448 Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) Avoid ax-12 2175. (Revised by Gino Giotto, 12-Oct-2024.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)
 
12-Oct-2024rabrabi 3396 Abstract builder restricted to another restricted abstract builder with implicit substitution. (Contributed by AV, 2-Aug-2022.) Avoid ax-10 2141, ax-11 2158 and ax-12 2175. (Revised by Gino Giotto, 12-Oct-2024.)
(𝑥 = 𝑦 → (𝜒 ↔ 𝜑))    ⇒   {𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ (𝜒 ∧ 𝜓)}
 
12-Oct-2024sbco4lem 2277 Lemma for sbco4 2279. It replaces the temporary variable 𝑣 with another temporary variable 𝑤. (Contributed by Jim Kingdon, 26-Sep-2018.) (Proof shortened by Wolf Lammen, 12-Oct-2024.)
([𝑥 / 𝑣][𝑦 / 𝑥][𝑣 / 𝑦]𝜑 ↔ [𝑥 / 𝑤][𝑦 / 𝑥][𝑤 / 𝑦]𝜑)
 
11-Oct-2024sticksstones12a 39790 Establish bijective mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 11-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ)    &   𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))    &   𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → ∀𝑑 ∈ 𝐵 (𝐹‘(𝐺‘𝑑)) = 𝑑)
 
10-Oct-2024dfuniv2 41545 Alternative definition of Univ using only simple defined symbols. (Contributed by Rohan Ridenour, 10-Oct-2024.)
Univ = {𝑦 ∣ ∀𝑧 ∈ 𝑦 ∀𝑓 ∈ 𝒫 𝑦∃𝑤 ∈ 𝑦 (𝒫 𝑧 ⊆ (𝑦 ∩ 𝑤) ∧ (𝑧 ∩ ∪ 𝑓) ⊆ ∪ (𝑓 ∩ 𝒫 𝒫 𝑤))}
 
10-Oct-2024ismnushort 41544 Express the predicate on 𝑈 and 𝑧 in ismnu 41504 in a shorter form while avoiding complicated definitions. (Contributed by Rohan Ridenour, 10-Oct-2024.)
(∀𝑓 ∈ 𝒫 𝑈∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ (𝑈 ∩ 𝑤) ∧ (𝑧 ∩ ∪ 𝑓) ⊆ ∪ (𝑓 ∩ 𝒫 𝒫 𝑤)) ↔ (𝒫 𝑧 ⊆ 𝑈 ∧ ∀𝑓∃𝑤 ∈ 𝑈 (𝒫 𝑧 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝑧 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓) → ∃𝑢 ∈ 𝑓 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))))
 
9-Oct-2024no3inds 33809 Triple induction over surreal numbers. (Contributed by Scott Fenton, 9-Oct-2024.)
(𝑎 = 𝑑 → (𝜑 ↔ 𝜓))    &   (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))    &   (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))    &   (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))    &   (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))    &   (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))    &   (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))    &   (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))    &   (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))    &   (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))    &   ((𝑎 ∈ No ∧ 𝑏 ∈ No ∧ 𝑐 ∈ No ) → (((∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜃 ∧ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜒 ∧ ∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜁) ∧ (∀𝑑 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))𝜓 ∧ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜏 ∧ ∀𝑒 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))𝜎) ∧ ∀𝑓 ∈ (( L ‘𝑐) ∪ ( R ‘𝑐))𝜂) → 𝜑))    ⇒   ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No ) → 𝜆)
 
9-Oct-2024lrold 33771 The union of the left and right options of a surreal make its old set. (Contributed by Scott Fenton, 9-Oct-2024.)
(( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴))
 
9-Oct-2024rightirr 33768 No surreal is a member of its right set. (Contributed by Scott Fenton, 9-Oct-2024.)
¬ 𝑋 ∈ ( R ‘𝑋)
 
9-Oct-2024leftirr 33767 No surreal is a member of its left set. (Contributed by Scott Fenton, 9-Oct-2024.)
¬ 𝑋 ∈ ( L ‘𝑋)
 
9-Oct-2024madeun 33760 The made set is the union of the old set and the new set. (Contributed by Scott Fenton, 9-Oct-2024.)
( M ‘𝐴) = (( O ‘𝐴) ∪ ( N ‘𝐴))
 
9-Oct-2024rightssno 33758 The right set of a surreal number is a subset of the surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( R ‘𝐴) ⊆ No
 
9-Oct-2024leftssno 33757 The left set of a surreal number is a subset of the surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( L ‘𝐴) ⊆ No
 
9-Oct-2024rightssold 33756 The right options are a subset of the old set. (Contributed by Scott Fenton, 9-Oct-2024.)
( R ‘𝑋) ⊆ ( O ‘( bday ‘𝑋))
 
9-Oct-2024leftssold 33755 The left options are a subset of the old set. (Contributed by Scott Fenton, 9-Oct-2024.)
( L ‘𝑋) ⊆ ( O ‘( bday ‘𝑋))
 
9-Oct-2024oldssmade 33754 The older-than set is a subset of the made set. (Contributed by Scott Fenton, 9-Oct-2024.)
( O ‘𝐴) ⊆ ( M ‘𝐴)
 
9-Oct-2024madess 33753 If 𝐴 is less than or equal to ordinal 𝐵, then the made set of 𝐴 is included in the made set of 𝐵. (Contributed by Scott Fenton, 9-Oct-2024.)
((𝐵 ∈ On ∧ 𝐴 ⊆ 𝐵) → ( M ‘𝐴) ⊆ ( M ‘𝐵))
 
9-Oct-2024rightval 33742 The value of the right options function. (Contributed by Scott Fenton, 9-Oct-2024.)
( R ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝐴 <s 𝑥}
 
9-Oct-2024leftval 33741 The value of the left options function. (Contributed by Scott Fenton, 9-Oct-2024.)
( L ‘𝐴) = {𝑥 ∈ ( O ‘( bday ‘𝐴)) ∣ 𝑥 <s 𝐴}
 
9-Oct-2024newssno 33740 New sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( N ‘𝐴) ⊆ No
 
9-Oct-2024oldssno 33739 Old sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( O ‘𝐴) ⊆ No
 
9-Oct-2024madessno 33738 Made sets are surreals. (Contributed by Scott Fenton, 9-Oct-2024.)
( M ‘𝐴) ⊆ No
 
9-Oct-2024newval 33733 The value of the new options function. (Contributed by Scott Fenton, 9-Oct-2024.)
( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))
 
9-Oct-2024dtru 5252 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 2036.

This theorem is proved directly from set theory axioms (no set theory definitions) and does not use ax-ext 2706 or ax-sep 5181. See dtruALT 5270 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 2369. (Revised by Gino Giotto, 5-Sep-2023.) Avoid ax-12 2175. (Revised by Rohan Ridenour, 9-Oct-2024.)

¬ ∀𝑥 𝑥 = 𝑦
 
8-Oct-2024rr-grothshort 41547 A shorter equivalent of ax-groth 10420 than rr-groth 41542 using a few more simple defined symbols. (Contributed by Rohan Ridenour, 8-Oct-2024.)
∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 ∀𝑓 ∈ 𝒫 𝑦∃𝑤 ∈ 𝑦 (𝒫 𝑧 ⊆ (𝑦 ∩ 𝑤) ∧ (𝑧 ∩ ∪ 𝑓) ⊆ ∪ (𝑓 ∩ 𝒫 𝒫 𝑤)))
 
8-Oct-2024rr-grothshortbi 41546 Express "every set is contained in a Grothendieck universe" in a short form while avoiding complicated definitions. (Contributed by Rohan Ridenour, 8-Oct-2024.)
(∀𝑥∃𝑦 ∈ Univ 𝑥 ∈ 𝑦 ↔ ∀𝑥∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧 ∈ 𝑦 ∀𝑓 ∈ 𝒫 𝑦∃𝑤 ∈ 𝑦 (𝒫 𝑧 ⊆ (𝑦 ∩ 𝑤) ∧ (𝑧 ∩ ∪ 𝑓) ⊆ ∪ (𝑓 ∩ 𝒫 𝒫 𝑤))))
 
7-Oct-2024f1ocof1ob2 44200 If the range of 𝐹 equals the domain of 𝐺, then the composition (𝐺 ∘ 𝐹) is bijective iff 𝐹 and 𝐺 are both bijective. Symmetric version of f1ocof1ob 44199 including the fact that 𝐹 is a surjection onto its range. (Contributed by GL and AV, 20-Sep-2024.) (Proof shortened by AV, 7-Oct-2024.)
((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1-onto→𝐷 ↔ (𝐹:𝐴–1-1-onto→𝐶 ∧ 𝐺:𝐶–1-1-onto→𝐷)))
 
7-Oct-2024f1ocof1ob 44199 If the range of 𝐹 equals the domain of 𝐺, then the composition (𝐺 ∘ 𝐹) is bijective iff 𝐹 and 𝐺 are both bijective. (Contributed by GL and AV, 7-Oct-2024.)
((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1-onto→𝐷 ↔ (𝐹:𝐴–1-1→𝐶 ∧ 𝐺:𝐶–1-1-onto→𝐷)))
 
7-Oct-2024fcoresf1ob 44193 A composition is bijective iff the restriction of its first component to the minimum domain is bijective and the restriction of its second component to the minimum domain is injective. (Contributed by GL and AV, 7-Oct-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    &   (𝜑 → 𝐺:𝐶⟶𝐷)    &   𝑌 = (𝐺 ↾ 𝐸)    ⇒   (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1-onto→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1-onto→𝐷)))
 
7-Oct-2024fcoresfob 44192 A composition is surjective iff the restriction of its first component to the minimum domain is surjective. (Contributed by GL and AV, 7-Oct-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    &   (𝜑 → 𝐺:𝐶⟶𝐷)    &   𝑌 = (𝐺 ↾ 𝐸)    ⇒   (𝜑 → ((𝐺 ∘ 𝐹):𝑃–onto→𝐷 ↔ 𝑌:𝐸–onto→𝐷))
 
7-Oct-2024fcoresf1b 44190 A composition is injective iff the restrictions of its components to the minimum domains are injective. (Contributed by GL and AV, 7-Oct-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    &   (𝜑 → 𝐺:𝐶⟶𝐷)    &   𝑌 = (𝐺 ↾ 𝐸)    ⇒   (𝜑 → ((𝐺 ∘ 𝐹):𝑃–1-1→𝐷 ↔ (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷)))
 
7-Oct-2024fcoresf1 44189 If a composition is injective, then the restrictions of its components to the minimum domains are injective. (Contributed by GL and AV, 18-Sep-2024.) (Revised by AV, 7-Oct-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    &   (𝜑 → 𝐺:𝐶⟶𝐷)    &   𝑌 = (𝐺 ↾ 𝐸)    &   (𝜑 → (𝐺 ∘ 𝐹):𝑃–1-1→𝐷)    ⇒   (𝜑 → (𝑋:𝑃–1-1→𝐸 ∧ 𝑌:𝐸–1-1→𝐷))
 
7-Oct-2024sticksstones15 39794 Sticks and stones with almost collapsed definitions for positive integers. (Contributed by metakunt, 7-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}    ⇒   (𝜑 → (♯‘𝐴) = ((𝑁 + 𝐾)C𝐾))
 
7-Oct-2024sticksstones14 39793 Sticks and stones with definitions as hypotheses. (Contributed by metakunt, 7-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))    &   𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → (♯‘𝐴) = ((𝑁 + 𝐾)C𝐾))
 
7-Oct-2024bj-sbievwd 34658 Variant of sbievw 2099. (Contributed by BJ, 7-Oct-2024.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → Ⅎ'𝑥𝜒)    &   ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → ([𝑦 / 𝑥]𝜓 ↔ 𝜒))
 
7-Oct-2024bj-equsexvwd 34657 Variant of equsexvw 2013. (Contributed by BJ, 7-Oct-2024.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → Ⅎ'𝑥𝜒)    &   ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜒))
 
7-Oct-2024bj-equsalvwd 34656 Variant of equsalvw 2012. (Contributed by BJ, 7-Oct-2024.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → Ⅎ'𝑥𝜒)    &   ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜓) ↔ 𝜒))
 
7-Oct-2024bj-equsvt 34655 A variant of equsv 2011. (Contributed by BJ, 7-Oct-2024.)
(Ⅎ'𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
 
7-Oct-2024bj-pm11.53a 34654 A variant of pm11.53v 1952. One can similarly prove a variant with DV (𝑦, 𝜑) and ∀𝑦Ⅎ'𝑥𝜓 instead of DV (𝑥, 𝜓) and ∀𝑥Ⅎ'𝑦𝜑. (Contributed by BJ, 7-Oct-2024.)
(∀𝑥Ⅎ'𝑦𝜑 → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
 
7-Oct-2024bj-pm11.53v 34653 Version of pm11.53v 1952 with nonfreeness antecedents. (Contributed by BJ, 7-Oct-2024.)
((∀𝑥Ⅎ'𝑦𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
 
7-Oct-2024bj-pm11.53vw 34652 Version of pm11.53v 1952 with nonfreeness antecedents. One can also prove the theorem with antecedent (Ⅎ'𝑦∀𝑥𝜑 ∧ ∀𝑦Ⅎ'𝑥𝜓). (Contributed by BJ, 7-Oct-2024.)
((∀𝑥Ⅎ'𝑦𝜑 ∧ Ⅎ'𝑥∀𝑦𝜓) → (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)))
 
6-Oct-2024sticksstones13 39792 Establish bijective mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))    &   𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
 
6-Oct-2024sticksstones12 39791 Establish bijective mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ)    &   𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))    &   𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
 
6-Oct-2024sticksstones11 39789 Establish bijective mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 = 0)    &   𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))    &   𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
 
6-Oct-2024sticksstones10 39788 Establish mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ)    &   𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → 𝐺:𝐵⟶𝐴)
 
6-Oct-2024sticksstones9 39787 Establish mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 6-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 = 0)    &   𝐺 = (𝑏 ∈ 𝐵 ↦ if(𝐾 = 0, {⟨1, 𝑁⟩}, (𝑘 ∈ (1...(𝐾 + 1)) ↦ if(𝑘 = (𝐾 + 1), ((𝑁 + 𝐾) − (𝑏‘𝐾)), if(𝑘 = 1, ((𝑏‘1) − 1), (((𝑏‘𝑘) − (𝑏‘(𝑘 − 1))) − 1))))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → 𝐺:𝐵⟶𝐴)
 
6-Oct-2024nfnbi 1862 A variable is nonfree in a proposition if and only if it is so in its negation. (Contributed by BJ, 6-May-2019.) (Proof shortened by Wolf Lammen, 6-Oct-2024.)
(Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)
 
5-Oct-2024bj-ntrufal 34444 The negation of a theorem is equivalent to false. This can shorten dfnul2 4230. (Contributed by BJ, 5-Oct-2024.)
𝜑    ⇒   (¬ 𝜑 ↔ ⊥)
 
5-Oct-2024bj-mt2bi 34443 Version of mt2 203 where the major premise is a biconditional. Another proof is also possible via con2bii 361 and mpbi 233. The current mt2bi 367 should be relabeled, maybe to imfal. (Contributed by BJ, 5-Oct-2024.)
𝜑    &   (𝜓 ↔ ¬ 𝜑)    ⇒    ¬ 𝜓
 
5-Oct-2024sbc6g 3717 An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by SN, 5-Oct-2024.)
(𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
 
5-Oct-2024elab 3580 Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.) Avoid ax-10 2141, ax-11 2158, ax-12 2175. (Revised by SN, 5-Oct-2024.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
 
5-Oct-2024elabg 3578 Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.) Avoid ax-13 2369. (Revised by SN, 23-Nov-2022.) Avoid ax-10 2141, ax-11 2158, ax-12 2175. (Revised by SN, 5-Oct-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
 
5-Oct-2024elab6g 3571 Membership in a class abstraction. Class version of sb6 2091. (Contributed by SN, 5-Oct-2024.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
 
4-Oct-2024bj-gabima 34822 Generalized class abstraction as a direct image.

TODO: improve the support lemmas elimag 5922 and fvelima 6767 to nonfreeness hypothesis (and for the latter, biconditional). (Contributed by BJ, 4-Oct-2024.)

(𝜑 → ∀𝑥𝜑)    &   (𝜑 → Ⅎ𝑥𝐹)    &   (𝜑 → Fun 𝐹)    &   (𝜑 → {𝑥 ∣ 𝜓} ⊆ dom 𝐹)    ⇒   (𝜑 → {(𝐹‘𝑥) ∣ 𝑥 ∣ 𝜓} = (𝐹 “ {𝑥 ∣ 𝜓}))
 
4-Oct-2024bj-elgab 34821 Elements of a generalized class abstraction. (Contributed by BJ, 4-Oct-2024.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → Ⅎ𝑥𝐴)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → (∃𝑥(𝐴 = 𝐵 ∧ 𝜓) ↔ 𝜒))    ⇒   (𝜑 → (𝐴 ∈ {𝐵 ∣ 𝑥 ∣ 𝜓} ↔ 𝜒))
 
4-Oct-2024bj-gabeqis 34820 Equality of generalized class abstractions, with implicit substitution. (Contributed by BJ, 4-Oct-2024.)
(𝑥 = 𝑦 → 𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   {𝐴 ∣ 𝑥 ∣ 𝜑} = {𝐵 ∣ 𝑦 ∣ 𝜓}
 
4-Oct-2024bj-gabeqd 34819 Equality of generalized class abstractions. Deduction form. (Contributed by BJ, 4-Oct-2024.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → 𝐴 = 𝐵)    &   (𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} = {𝐵 ∣ 𝑥 ∣ 𝜒})
 
4-Oct-2024bj-gabssd 34818 Inclusion of generalized class abstractions. Deduction form. (Contributed by BJ, 4-Oct-2024.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → 𝐴 = 𝐵)    &   (𝜑 → (𝜓 → 𝜒))    ⇒   (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜒})
 
4-Oct-2024bj-gabss 34817 Inclusion of generalized class abstractions. (Contributed by BJ, 4-Oct-2024.)
(∀𝑥(𝐴 = 𝐵 ∧ (𝜑 → 𝜓)) → {𝐴 ∣ 𝑥 ∣ 𝜑} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜓})
 
4-Oct-2024df-bj-gab 34816 Definition of generalized class abstractions: typically, 𝑥 is a bound variable in 𝐴 and 𝜑 and {𝐴 ∣ 𝑥 ∣ 𝜑} denotes "the class of 𝐴(𝑥)'s such that 𝜑(𝑥)". (Contributed by BJ, 4-Oct-2024.)
{𝐴 ∣ 𝑥 ∣ 𝜑} = {𝑦 ∣ ∃𝑥(𝐴 = 𝑦 ∧ 𝜑)}
 
3-Oct-2024itg1addlem4 24568 Lemma for itg1add . (Contributed by Mario Carneiro, 28-Jun-2014.) (Proof shortened by SN, 3-Oct-2024.)
(𝜑 → 𝐹 ∈ dom ∫1)    &   (𝜑 → 𝐺 ∈ dom ∫1)    &   𝐼 = (𝑖 ∈ ℝ, 𝑗 ∈ ℝ ↦ if((𝑖 = 0 ∧ 𝑗 = 0), 0, (vol‘((◡𝐹 “ {𝑖}) ∩ (◡𝐺 “ {𝑗})))))    &   𝑃 = ( + ↾ (ran 𝐹 × ran 𝐺))    ⇒   (𝜑 → (∫1‘(𝐹 ∘f + 𝐺)) = Σ𝑦 ∈ ran 𝐹Σ𝑧 ∈ ran 𝐺((𝑦 + 𝑧) · (𝑦𝐼𝑧)))
 
3-Oct-2024peano5 7660 The induction postulate: any class containing zero and closed under the successor operation contains all natural numbers. One of Peano's five postulates for arithmetic. Proposition 7.30(5) of [TakeutiZaring] p. 43, except our proof does not require the Axiom of Infinity. The more traditional statement of mathematical induction as a theorem schema, with a basis and an induction step, is derived from this theorem as Theorem findes 7669. (Contributed by NM, 18-Feb-2004.) Avoid ax-10 2141, ax-12 2175. (Revised by Gino Giotto, 3-Oct-2024.)
((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴)
 
3-Oct-2024dffr2 5504 Alternate definition of well-founded relation. Similar to Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof shortened by Mario Carneiro, 23-Jun-2015.) Avoid ax-10 2141, ax-11 2158, ax-12 2175, but use ax-8 2112. (Revised by Gino Giotto, 3-Oct-2024.)
(𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 {𝑧 ∈ 𝑥 ∣ 𝑧𝑅𝑦} = ∅))
 
3-Oct-2024pocl 5464 Characteristic properties of a partial order in class notation. (Contributed by NM, 27-Mar-1997.) Reduce axiom usage and shorten proof. (Revised by Gino Giotto, 3-Oct-2024.)
(𝑅 Po 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))))
 
3-Oct-20240nelopab 5435 The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.) Reduce axiom usage and shorten proof. (Revised by Gino Giotto, 3-Oct-2024.)
¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
3-Oct-2024ab0w 4278 The class of sets verifying a property is the empty class if and only if that property is a contradiction. Version of ab0 4279 using implicit substitution, which requires fewer axioms. (Contributed by Gino Giotto, 3-Oct-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑦 ¬ 𝜓)
 
3-Oct-2024vtocl3ga 3486 Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.) Reduce axiom usage. (Revised by Gino Giotto, 3-Oct-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))    &   ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑)    ⇒   ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃)
 
3-Oct-2024vtocl3g 3480 Implicit substitution of a class for a setvar variable. Version of vtocl3gf 3478 with disjoint variable conditions instead of nonfreeness hypotheses, requiring fewer axioms. (Contributed by Gino Giotto, 3-Oct-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))    &   𝜑    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝜃)
 
3-Oct-2024rspw 3119 Restricted specialization. Weak version of rsp 3120, requiring ax-8 2112, but not ax-12 2175. (Contributed by Gino Giotto, 3-Oct-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑))
 
1-Oct-2024thincfth 45956 A functor from a thin category is faithful. (Contributed by Zhi Wang, 1-Oct-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)
 
1-Oct-2024fullthinc2 45955 A full functor to a thin category maps empty hom-sets to empty hom-sets. (Contributed by Zhi Wang, 1-Oct-2024.)
𝐵 = (Base‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝐷 ∈ ThinCat)    &   (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋𝐻𝑌) = ∅ ↔ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))
 
1-Oct-2024fullthinc 45954 A functor to a thin category is full iff empty hom-sets are mapped to empty hom-sets. (Contributed by Zhi Wang, 1-Oct-2024.)
𝐵 = (Base‘𝐶)    &   𝐽 = (Hom ‘𝐷)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝐷 ∈ ThinCat)    &   (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)    ⇒   (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)))
 
1-Oct-2024functhinc 45953 A functor to a thin category is determined entirely by the object part. The hypothesis "functhinc.1" is related to a monotone function if preorders induced by the categories are considered (catprs2 45920). (Contributed by Zhi Wang, 1-Oct-2024.)
𝐵 = (Base‘𝐷)    &   𝐶 = (Base‘𝐸)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &   (𝜑 → 𝐷 ∈ Cat)    &   (𝜑 → 𝐸 ∈ ThinCat)    &   (𝜑 → 𝐹:𝐵⟶𝐶)    &   𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))    &   (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))    ⇒   (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ 𝐺 = 𝐾))
 
1-Oct-2024functhinclem4 45952 Lemma for functhinc 45953. Other requirements on the morphism part are automatically satisfied. (Contributed by Zhi Wang, 1-Oct-2024.)
𝐵 = (Base‘𝐷)    &   𝐶 = (Base‘𝐸)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &   (𝜑 → 𝐷 ∈ Cat)    &   (𝜑 → 𝐸 ∈ ThinCat)    &   (𝜑 → 𝐹:𝐵⟶𝐶)    &   𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))    &   (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))    &    1 = (Id‘𝐷)    &   𝐼 = (Id‘𝐸)    &    · = (comp‘𝐷)    &   𝑂 = (comp‘𝐸)    ⇒   ((𝜑 ∧ 𝐺 = 𝐾) → ∀𝑎 ∈ 𝐵 (((𝑎𝐺𝑎)‘( 1 ‘𝑎)) = (𝐼‘(𝐹‘𝑎)) ∧ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ∀𝑚 ∈ (𝑎𝐻𝑏)∀𝑛 ∈ (𝑏𝐻𝑐)((𝑎𝐺𝑐)‘(𝑛(⟨𝑎, 𝑏⟩ · 𝑐)𝑚)) = (((𝑏𝐺𝑐)‘𝑛)(⟨(𝐹‘𝑎), (𝐹‘𝑏)⟩𝑂(𝐹‘𝑐))((𝑎𝐺𝑏)‘𝑚))))
 
1-Oct-2024functhinclem3 45951 Lemma for functhinc 45953. The mapped morphism is in its corresponding hom-set. (Contributed by Zhi Wang, 1-Oct-2024.)
(𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))    &   (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))))    &   (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅))    &   (𝜑 → ∃*𝑛 𝑛 ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))    ⇒   (𝜑 → ((𝑋𝐺𝑌)‘𝑀) ∈ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
 
1-Oct-2024functhinclem2 45950 Lemma for functhinc 45953. (Contributed by Zhi Wang, 1-Oct-2024.)
(𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))    ⇒   (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅))
 
1-Oct-2024functhinclem1 45949 Lemma for functhinc 45953. Given the object part, there is only one possible morphism part such that the mapped morphism is in its corresponding hom-set. (Contributed by Zhi Wang, 1-Oct-2024.)
𝐵 = (Base‘𝐷)    &   𝐶 = (Base‘𝐸)    &   𝐻 = (Hom ‘𝐷)    &   𝐽 = (Hom ‘𝐸)    &   (𝜑 → 𝐸 ∈ ThinCat)    &   (𝜑 → 𝐹:𝐵⟶𝐶)    &   𝐾 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((𝑥𝐻𝑦) × ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))    &   ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (((𝐹‘𝑧)𝐽(𝐹‘𝑤)) = ∅ → (𝑧𝐻𝑤) = ∅))    ⇒   (𝜑 → ((𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧𝐺𝑤):(𝑧𝐻𝑤)⟶((𝐹‘𝑧)𝐽(𝐹‘𝑤))) ↔ 𝐺 = 𝐾))
 
1-Oct-2024funcf2lem 45926 A utility theorem for proving equivalence of "is a functor". (Contributed by Zhi Wang, 1-Oct-2024.)
(𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ↔ (𝐺 ∈ V ∧ 𝐺 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
 
1-Oct-2024map0cor 45809 A function exists iff an empty codomain is accompanied with an empty domain. (Contributed by Zhi Wang, 1-Oct-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → ((𝐵 = ∅ → 𝐴 = ∅) ↔ ∃𝑓 𝑓:𝐴⟶𝐵))
 
1-Oct-2024f002 45808 A function with an empty codomain must have empty domain. (Contributed by Zhi Wang, 1-Oct-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   (𝜑 → (𝐵 = ∅ → 𝐴 = ∅))
 
1-Oct-2024f1mo 45807 A function that maps a set with at most one element to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.)
((∃*𝑥 𝑥 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)
 
1-Oct-2024f102g 45806 A function that maps the empty set to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.)
((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)
 
1-Oct-2024f1sn2g 45805 A function that maps a singleton to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → 𝐹:{𝐴}–1-1→𝐵)
 
1-Oct-2024fdomne0 45804 A function with non-empty domain is non-empty and has non-empty codomain. (Contributed by Zhi Wang, 1-Oct-2024.)
((𝐹:𝑋⟶𝑌 ∧ 𝑋 ≠ ∅) → (𝐹 ≠ ∅ ∧ 𝑌 ≠ ∅))
 
1-Oct-2024mofeu 45802 The uniqueness of a function into a set with at most one element. (Contributed by Zhi Wang, 1-Oct-2024.)
𝐺 = (𝐴 × 𝐵)    &   (𝜑 → (𝐵 = ∅ → 𝐴 = ∅))    &   (𝜑 → ∃*𝑥 𝑥 ∈ 𝐵)    ⇒   (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
 
1-Oct-2024sticksstones8 39786 Establish mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 1-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑗 ∈ (1...𝐾) ↦ (𝑗 + Σ𝑙 ∈ (1...𝑗)(𝑎‘𝑙))))    &   𝐴 = {𝑔 ∣ (𝑔:(1...(𝐾 + 1))⟶ℕ0 ∧ Σ𝑖 ∈ (1...(𝐾 + 1))(𝑔‘𝑖) = 𝑁)}    &   𝐵 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...(𝑁 + 𝐾)) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → 𝐹:𝐴⟶𝐵)
 
1-Oct-2024sticksstones7 39785 Closure property of sticks and stones function. (Contributed by metakunt, 1-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   (𝜑 → 𝐺:(1...(𝐾 + 1))⟶ℕ0)    &   (𝜑 → 𝑋 ∈ (1...𝐾))    &   𝐹 = (𝑥 ∈ (1...𝐾) ↦ (𝑥 + Σ𝑖 ∈ (1...𝑥)(𝐺‘𝑖)))    &   (𝜑 → Σ𝑖 ∈ (1...(𝐾 + 1))(𝐺‘𝑖) = 𝑁)    ⇒   (𝜑 → (𝐹‘𝑋) ∈ (1...(𝑁 + 𝐾)))
 
1-Oct-2024sticksstones6 39784 Function induces an order isomorphism for sticks and stones theorem. (Contributed by metakunt, 1-Oct-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   (𝜑 → 𝐺:(1...(𝐾 + 1))⟶ℕ0)    &   (𝜑 → 𝑋 ∈ (1...𝐾))    &   (𝜑 → 𝑌 ∈ (1...𝐾))    &   (𝜑 → 𝑋 < 𝑌)    &   𝐹 = (𝑥 ∈ (1...𝐾) ↦ (𝑥 + Σ𝑖 ∈ (1...𝑥)(𝐺‘𝑖)))    ⇒   (𝜑 → (𝐹‘𝑋) < (𝐹‘𝑌))
 
30-Sep-2024subthinc 45948 A subcategory of a thin category is thin. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐷 = (𝐶 ↾cat 𝐽)    &   (𝜑 → 𝐽 ∈ (Subcat‘𝐶))    &   (𝜑 → 𝐶 ∈ ThinCat)    ⇒   (𝜑 → 𝐷 ∈ ThinCat)
 
30-Sep-2024topdlat 45917 A topology is a distributive lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐽)    ⇒   (𝐽 ∈ Top → 𝐼 ∈ DLat)
 
30-Sep-2024toplatmeet 45916 Meets in a topology are realized by intersections. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐽)    &   (𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝐴 ∈ 𝐽)    &   (𝜑 → 𝐵 ∈ 𝐽)    &    ∧ = (meet‘𝐼)    ⇒   (𝜑 → (𝐴 ∧ 𝐵) = (𝐴 ∩ 𝐵))
 
30-Sep-2024toplatjoin 45915 Joins in a topology are realized by unions. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐽)    &   (𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝐴 ∈ 𝐽)    &   (𝜑 → 𝐵 ∈ 𝐽)    &    ∨ = (join‘𝐼)    ⇒   (𝜑 → (𝐴 ∨ 𝐵) = (𝐴 ∪ 𝐵))
 
30-Sep-2024toplatglb 45914 Greatest lower bounds in a topology are realized by the interior of the intersection. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐽)    &   (𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑆 ⊆ 𝐽)    &   𝐺 = (glb‘𝐼)    &   (𝜑 → 𝑆 ≠ ∅)    ⇒   (𝜑 → (𝐺‘𝑆) = ((int‘𝐽)‘∩ 𝑆))
 
30-Sep-2024toplatlub 45913 Least upper bounds in a topology are realized by unions. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐽)    &   (𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑆 ⊆ 𝐽)    &   𝑈 = (lub‘𝐼)    ⇒   (𝜑 → (𝑈‘𝑆) = ∪ 𝑆)
 
30-Sep-2024toplatglb0 45912 The empty intersection in a topology is realized by the base set. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐽)    &   (𝜑 → 𝐽 ∈ Top)    &   𝐺 = (glb‘𝐼)    ⇒   (𝜑 → (𝐺‘∅) = ∪ 𝐽)
 
30-Sep-2024topclat 45911 A topology is a complete lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐽)    ⇒   (𝐽 ∈ Top → 𝐼 ∈ CLat)
 
30-Sep-2024mreclat 45910 A Moore space is a complete lattice under inclusion. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐶)    ⇒   (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ CLat)
 
30-Sep-2024ipoglb0 45907 The GLB of the empty set is the union of the base. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐹)    &   (𝜑 → 𝐺 = (glb‘𝐼))    &   (𝜑 → ∪ 𝐹 ∈ 𝐹)    ⇒   (𝜑 → (𝐺‘∅) = ∪ 𝐹)
 
30-Sep-2024ipolub00 45906 The LUB of the empty set is the empty set if it is contained. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐹)    &   (𝜑 → 𝑈 = (lub‘𝐼))    &   (𝜑 → ∅ ∈ 𝐹)    ⇒   (𝜑 → (𝑈‘∅) = ∅)
 
30-Sep-2024ipolub0 45905 The LUB of the empty set is the intersection of the base. (Contributed by Zhi Wang, 30-Sep-2024.)
𝐼 = (toInc‘𝐹)    &   (𝜑 → 𝑈 = (lub‘𝐼))    &   (𝜑 → ∩ 𝐹 ∈ 𝐹)    &   (𝜑 → 𝐹 ∈ 𝑉)    ⇒   (𝜑 → (𝑈‘∅) = ∩ 𝐹)
 
30-Sep-2024cbvriotavw 7169 Change bound variable in a restricted description binder. Version of cbvriotav 7174 with a disjoint variable condition, which requires fewer axioms . (Contributed by NM, 18-Mar-2013.) (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)
 
30-Sep-2024cbviotavw 6335 Change bound variables in a description binder. Version of cbviotav 6338 with a disjoint variable condition, which requires fewer axioms . (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (℩𝑥𝜑) = (℩𝑦𝜓)
 
30-Sep-2024rexprg 4602 Convert a restricted existential quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) Avoid ax-10 2141, ax-12 2175. (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒)))
 
30-Sep-2024ralprg 4600 Convert a restricted universal quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) Avoid ax-10 2141, ax-12 2175. (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒)))
 
30-Sep-2024rexsng 4580 Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) Avoid ax-10 2141, ax-12 2175. (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
 
30-Sep-2024ralsng 4579 Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.) Avoid ax-10 2141, ax-12 2175. (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓))
 
30-Sep-2024ralidmw 4409 Idempotent law for restricted quantifier. Weak version of ralidm 4413, which does not require ax-10 2141, ax-12 2175, but requires ax-8 2112. (Contributed by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)
 
30-Sep-2024rabeq0w 4288 Condition for a restricted class abstraction to be empty. Version of rabeq0 4289 using implicit substitution, which does not require ax-10 2141, ax-11 2158, ax-12 2175, but requires ax-8 2112. (Contributed by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ({𝑥 ∈ 𝐴 ∣ 𝜑} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ 𝜓)
 
30-Sep-2024cbvreuvw 3354 Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreuv 3358 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 5-Apr-2004.) (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)
 
30-Sep-2024cbvrmovw 3353 Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Version of cbvrmov 3359 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 16-Jun-2017.) (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑦 ∈ 𝐴 𝜓)
 
30-Sep-2024cbveuvw 2603 Change bound variable. Uses only Tarski's FOL axiom schemes. See cbveu 2606 for a version with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 25-Nov-1994.) (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃!𝑥𝜑 ↔ ∃!𝑦𝜓)
 
30-Sep-2024cbvmovw 2599 Change bound variable. Uses only Tarski's FOL axiom schemes. See cbvmo 2602 and cbvmow 2600 for versions with fewer disjoint variable conditions but requiring more axioms. (Contributed by NM, 9-Mar-1995.) (Revised by Gino Giotto, 30-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃*𝑥𝜑 ↔ ∃*𝑦𝜓)
 
29-Sep-2024oppcthin 45947 The opposite category of a thin category is thin. (Contributed by Zhi Wang, 29-Sep-2024.)
𝑂 = (oppCat‘𝐶)    ⇒   (𝐶 ∈ ThinCat → 𝑂 ∈ ThinCat)
 
29-Sep-2024mrelatglbALT 45909 Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Zhi Wang, 29-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐼 = (toInc‘𝐶)    &   𝐺 = (glb‘𝐼)    ⇒   ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → (𝐺‘𝑈) = ∩ 𝑈)
 
29-Sep-2024mrelatlubALT 45908 Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Proof shortened by Zhi Wang, 29-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐼 = (toInc‘𝐶)    &   𝐹 = (mrCls‘𝐶)    &   𝐿 = (lub‘𝐼)    ⇒   ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈))
 
29-Sep-2024ipoglb 45904 The GLB of the inclusion poset. (hypotheses "ipolub.s" and "ipoglb.t" could be eliminated with 𝑆 ∈ dom 𝐺.) Could be significantly shortened if posglbdg 17893 is in quantified form. mrelatglb 18038 could potentially be shortened using this. See mrelatglbALT 45909. (Contributed by Zhi Wang, 29-Sep-2024.)
𝐼 = (toInc‘𝐹)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝑆 ⊆ 𝐹)    &   (𝜑 → 𝐺 = (glb‘𝐼))    &   (𝜑 → 𝑇 = ∪ {𝑥 ∈ 𝐹 ∣ 𝑥 ⊆ ∩ 𝑆})    &   (𝜑 → 𝑇 ∈ 𝐹)    ⇒   (𝜑 → (𝐺‘𝑆) = 𝑇)
 
29-Sep-2024ipoglbdm 45903 The domain of the GLB of the inclusion poset. (Contributed by Zhi Wang, 29-Sep-2024.)
𝐼 = (toInc‘𝐹)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝑆 ⊆ 𝐹)    &   (𝜑 → 𝐺 = (glb‘𝐼))    &   (𝜑 → 𝑇 = ∪ {𝑥 ∈ 𝐹 ∣ 𝑥 ⊆ ∩ 𝑆})    ⇒   (𝜑 → (𝑆 ∈ dom 𝐺 ↔ 𝑇 ∈ 𝐹))
 
29-Sep-2024ipoglblem 45902 Lemma for ipoglbdm 45903 and ipoglb 45904. (Contributed by Zhi Wang, 29-Sep-2024.)
𝐼 = (toInc‘𝐹)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝑆 ⊆ 𝐹)    &    ≤ = (le‘𝐼)    ⇒   ((𝜑 ∧ 𝑋 ∈ 𝐹) → ((𝑋 ⊆ ∩ 𝑆 ∧ ∀𝑧 ∈ 𝐹 (𝑧 ⊆ ∩ 𝑆 → 𝑧 ⊆ 𝑋)) ↔ (∀𝑦 ∈ 𝑆 𝑋 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐹 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑋))))
 
29-Sep-2024unilbeu 45898 Existential uniqueness of the greatest lower bound. (Contributed by Zhi Wang, 29-Sep-2024.)
(𝐶 ∈ 𝐵 → ((𝐶 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝐵 (𝑦 ⊆ 𝐴 → 𝑦 ⊆ 𝐶)) ↔ 𝐶 = ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴}))
 
29-Sep-2024isclatd 45896 The predicate "is a complete lattice" (deduction form). (Contributed by Zhi Wang, 29-Sep-2024.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → 𝑈 = (lub‘𝐾))    &   (𝜑 → 𝐺 = (glb‘𝐾))    &   (𝜑 → 𝐾 ∈ Poset)    &   ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝑈)    &   ((𝜑 ∧ 𝑠 ⊆ 𝐵) → 𝑠 ∈ dom 𝐺)    ⇒   (𝜑 → 𝐾 ∈ CLat)
 
29-Sep-2024glbeldm2d 45880 Member of the domain of the greatest lower bound function of a poset. (Contributed by Zhi Wang, 29-Sep-2024.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → ≤ = (le‘𝐾))    &   (𝜑 → 𝐺 = (glb‘𝐾))    &   ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))    &   (𝜑 → 𝐾 ∈ Poset)    ⇒   (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
 
29-Sep-2024mreuniss 45820 The union of a collection of closed sets is a subset. (Contributed by Zhi Wang, 29-Sep-2024.)
((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → ∪ 𝑆 ⊆ 𝑋)
 
29-Sep-2024unilbss 45790 Superclass of the greatest lower bound. A dual statement of ssintub 4867. (Contributed by Zhi Wang, 29-Sep-2024.)
∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴
 
29-Sep-2024focofob 44198 If the domain of a function 𝐺 equals the range of a function 𝐹, then the composition (𝐺 ∘ 𝐹) is surjective iff 𝐺 and 𝐹 as function to the domain of 𝐺 are both surjective. Symmetric version of fnfocofob 44197 including the fact that 𝐹 is a surjection onto its range. (Contributed by GL and AV, 20-Sep-2024.) (Proof shortened by AV, 29-Sep-2024.)
((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐷 ↔ (𝐹:𝐴–onto→𝐶 ∧ 𝐺:𝐶–onto→𝐷)))
 
29-Sep-2024fnfocofob 44197 If the domain of a function 𝐺 equals the range of a function 𝐹, then the composition (𝐺 ∘ 𝐹) is surjective iff 𝐺 is surjective. (Contributed by GL and AV, 29-Sep-2024.)
((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐶))
 
29-Sep-2024funfocofob 44196 If the domain of a function 𝐺 is a subset of the range of a function 𝐹, then the composition (𝐺 ∘ 𝐹) is surjective iff 𝐺 is surjective. (Contributed by GL and AV, 29-Sep-2024.)
((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵))
 
29-Sep-2024fresfo 44168 Conditions for a restriction to be an onto function. Part of fresf1o 30657. (Contributed by AV, 29-Sep-2024.)
((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹) → (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–onto→𝐶)
 
29-Sep-2024rescnvimafod 6883 The restriction of a function to a preimage of a class is a function onto the intersection of this class and the range of the function. (Contributed by AV, 13-Sep-2024.) (Revised by AV, 29-Sep-2024.)
(𝜑 → 𝐹 Fn 𝐴)    &   (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐵))    &   (𝜑 → 𝐷 = (◡𝐹 “ 𝐵))    ⇒   (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→𝐸)
 
29-Sep-2024foco 6636 Composition of onto functions. (Contributed by NM, 22-Mar-2006.) (Proof shortened by AV, 29-Sep-2024.)
((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶)
 
29-Sep-2024focofo 6635 Composition of onto functions. Generalisation of foco 6636. (Contributed by AV, 29-Sep-2024.)
((𝐹:𝐴–onto→𝐵 ∧ Fun 𝐺 ∧ 𝐴 ⊆ ran 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)–onto→𝐵)
 
29-Sep-2024focnvimacdmdm 6634 The preimage of the codomain of a surjection is its domain. (Contributed by AV, 29-Sep-2024.)
(𝐺:𝐴–onto→𝐵 → (◡𝐺 “ 𝐵) = 𝐴)
 
28-Sep-2024ipolub 45901 The LUB of the inclusion poset. (hypotheses "ipolub.s" and "ipolub.t" could be eliminated with 𝑆 ∈ dom 𝑈.) Could be significantly shortened if poslubdg 17892 is in quantified form. mrelatlub 18040 could potentially be shortened using this. See mrelatlubALT 45908. (Contributed by Zhi Wang, 28-Sep-2024.)
𝐼 = (toInc‘𝐹)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝑆 ⊆ 𝐹)    &   (𝜑 → 𝑈 = (lub‘𝐼))    &   (𝜑 → 𝑇 = ∩ {𝑥 ∈ 𝐹 ∣ ∪ 𝑆 ⊆ 𝑥})    &   (𝜑 → 𝑇 ∈ 𝐹)    ⇒   (𝜑 → (𝑈‘𝑆) = 𝑇)
 
28-Sep-2024ipolubdm 45900 The domain of the LUB of the inclusion poset. (Contributed by Zhi Wang, 28-Sep-2024.)
𝐼 = (toInc‘𝐹)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝑆 ⊆ 𝐹)    &   (𝜑 → 𝑈 = (lub‘𝐼))    &   (𝜑 → 𝑇 = ∩ {𝑥 ∈ 𝐹 ∣ ∪ 𝑆 ⊆ 𝑥})    ⇒   (𝜑 → (𝑆 ∈ dom 𝑈 ↔ 𝑇 ∈ 𝐹))
 
28-Sep-2024ipolublem 45899 Lemma for ipolubdm 45900 and ipolub 45901. (Contributed by Zhi Wang, 28-Sep-2024.)
𝐼 = (toInc‘𝐹)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝑆 ⊆ 𝐹)    &    ≤ = (le‘𝐼)    ⇒   ((𝜑 ∧ 𝑋 ∈ 𝐹) → ((∪ 𝑆 ⊆ 𝑋 ∧ ∀𝑧 ∈ 𝐹 (∪ 𝑆 ⊆ 𝑧 → 𝑋 ⊆ 𝑧)) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑋 ∧ ∀𝑧 ∈ 𝐹 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑋 ≤ 𝑧))))
 
28-Sep-2024intubeu 45897 Existential uniqueness of the least upper bound. (Contributed by Zhi Wang, 28-Sep-2024.)
(𝐶 ∈ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ ∀𝑦 ∈ 𝐵 (𝐴 ⊆ 𝑦 → 𝐶 ⊆ 𝑦)) ↔ 𝐶 = ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}))
 
28-Sep-2024lubeldm2d 45879 Member of the domain of the least upper bound function of a poset. (Contributed by Zhi Wang, 28-Sep-2024.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → ≤ = (le‘𝐾))    &   (𝜑 → 𝑈 = (lub‘𝐾))    &   ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))    &   (𝜑 → 𝐾 ∈ Poset)    ⇒   (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
 
28-Sep-2024rspceb2dv 45775 Restricted existential specialization, using implicit substitution in both directions. (Contributed by Zhi Wang, 28-Sep-2024.)
((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 → 𝜒))    &   ((𝜑 ∧ 𝜒) → 𝐴 ∈ 𝐵)    &   ((𝜑 ∧ 𝜒) → 𝜃)    &   (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    ⇒   (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ 𝜒))
 
28-Sep-2024sticksstones5 39783 Count the number of strictly monotonely increasing functions on finite domains and codomains. (Contributed by metakunt, 28-Sep-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → (♯‘𝐴) = (𝑁C𝐾))
 
28-Sep-2024sticksstones4 39782 Equinumerosity lemma for sticks and stones. (Contributed by metakunt, 28-Sep-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐵 = {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾}    &   𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    ⇒   (𝜑 → 𝐴 ≈ 𝐵)
 
28-Sep-2024sticksstones3 39781 The range function on strictly monotone functions with finite domain and codomain is an surjective mapping onto 𝐾-elemental sets. (Contributed by metakunt, 28-Sep-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐵 = {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾}    &   𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    &   𝐹 = (𝑧 ∈ 𝐴 ↦ ran 𝑧)    ⇒   (𝜑 → 𝐹:𝐴–onto→𝐵)
 
27-Sep-2024posmidm 45894 Poset meet is idempotent. latmidm 17952 could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024.)
𝐵 = (Base‘𝐾)    &    ∧ = (meet‘𝐾)    ⇒   ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 𝑋) = 𝑋)
 
27-Sep-2024posjidm 45893 Poset join is idempotent. latjidm 17940 could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024.)
𝐵 = (Base‘𝐾)    &    ∨ = (join‘𝐾)    ⇒   ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 𝑋) = 𝑋)
 
27-Sep-2024inpw 45791 Two ways of expressing a collection of subsets as seen in df-ntr 21889, unimax 4847, and others (Contributed by Zhi Wang, 27-Sep-2024.)
(𝐵 ∈ 𝑉 → (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵})
 
27-Sep-2024mpbiran4d 45770 Equivalence with a conjunction one of whose conjuncts is a consequence of the other. Deduction form. (Contributed by Zhi Wang, 27-Sep-2024.)
(𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    &   ((𝜑 ∧ 𝜃) → 𝜒)    ⇒   (𝜑 → (𝜓 ↔ 𝜃))
 
27-Sep-2024sticksstones2 39780 The range function on strictly monotone functions with finite domain and codomain is an injective mapping onto 𝐾-elemental sets. (Contributed by metakunt, 27-Sep-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐵 = {𝑎 ∈ 𝒫 (1...𝑁) ∣ (♯‘𝑎) = 𝐾}    &   𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    &   𝐹 = (𝑧 ∈ 𝐴 ↦ ran 𝑧)    ⇒   (𝜑 → 𝐹:𝐴–1-1→𝐵)
 
27-Sep-2024sticksstones1 39779 Different strictly monotone functions have different ranges. (Contributed by metakunt, 27-Sep-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   𝐴 = {𝑓 ∣ (𝑓:(1...𝐾)⟶(1...𝑁) ∧ ∀𝑥 ∈ (1...𝐾)∀𝑦 ∈ (1...𝐾)(𝑥 < 𝑦 → (𝑓‘𝑥) < (𝑓‘𝑦)))}    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐴)    &   (𝜑 → 𝑋 ≠ 𝑌)    &   𝐼 = inf({𝑧 ∈ (1...𝐾) ∣ (𝑋‘𝑧) ≠ (𝑌‘𝑧)}, ℝ, < )    ⇒   (𝜑 → ran 𝑋 ≠ ran 𝑌)
 
27-Sep-2024cofcutrtime 33787 If 𝑋 is the cut of 𝐴 and 𝐵 and all of 𝐴 and 𝐵 are older than 𝑋, then ( L ‘𝑋) is cofinal with 𝐴 and ( R ‘𝑋) is coinitial with 𝐵. Note: we will call a cut where all of the elements of the cut are older than the cut itself a "timely" cut. Part of Theorem 4.02(12) of [Alling] p. 125. (Contributed by Scott Fenton, 27-Sep-2024.)
(((𝐴 ∪ 𝐵) ⊆ ( O ‘( bday ‘𝑋)) ∧ 𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ( L ‘𝑋)𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ ( R ‘𝑋)𝑤 ≤s 𝑧))
 
27-Sep-2024imaeqsalv 33379 Substitute a function value into a universal quantifier over an image. (Contributed by Scott Fenton, 27-Sep-2024.)
(𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))    ⇒   ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))
 
27-Sep-2024imaeqsexv 33378 Substitute a function value into an existential quantifier over an image. (Contributed by Scott Fenton, 27-Sep-2024.)
(𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))    ⇒   ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓))
 
27-Sep-2024rlimcn3 15134 Image of a limit under a continuous map, two-arg version. Originally a subproof of rlimcn2 15135. (Contributed by SN, 27-Sep-2024.)
((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐵 ∈ 𝑋)    &   ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝐶 ∈ 𝑌)    &   ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐵𝐹𝐶) ∈ ℂ)    &   (𝜑 → (𝑅𝐹𝑆) ∈ ℂ)    &   (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝑅)    &   (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐶) ⇝𝑟 𝑆)    &   ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ ∀𝑢 ∈ 𝑋 ∀𝑣 ∈ 𝑌 (((abs‘(𝑢 − 𝑅)) < 𝑟 ∧ (abs‘(𝑣 − 𝑆)) < 𝑠) → (abs‘((𝑢𝐹𝑣) − (𝑅𝐹𝑆))) < 𝑥))    ⇒   (𝜑 → (𝑧 ∈ 𝐴 ↦ (𝐵𝐹𝐶)) ⇝𝑟 (𝑅𝐹𝑆))
 
26-Sep-2024postcpos 45986 The converted category is a poset iff the original proset is a poset. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    ⇒   (𝜑 → (𝐾 ∈ Poset ↔ 𝐶 ∈ Poset))
 
26-Sep-2024toslat 45895 A toset is a lattice. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝐾 ∈ Toset → 𝐾 ∈ Lat)
 
26-Sep-2024glbpr 45888 The GLB of the set of two comparable elements in a poset is the less one of the two. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝜑 → 𝐾 ∈ Poset)    &   𝐵 = (Base‘𝐾)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &    ≤ = (le‘𝐾)    &   (𝜑 → 𝑋 ≤ 𝑌)    &   (𝜑 → 𝑆 = {𝑋, 𝑌})    &   𝐺 = (glb‘𝐾)    ⇒   (𝜑 → (𝐺‘𝑆) = 𝑋)
 
26-Sep-2024glbprdm 45887 The set of two comparable elements in a poset has GLB. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝜑 → 𝐾 ∈ Poset)    &   𝐵 = (Base‘𝐾)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &    ≤ = (le‘𝐾)    &   (𝜑 → 𝑋 ≤ 𝑌)    &   (𝜑 → 𝑆 = {𝑋, 𝑌})    &   𝐺 = (glb‘𝐾)    ⇒   (𝜑 → 𝑆 ∈ dom 𝐺)
 
26-Sep-2024glbprlem 45886 Lemma for glbprdm 45887 and glbpr 45888. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝜑 → 𝐾 ∈ Poset)    &   𝐵 = (Base‘𝐾)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &    ≤ = (le‘𝐾)    &   (𝜑 → 𝑋 ≤ 𝑌)    &   (𝜑 → 𝑆 = {𝑋, 𝑌})    &   𝐺 = (glb‘𝐾)    ⇒   (𝜑 → (𝑆 ∈ dom 𝐺 ∧ (𝐺‘𝑆) = 𝑋))
 
26-Sep-2024lubpr 45885 The LUB of the set of two comparable elements in a poset is the greater one of the two. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝜑 → 𝐾 ∈ Poset)    &   𝐵 = (Base‘𝐾)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &    ≤ = (le‘𝐾)    &   (𝜑 → 𝑋 ≤ 𝑌)    &   (𝜑 → 𝑆 = {𝑋, 𝑌})    &   𝑈 = (lub‘𝐾)    ⇒   (𝜑 → (𝑈‘𝑆) = 𝑌)
 
26-Sep-2024lubprdm 45884 The set of two comparable elements in a poset has LUB. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝜑 → 𝐾 ∈ Poset)    &   𝐵 = (Base‘𝐾)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &    ≤ = (le‘𝐾)    &   (𝜑 → 𝑋 ≤ 𝑌)    &   (𝜑 → 𝑆 = {𝑋, 𝑌})    &   𝑈 = (lub‘𝐾)    ⇒   (𝜑 → 𝑆 ∈ dom 𝑈)
 
26-Sep-2024lubprlem 45883 Lemma for lubprdm 45884 and lubpr 45885. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝜑 → 𝐾 ∈ Poset)    &   𝐵 = (Base‘𝐾)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &    ≤ = (le‘𝐾)    &   (𝜑 → 𝑋 ≤ 𝑌)    &   (𝜑 → 𝑆 = {𝑋, 𝑌})    &   𝑈 = (lub‘𝐾)    ⇒   (𝜑 → (𝑆 ∈ dom 𝑈 ∧ (𝑈‘𝑆) = 𝑌))
 
26-Sep-2024glbsscl 45882 If a subset of 𝑆 contains the GLB of 𝑆, then the two sets have the same GLB. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝜑 → 𝐾 ∈ Poset)    &   (𝜑 → 𝑇 ⊆ 𝑆)    &   𝐺 = (glb‘𝐾)    &   (𝜑 → 𝑆 ∈ dom 𝐺)    &   (𝜑 → (𝐺‘𝑆) ∈ 𝑇)    ⇒   (𝜑 → (𝑇 ∈ dom 𝐺 ∧ (𝐺‘𝑇) = (𝐺‘𝑆)))
 
26-Sep-2024lubsscl 45881 If a subset of 𝑆 contains the LUB of 𝑆, then the two sets have the same LUB. (Contributed by Zhi Wang, 26-Sep-2024.)
(𝜑 → 𝐾 ∈ Poset)    &   (𝜑 → 𝑇 ⊆ 𝑆)    &   𝑈 = (lub‘𝐾)    &   (𝜑 → 𝑆 ∈ dom 𝑈)    &   (𝜑 → (𝑈‘𝑆) ∈ 𝑇)    ⇒   (𝜑 → (𝑇 ∈ dom 𝑈 ∧ (𝑈‘𝑇) = (𝑈‘𝑆)))
 
26-Sep-2024glbeldm2 45878 Member of the domain of the greatest lower bound function of a poset. (Contributed by Zhi Wang, 26-Sep-2024.)
𝐵 = (Base‘𝐾)    &    ≤ = (le‘𝐾)    &   𝐺 = (glb‘𝐾)    &   (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))    &   (𝜑 → 𝐾 ∈ Poset)    ⇒   (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
 
26-Sep-2024lubeldm2 45877 Member of the domain of the least upper bound function of a poset. (Contributed by Zhi Wang, 26-Sep-2024.)
𝐵 = (Base‘𝐾)    &    ≤ = (le‘𝐾)    &   𝑈 = (lub‘𝐾)    &   (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))    &   (𝜑 → 𝐾 ∈ Poset)    ⇒   (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))
 
26-Sep-202419.8aw 39852 This is to 19.8a 2178 what spw 2042 is to sp 2180. (Contributed by SN, 26-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (𝜑 → ∃𝑥𝜑)
 
26-Sep-2024sn-wcdeq 39843 Alternative to wcdeq 3669 and df-cdeq 3670. This flattens the syntax representation ( wi ( weq vx vy ) wph ) to ( sn-wcdeq vx vy wph ), illustrating the comment of df-cdeq 3670. (Contributed by SN, 26-Sep-2024.) (New usage is discouraged.)
wff (𝑥 = 𝑦 → 𝜑)
 
25-Sep-2024postc 45988 The converted category is a poset iff no distinct objects are isomorphic. (Contributed by Zhi Wang, 25-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   𝐵 = (Base‘𝐶)    ⇒   (𝜑 → (𝐶 ∈ Poset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥( ≃𝑐 ‘𝐶)𝑦 → 𝑥 = 𝑦)))
 
25-Sep-2024postcposALT 45987 Alternate proof for postcpos 45986. (Contributed by Zhi Wang, 25-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    ⇒   (𝜑 → (𝐾 ∈ Poset ↔ 𝐶 ∈ Poset))
 
25-Sep-2024thinccic 45969 In a thin category, two objects are isomorphic iff there are morphisms between them in both directions. (Contributed by Zhi Wang, 25-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝐻 = (Hom ‘𝐶)    ⇒   (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑋) ≠ ∅)))
 
25-Sep-2024thinciso 45968 In a thin category, 𝐹:𝑋⟶𝑌 is an isomorphism iff there is a morphism from 𝑌 to 𝑋. (Contributed by Zhi Wang, 25-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝐻 = (Hom ‘𝐶)    &   𝐼 = (Iso‘𝐶)    &   (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    ⇒   (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ (𝑌𝐻𝑋) ≠ ∅))
 
25-Sep-2024endmndlem 45923 A diagonal hom-set in a category equipped with the restriction of the composition has a structure of monoid. See also df-mndtc 45990 for converting a monoid to a category. Lemma for bj-endmnd 35180. (Contributed by Zhi Wang, 25-Sep-2024.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → (𝑋𝐻𝑋) = (Base‘𝑀))    &   (𝜑 → (⟨𝑋, 𝑋⟩ · 𝑋) = (+g‘𝑀))    ⇒   (𝜑 → 𝑀 ∈ Mnd)
 
25-Sep-2024meetdm3 45892 The meet of any two elements always exists iff all unordered pairs have GLB (expanded version). (Contributed by Zhi Wang, 25-Sep-2024.)
𝐵 = (Base‘𝐾)    &   (𝜑 → 𝐾 ∈ 𝑉)    &   𝐺 = (glb‘𝐾)    &    ∧ = (meet‘𝐾)    &    ≤ = (le‘𝐾)    ⇒   (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑥 ∧ 𝑧 ≤ 𝑦) ∧ ∀𝑤 ∈ 𝐵 ((𝑤 ≤ 𝑥 ∧ 𝑤 ≤ 𝑦) → 𝑤 ≤ 𝑧))))
 
25-Sep-2024meetdm2 45891 The meet of any two elements always exists iff all unordered pairs have GLB. (Contributed by Zhi Wang, 25-Sep-2024.)
𝐵 = (Base‘𝐾)    &   (𝜑 → 𝐾 ∈ 𝑉)    &   𝐺 = (glb‘𝐾)    &    ∧ = (meet‘𝐾)    ⇒   (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺))
 
25-Sep-2024joindm3 45890 The join of any two elements always exists iff all unordered pairs have LUB (expanded version). (Contributed by Zhi Wang, 25-Sep-2024.)
𝐵 = (Base‘𝐾)    &   (𝜑 → 𝐾 ∈ 𝑉)    &   𝑈 = (lub‘𝐾)    &    ∨ = (join‘𝐾)    &    ≤ = (le‘𝐾)    ⇒   (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃!𝑧 ∈ 𝐵 ((𝑥 ≤ 𝑧 ∧ 𝑦 ≤ 𝑧) ∧ ∀𝑤 ∈ 𝐵 ((𝑥 ≤ 𝑤 ∧ 𝑦 ≤ 𝑤) → 𝑧 ≤ 𝑤))))
 
25-Sep-2024joindm2 45889 The join of any two elements always exists iff all unordered pairs have LUB. (Contributed by Zhi Wang, 25-Sep-2024.)
𝐵 = (Base‘𝐾)    &   (𝜑 → 𝐾 ∈ 𝑉)    &   𝑈 = (lub‘𝐾)    &    ∨ = (join‘𝐾)    ⇒   (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝑈))
 
25-Sep-2024cofcutr 33786 If 𝑋 is the cut of 𝐴 and 𝐵, then 𝐴 is cofinal with ( L ‘𝑋) and 𝐵 is coinitial with ( R ‘𝑋). Theorem 2.9 of [Gonshor] p. 12. (Contributed by Scott Fenton, 25-Sep-2024.)
((𝐴 <<s 𝐵 ∧ 𝑋 = (𝐴 |s 𝐵)) → (∀𝑥 ∈ ( L ‘𝑋)∃𝑦 ∈ 𝐴 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ ( R ‘𝑋)∃𝑤 ∈ 𝐵 𝑤 ≤s 𝑧))
 
25-Sep-2024cofcut2 33785 If 𝐴 and 𝐶 are mutually cofinal and 𝐵 and 𝐷 are mutually coinitial, then the cut of 𝐴 and 𝐵 is equal to the cut of 𝐶 and 𝐷. Theorem 2.7 of [Gonshor] p. 10. (Contributed by Scott Fenton, 25-Sep-2024.)
(((𝐴 <<s 𝐵 ∧ 𝐶 ∈ 𝒫 No ∧ 𝐷 ∈ 𝒫 No ) ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (∀𝑡 ∈ 𝐶 ∃𝑢 ∈ 𝐴 𝑡 ≤s 𝑢 ∧ ∀𝑟 ∈ 𝐷 ∃𝑠 ∈ 𝐵 𝑠 ≤s 𝑟)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
 
25-Sep-2024cofcut1 33784 If 𝐶 is cofinal with 𝐴 and 𝐷 is coinitial with 𝐵 and the cut of 𝐴 and 𝐵 lies between 𝐶 and 𝐷. Then the cut of 𝐶 and 𝐷 is equal to the cut of 𝐴 and 𝐵. Theorem 2.6 of [Gonshor] p. 10. (Contributed by Scott Fenton, 25-Sep-2024.)
((𝐴 <<s 𝐵 ∧ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐷 𝑤 ≤s 𝑧) ∧ (𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷)) → (𝐴 |s 𝐵) = (𝐶 |s 𝐷))
 
25-Sep-2024ssltsepcd 33682 Two elements of separated sets obey less than. Deduction form of ssltsepc 33681. (Contributed by Scott Fenton, 25-Sep-2024.)
(𝜑 → 𝐴 <<s 𝐵)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → 𝑋 <s 𝑌)
 
25-Sep-2024en2sn 8707 Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.) Avoid ax-pow 5247. (Revised by BTernaryTau, 31-Jul-2024.) Avoid ax-un 7512. (Revised by BTernaryTau, 25-Sep-2024.)
((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵})
 
25-Sep-2024rexeqbidvv 3309 Version of rexeqbidv 3307 with additional disjoint variable conditions, not requiring ax-8 2112 nor df-clel 2812. (Contributed by Wolf Lammen, 25-Sep-2024.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))
 
25-Sep-2024nfnaew 2149 All variables are effectively bound in a distinct variable specifier. Version of nfnae 2431 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by Mario Carneiro, 11-Aug-2016.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 25-Sep-2024.)
Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦
 
24-Sep-2024mndtcbas2 45995 Two objects in a category built from a monoid are identical. (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → 𝑋 = 𝑌)
 
24-Sep-2024thincinv 45967 In a thin category, 𝐹 is an inverse of 𝐺 iff 𝐹 is a section of 𝐺 (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝑆 = (Sect‘𝐶)    &   𝑁 = (Inv‘𝐶)    ⇒   (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ 𝐹(𝑋𝑆𝑌)𝐺))
 
24-Sep-2024thincsect2 45966 In a thin category, 𝐹 is a section of 𝐺 iff 𝐺 is a section of 𝐹. (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝑆 = (Sect‘𝐶)    ⇒   (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ 𝐺(𝑌𝑆𝑋)𝐹))
 
24-Sep-2024thincsect 45965 In a thin category, one morphism is a section of another iff they are pointing towards each other. (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   𝐵 = (Base‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝑆 = (Sect‘𝐶)    &   𝐻 = (Hom ‘𝐶)    ⇒   (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋))))
 
24-Sep-2024thincepi 45943 In a thin category, all morphisms are epimorphisms. The converse does not hold. See grptcepi 46003. (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝐸 = (Epi‘𝐶)    ⇒   (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌))
 
24-Sep-2024thincmon 45942 In a thin category, all morphisms are monomorphisms. The converse does not hold. See grptcmon 46002. (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝑀 = (Mono‘𝐶)    ⇒   (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))
 
24-Sep-2024thincid 45941 In a thin category, a morphism from an object to itself is an identity morphism. (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝑋 ∈ 𝐵)    &    1 = (Id‘𝐶)    &   (𝜑 → 𝐹 ∈ (𝑋𝐻𝑋))    ⇒   (𝜑 → 𝐹 = ( 1 ‘𝑋))
 
24-Sep-2024thinccd 45933 A thin category is a category (deduction form). (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    ⇒   (𝜑 → 𝐶 ∈ Cat)
 
24-Sep-2024mofsssn 45800 There is at most one function into a subclass of a singleton. (Contributed by Zhi Wang, 24-Sep-2024.)
(𝐵 ⊆ {𝑌} → ∃*𝑓 𝑓:𝐴⟶𝐵)
 
24-Sep-2024mpbiran3d 45769 Equivalence with a conjunction one of whose conjuncts is a consequence of the other. Deduction form. (Contributed by Zhi Wang, 24-Sep-2024.)
(𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    &   ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   (𝜑 → (𝜓 ↔ 𝜒))
 
24-Sep-2024coinitsslt 33783 If 𝐵 is coinitial with 𝐶 and 𝐴 precedes 𝐶, then 𝐴 precedes 𝐵. (Contributed by Scott Fenton, 24-Sep-2024.)
((𝐵 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝑦 ≤s 𝑥 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s 𝐵)
 
24-Sep-2024cofsslt 33782 If every element of 𝐴 is bounded by some element of 𝐵 and 𝐵 precedes 𝐶, then 𝐴 precedes 𝐶. Note - we will often use the term "cofinal" to denote that every element of 𝐴 is bounded above by some element of 𝐵. Similarly, we will use the term "coinitial" to denote that every element of 𝐴 is bounded below by some element of 𝐵. (Contributed by Scott Fenton, 24-Sep-2024.)
((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 <<s 𝐶)
 
24-Sep-2024ssltd 33680 Deduce surreal set less than. (Contributed by Scott Fenton, 24-Sep-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐴 ⊆ No )    &   (𝜑 → 𝐵 ⊆ No )    &   ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑥 <s 𝑦)    ⇒   (𝜑 → 𝐴 <<s 𝐵)
 
24-Sep-2024setc2ohom 17573 (SetCat‘2o) is a category (provable from setccat 17563 and 2oex 8208) that does not have pairwise disjoint hom-sets, proved by this theorem combined with setc2obas 17572. Notably, the empty set ∅ is simultaneously an object (setc2obas 17572) , an identity morphism from ∅ to ∅ (setcid 17564 or thincid 45941) , and a non-identity morphism from ∅ to 1o. See cat1lem 17574 and cat1 17575 for a more general statement. This category is also thin (setc2othin 45964), and therefore is "equivalent" to a preorder (actually a partial order). See prsthinc 45962 for more details on the "equivalence". (Contributed by Zhi Wang, 24-Sep-2024.)
𝐶 = (SetCat‘2o)    &   𝐻 = (Hom ‘𝐶)    ⇒   ∅ ∈ ((∅𝐻∅) ∩ (∅𝐻1o))
 
24-Sep-2024setc2obas 17572 ∅ and 1o are distinct objects in (SetCat‘2o). This combined with setc2ohom 17573 demonstrates that the category does not have pairwise disjoint hom-sets. See also df-cat 17143 and cat1 17575. (Contributed by Zhi Wang, 24-Sep-2024.)
𝐶 = (SetCat‘2o)    &   𝐵 = (Base‘𝐶)    ⇒   (∅ ∈ 𝐵 ∧ 1o ∈ 𝐵 ∧ 1o ≠ ∅)
 
24-Sep-2024en1uniel 8694 A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.) Avoid ax-un 7512. (Revised by BTernaryTau, 24-Sep-2024.)
(𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆)
 
24-Sep-2024en1b 8689 A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) Avoid ax-un 7512. (Revised by BTernaryTau, 24-Sep-2024.)
(𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴})
 
24-Sep-2024eqsnuniex 5241 If a class is equal to the singleton of its union, then its union exists. (Contributed by BTernaryTau, 24-Sep-2024.)
(𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V)
 
24-Sep-2024nfra2w 3142 Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 42105. Version of nfra2 3144 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by Alan Sare, 31-Dec-2011.) (Revised by Gino Giotto, 24-Sep-2024.)
Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑
 
24-Sep-2024nfraldw 3137 Deduction version of nfralw 3140. Version of nfrald 3139 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 24-Sep-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝐴)    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)
 
23-Sep-2024grptcepi 46003 All morphisms in a category converted from a group are epimorphisms. (Contributed by Zhi Wang, 23-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝐺))    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   (𝜑 → 𝐸 = (Epi‘𝐶))    ⇒   (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌))
 
23-Sep-2024grptcmon 46002 All morphisms in a category converted from a group are monomorphisms. (Contributed by Zhi Wang, 23-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝐺))    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   (𝜑 → 𝑀 = (Mono‘𝐶))    ⇒   (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))
 
23-Sep-2024catprsc2 45922 An alternate construction of the preorder induced by a category. See catprs2 45920 for details. See also catprsc 45921 for a different construction. The two constructions are different because df-cat 17143 does not require the domain of 𝐻 to be 𝐵 × 𝐵. (Contributed by Zhi Wang, 23-Sep-2024.)
(𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐻𝑦) ≠ ∅})    ⇒   (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
 
23-Sep-2024mosssn2 45789 Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 23-Sep-2024.)
(∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝐴 ⊆ {𝑦})
 
23-Sep-2024mosssn 45787 "At most one" element in a subclass of a singleton. (Contributed by Zhi Wang, 23-Sep-2024.)
(𝐴 ⊆ {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
 
23-Sep-2024reutruALT 45779 Alternate proof for reutru 45778. (Contributed by Zhi Wang, 23-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤)
 
23-Sep-2024reutru 45778 Two ways of expressing "exactly one" element. (Contributed by Zhi Wang, 23-Sep-2024.)
(∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤)
 
23-Sep-2024rmotru 45777 Two ways of expressing "at most one" element. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof shortened by BJ, 23-Sep-2024.)
(∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 ∈ 𝐴 ⊤)
 
23-Sep-2024rextru 45776 Two ways of expressing "at least one" element. (Contributed by Zhi Wang, 23-Sep-2024.)
(∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤)
 
23-Sep-2024dtrucor3 45771 An example of how ax-5 1918 without a distinct variable condition causes paradox in models of at least two objects. The hypothesis "dtrucor3.1" is provable from dtru 5252 in the ZF set theory. axc16nf 2260 and euae 2658 demonstrate that the violation of dtru 5252 leads to a model with only one object assuming its existence (ax-6 1976). The conclusion is also provable in the empty model ( see emptyal 1916). See also nf5 2283 and nf5i 2146 for the relation between unconditional ax-5 1918 and being not free. (Contributed by Zhi Wang, 23-Sep-2024.)
¬ ∀𝑥 𝑥 = 𝑦    &   (𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)    ⇒   ∀𝑥 𝑥 = 𝑦
 
23-Sep-2024entrfir 8857 Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 8669). (Contributed by BTernaryTau, 23-Sep-2024.)
((𝐶 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)
 
23-Sep-2024entrfi 8856 Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 8669). (Contributed by BTernaryTau, 23-Sep-2024.)
((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)
 
23-Sep-2024enfi 8854 Equinumerous sets have the same finiteness. For a shorter proof using ax-pow 5247, see enfiALT 8855. (Contributed by NM, 22-Aug-2008.) Avoid ax-pow 5247. (Revised by BTernaryTau, 23-Sep-2024.)
(𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
 
23-Sep-2024enfii 8853 A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.) Avoid ax-pow 5247. (Revised by BTernaryTau, 23-Sep-2024.)
((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
 
23-Sep-2024ssnnfi 8836 A subset of a natural number is finite. (Contributed by NM, 24-Jun-1998.) (Proof shortened by BTernaryTau, 23-Sep-2024.)
((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)
 
23-Sep-2024nnfi 8834 Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.) Avoid ax-pow 5247. (Revised by BTernaryTau, 23-Sep-2024.)
(𝐴 ∈ ω → 𝐴 ∈ Fin)
 
23-Sep-2024en1 8687 A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) Avoid ax-un 7512. (Revised by BTernaryTau, 23-Sep-2024.)
(𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
 
23-Sep-2024ensn1 8683 A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) Avoid ax-un 7512. (Revised by BTernaryTau, 23-Sep-2024.)
𝐴 ∈ V    ⇒   {𝐴} ≈ 1o
 
23-Sep-2024en0 8680 The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5247, ax-un 7512. (Revised by BTernaryTau, 23-Sep-2024.)
(𝐴 ≈ ∅ ↔ 𝐴 = ∅)
 
23-Sep-2024bren 8625 Equinumerosity relation. (Contributed by NM, 15-Jun-1998.) (Proof shortened by BTernaryTau, 23-Sep-2024.)
(𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
 
23-Sep-2024breng 8624 Equinumerosity relation. This variation of bren 8625 does not require the Axiom of Union. (Contributed by BTernaryTau, 23-Sep-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵))
 
23-Sep-2024noel 4235 The empty set has no elements. Theorem 6.14 of [Quine] p. 44. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Mario Carneiro, 1-Sep-2015.) Remove dependency on ax-10 2141, ax-11 2158, and ax-12 2175. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.)
¬ 𝐴 ∈ ∅
 
23-Sep-2024dfnul3 4231 Alternate definition of the empty set. (Contributed by NM, 25-Mar-2004.) (Proof shortened by BJ, 23-Sep-2024.)
∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴}
 
23-Sep-2024dfnul2 4230 Alternate definition of the empty set. Definition 5.14 of [TakeutiZaring] p. 20. (Contributed by NM, 26-Dec-1996.) Remove dependency on ax-10 2141, ax-11 2158, and ax-12 2175. (Revised by Steven Nguyen, 3-May-2023.) (Proof shortened by BJ, 23-Sep-2024.)
∅ = {𝑥 ∣ ¬ 𝑥 = 𝑥}
 
23-Sep-2024dfnul4 4229 Alternate definition of the empty class/set. (Contributed by BJ, 30-Nov-2019.) Avoid ax-8 2112, df-clel 2812. (Revised by Gino Giotto, 3-Sep-2024.) Prove directly from definition to allow shortening dfnul2 4230. (Revised by BJ, 23-Sep-2024.)
∅ = {𝑥 ∣ ⊥}
 
23-Sep-2024nfabdw 2923 Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2925 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof shortened by Wolf Lammen, 23-Sep-2024.)
Ⅎ𝑦𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    ⇒   (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
 
22-Sep-2024mndtcid 46001 The identity morphism, or identity arrow, of the category built from a monoid is the identity element of the monoid. (Contributed by Zhi Wang, 22-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 1 = (Id‘𝐶))    ⇒   (𝜑 → ( 1 ‘𝑋) = (0g‘𝑀))
 
22-Sep-2024mndtccat 46000 The function value is a category. (Contributed by Zhi Wang, 22-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    ⇒   (𝜑 → 𝐶 ∈ Cat)
 
22-Sep-2024mndtccatid 45999 Lemma for mndtccat 46000 and mndtcid 46001. (Contributed by Zhi Wang, 22-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    ⇒   (𝜑 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑦 ∈ (Base‘𝐶) ↦ (0g‘𝑀))))
 
22-Sep-2024mndtcco2 45998 The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → · = (comp‘𝐶))    &   (𝜑 → ⚬ = (⟨𝑋, 𝑌⟩ · 𝑍))    ⇒   (𝜑 → (𝐺 ⚬ 𝐹) = (𝐺(+g‘𝑀)𝐹))
 
22-Sep-2024mndtcco 45997 The composition of the category built from a monoid is the monoid operation. (Contributed by Zhi Wang, 22-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → · = (comp‘𝐶))    ⇒   (𝜑 → (⟨𝑋, 𝑌⟩ · 𝑍) = (+g‘𝑀))
 
22-Sep-2024mndtchom 45996 The only hom-set of the category built from a monoid is the base set of the monoid. (Contributed by Zhi Wang, 22-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   (𝜑 → (𝑋𝐻𝑌) = (Base‘𝑀))
 
22-Sep-2024mndtcob 45994 Lemma for mndtchom 45996 and mndtcco 45997. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → 𝑋 = 𝑀)
 
22-Sep-2024mndtcbas 45993 The category built from a monoid contains precisely one object. (Contributed by Zhi Wang, 22-Sep-2024.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    &   (𝜑 → 𝐵 = (Base‘𝐶))    ⇒   (𝜑 → ∃!𝑥 𝑥 ∈ 𝐵)
 
22-Sep-2024mndtcbasval 45992 The base set of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    &   (𝜑 → 𝐵 = (Base‘𝐶))    ⇒   (𝜑 → 𝐵 = {𝑀})
 
22-Sep-2024mndtcval 45991 Value of the category built from a monoid. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)
(𝜑 → 𝐶 = (MndToCat‘𝑀))    &   (𝜑 → 𝑀 ∈ Mnd)    ⇒   (𝜑 → 𝐶 = {⟨(Base‘ndx), {𝑀}⟩, ⟨(Hom ‘ndx), {⟨𝑀, 𝑀, (Base‘𝑀)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑀, 𝑀, 𝑀⟩, (+g‘𝑀)⟩}⟩})
 
22-Sep-2024df-mndtc 45990 Definition of the function converting a monoid to a category. Example 3.3(4.e) of [Adamek] p. 24.

The definition of the base set is arbitrary. The whole extensible structure becomes the object here (see mndtcbasval 45992) , instead of just the base set, as is the case in Example 3.3(4.e) of [Adamek] p. 24.

The resulting category is defined entirely, up to isomorphism, by mndtcbas 45993, mndtchom 45996, mndtcco 45997. Use those instead.

See example 3.26(3) of [Adamek] p. 33 for more on isomorphism.

"MndToCat" was taken instead of "MndCat" because the latter might mean the category of monoids. (Contributed by Zhi Wang, 22-Sep-2024.) (New usage is discouraged.)

MndToCat = (𝑚 ∈ Mnd ↦ {⟨(Base‘ndx), {𝑚}⟩, ⟨(Hom ‘ndx), {⟨𝑚, 𝑚, (Base‘𝑚)⟩}⟩, ⟨(comp‘ndx), {⟨⟨𝑚, 𝑚, 𝑚⟩, (+g‘𝑚)⟩}⟩})
 
22-Sep-2024eleq2w2ALT 34914 Alternate proof of eleq2w2 2730 and special instance of eleq2 2822. (Contributed by BJ, 22-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
 
22-Sep-2024bj-nnfeai 34612 Nonfreeness implies the equivalent of ax5ea 1921, inference form. (Contributed by BJ, 22-Sep-2024.)
Ⅎ'𝑥𝜑    ⇒   (∃𝑥𝜑 → ∀𝑥𝜑)
 
22-Sep-2024bj-nnfei 34609 Nonfreeness implies the equivalent of ax5e 1920, inference form. (Contributed by BJ, 22-Sep-2024.)
Ⅎ'𝑥𝜑    ⇒   (∃𝑥𝜑 → 𝜑)
 
22-Sep-2024bj-nnfai 34606 Nonfreeness implies the equivalent of ax-5 1918, inference form. See nf5ri 2193. (Contributed by BJ, 22-Sep-2024.)
Ⅎ'𝑥𝜑    ⇒   (𝜑 → ∀𝑥𝜑)
 
22-Sep-2024cphpyth 24085 The pythagorean theorem for a subcomplex pre-Hilbert space. The square of the norm of the sum of two orthogonal vectors (i.e., whose inner product is 0) is the sum of the squares of their norms. Problem 2 in [Kreyszig] p. 135. This is Metamath 100 proof #4. (Contributed by NM, 17-Apr-2008.) (Revised by SN, 22-Sep-2024.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖‘𝑊)    &    + = (+g‘𝑊)    &   𝑁 = (norm‘𝑊)    &   (𝜑 → 𝑊 ∈ ℂPreHil)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝑁‘(𝐴 + 𝐵))↑2) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))
 
22-Sep-2024raleqbidvv 3308 Version of raleqbidv 3306 with additional disjoint variable conditions, not requiring ax-8 2112 nor df-clel 2812. (Contributed by BJ, 22-Sep-2024.)
(𝜑 → 𝐴 = 𝐵)    &   (𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))
 
22-Sep-2024drnfc2 2921 Formula-building lemma for use with the Distinctor Reduction Theorem. Proof revision is marked as discouraged because the minimizer replaces albidv 1928 with dral2 2435, leading to a one byte longer proof. However feel free to manually edit it according to conventions. (TODO: dral2 2435 depends on ax-13 2369, hence its usage during minimizing is discouraged. Check in the long run whether this is a permanent restriction). Usage of this theorem is discouraged because it depends on ax-13 2369. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-8 2112. (Revised by Wolf Lammen, 22-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝐴 ↔ Ⅎ𝑧𝐵))
 
22-Sep-2024drnfc1 2919 Formula-building lemma for use with the Distinctor Reduction Theorem. Usage of this theorem is discouraged because it depends on ax-13 2369. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-8 2112, ax-11 2158. (Revised by Wolf Lammen, 22-Sep-2024.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑦𝐵))
 
22-Sep-202419.36imv 1953 One direction of 19.36v 1996 that can be proven without ax-6 1976. (Contributed by Rohan Ridenour, 16-Apr-2022.) (Proof shortened by Wolf Lammen, 22-Sep-2024.)
(∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
 
21-Sep-2024prstchom2 45984 Hom-sets of the constructed category are dependent on the preorder.

Note that prstchom.x and prstchom.y are redundant here due to our definition of ProsetToCat ( see prstchom2ALT 45985). However, this should not be assumed as it is definition-dependent. Therefore, the two hypotheses are added for explicitness. (Contributed by Zhi Wang, 21-Sep-2024.)

(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ≤ = (le‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   (𝜑 → 𝑌 ∈ (Base‘𝐶))    ⇒   (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
 
21-Sep-2024thincn0eu 45940 In a thin category, a hom-set being non-empty is equivalent to having a unique element. (Contributed by Zhi Wang, 21-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   (𝜑 → ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
 
21-Sep-2024thincmod 45939 At most one morphism in each hom-set (deduction form). (Contributed by Zhi Wang, 21-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
 
21-Sep-2024thincmoALT 45938 Alternate proof for thincmo 45937. (Contributed by Zhi Wang, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝐶 ∈ ThinCat)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    ⇒   (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
 
21-Sep-2024thincmo 45937 There is at most one morphism in each hom-set. (Contributed by Zhi Wang, 21-Sep-2024.)
(𝜑 → 𝐶 ∈ ThinCat)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    ⇒   (𝜑 → ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑌))
 
21-Sep-2024idepi 45925 An identity arrow, or an identity morphism, is an epimorphism. (Contributed by Zhi Wang, 21-Sep-2024.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    1 = (Id‘𝐶)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   𝐸 = (Epi‘𝐶)    ⇒   (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐸𝑋))
 
21-Sep-2024idmon 45924 An identity arrow, or an identity morphism, is a monomorphism. (Contributed by Zhi Wang, 21-Sep-2024.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    1 = (Id‘𝐶)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   𝑀 = (Mono‘𝐶)    ⇒   (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝑀𝑋))
 
21-Sep-2024fineqvacALT 32752 Shorter proof of fineqvac 32751 using ax-rep 5168 and ax-pow 5247. (Contributed by BTernaryTau, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(Fin = V → CHOICE)
 
21-Sep-2024fineqvac 32751 If the Axiom of Infinity is negated, then the Axiom of Choice becomes redundant. For a shorter proof using ax-rep 5168 and ax-pow 5247, see fineqvacALT 32752. (Contributed by BTernaryTau, 21-Sep-2024.)
(Fin = V → CHOICE)
 
21-Sep-2024ffrnb 6549 Characterization of a function with domain and codomain (essentially using that the range is always included in the codomain). Generalization of ffrn 6548. (Contributed by BJ, 21-Sep-2024.)
(𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
 
21-Sep-2024sbalex 2240 Equivalence of two ways to express proper substitution of a setvar for another setvar disjoint from it in a formula. This proof of their equivalence does not use df-sb 2071.

That both sides of the biconditional express proper substitution is proved by sb5 2272 and sb6 2091. The implication "to the left" is equs4v 2008 and does not require ax-10 2141 nor ax-12 2175. It also holds without disjoint variable condition if we allow more axioms (see equs4 2413). Theorem 6.2 of [Quine] p. 40. Theorem equs5 2457 replaces the disjoint variable condition with a distinctor antecedent. Theorem equs45f 2456 replaces the disjoint variable condition on 𝑥, 𝑡 with the nonfreeness hypothesis of 𝑡 in 𝜑. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2265 in place of equsex 2415 in order to remove dependency on ax-13 2369. (Revised by BJ, 20-Dec-2020.) Revise to remove dependency on df-sb 2071. (Revised by BJ, 21-Sep-2024.)

(∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
 
21-Sep-2024cad0 1625 If one input is false, then the adder carry is true exactly when both of the other two inputs are true. (Contributed by Mario Carneiro, 8-Sep-2016.) (Proof shortened by Wolf Lammen, 21-Sep-2024.)
(¬ 𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∧ 𝜓)))
 
20-Sep-2024prstchom2ALT 45985 Hom-sets of the constructed category are dependent on the preorder. This proof depends on the definition df-prstc 45971. See prstchom2 45984 for a version that does not depend on the definition. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ≤ = (le‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    ⇒   (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)))
 
20-Sep-2024prstchom 45983 Hom-sets of the constructed category are dependent on the preorder.

Note that prstchom.x and prstchom.y are redundant here due to our definition of ProsetToCat. However, this should not be assumed as it is definition-dependent. Therefore, the two hypotheses are added for explicitness. (Contributed by Zhi Wang, 20-Sep-2024.)

(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ≤ = (le‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   (𝜑 → 𝑋 ∈ (Base‘𝐶))    &   (𝜑 → 𝑌 ∈ (Base‘𝐶))    ⇒   (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅))
 
20-Sep-2024prstcthin 45982 The preordered set is equipped with a thin category. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    ⇒   (𝜑 → 𝐶 ∈ ThinCat)
 
20-Sep-2024prstcprs 45981 The category is a preordered set. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    ⇒   (𝜑 → 𝐶 ∈ Proset )
 
20-Sep-2024prstchomval 45980 Hom-sets of the constructed category which depend on an arbitrary definition. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ≤ = (le‘𝐶))    ⇒   (𝜑 → ( ≤ × {1o}) = (Hom ‘𝐶))
 
20-Sep-2024prstcoc 45979 Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ⊥ = (oc‘𝐾))    ⇒   (𝜑 → ( ⊥ ‘𝑋) = ((oc‘𝐶)‘𝑋))
 
20-Sep-2024prstcocval 45978 Orthocomplementation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ⊥ = (oc‘𝐾))    ⇒   (𝜑 → ⊥ = (oc‘𝐶))
 
20-Sep-2024prstcle 45977 Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ≤ = (le‘𝐾))    ⇒   (𝜑 → (𝑋 ≤ 𝑌 ↔ 𝑋(le‘𝐶)𝑌))
 
20-Sep-2024prstcleval 45976 Value of the less-than-or-equal-to relation is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → ≤ = (le‘𝐾))    ⇒   (𝜑 → ≤ = (le‘𝐶))
 
20-Sep-2024prstcbas 45975 The base set is unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   (𝜑 → 𝐵 = (Base‘𝐾))    ⇒   (𝜑 → 𝐵 = (Base‘𝐶))
 
20-Sep-2024prstcnid 45974 Components other than Hom and comp are unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (comp‘ndx)    &   (𝐸‘ndx) ≠ (Hom ‘ndx)    ⇒   (𝜑 → (𝐸‘𝐾) = (𝐸‘𝐶))
 
20-Sep-2024prstcnidlem 45973 Lemma for prstcnid 45974 and prstchomval 45980. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    &   𝐸 = Slot (𝐸‘ndx)    &   (𝐸‘ndx) ≠ (comp‘ndx)    ⇒   (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)))
 
20-Sep-2024prstcval 45972 Lemma for prstcnidlem 45973 and prstcthin 45982. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
(𝜑 → 𝐶 = (ProsetToCat‘𝐾))    &   (𝜑 → 𝐾 ∈ Proset )    ⇒   (𝜑 → 𝐶 = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
 
20-Sep-2024df-prstc 45971 Definition of the function converting a preordered set to a category. Justified by prsthinc 45962.

This definition is somewhat arbitrary. Example 3.3(4.d) of [Adamek] p. 24 demonstrates an alternate definition with pairwise disjoint hom-sets. The behavior of the function is defined entirely, up to isomorphism, by prstcnid 45974, prstchom 45983, and prstcthin 45982. Other important properties include prstcbas 45975, prstcleval 45976, prstcle 45977, prstcocval 45978, prstcoc 45979, prstchom2 45984, and prstcprs 45981. Use those instead.

Note that the defining property prstchom 45983 is equivalent to prstchom2 45984 given prstcthin 45982. See thincn0eu 45940 for justification.

"ProsetToCat" was taken instead of "ProsetCat" because the latter might mean the category of preordered sets (classes). However, "ProsetToCat" seems too long. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)

ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
 
20-Sep-2024setc2othin 45964 The category (SetCat‘2o) is thin. A special case of setcthin 45963. (Contributed by Zhi Wang, 20-Sep-2024.)
(SetCat‘2o) ∈ ThinCat
 
20-Sep-2024setcthin 45963 A category of sets all of whose objects contain at most one element is thin. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐶 = (SetCat‘𝑈))    &   (𝜑 → 𝑈 ∈ 𝑉)    &   (𝜑 → ∀𝑥 ∈ 𝑈 ∃*𝑝 𝑝 ∈ 𝑥)    ⇒   (𝜑 → 𝐶 ∈ ThinCat)
 
20-Sep-2024fvconstrn0 45811 Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝜑 → 𝐹 = (𝑅 × {𝑌}))    &   (𝜑 → 𝑌 ∈ 𝑉)    &   (𝜑 → 𝑌 ≠ ∅)    ⇒   (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) ≠ ∅))
 
20-Sep-2024mof02 45793 A variant of mof0 45792. (Contributed by Zhi Wang, 20-Sep-2024.)
(𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵)
 
20-Sep-2024f1co 6616 Composition of one-to-one functions when the codomain of the first matches the domain of the second. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.) (Proof shortened by AV, 20-Sep-2024.)
((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶)
 
20-Sep-2024funcofd 6567 Composition of two functions as a function with domain and codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Proof shortened by AV, 20-Sep-2024.)
(𝜑 → Fun 𝐹)    &   (𝜑 → Fun 𝐺)    ⇒   (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)
 
20-Sep-2024fco 6558 Composition of two functions with domain and codomain as a function with domain and codomain. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Proof shortened by AV, 20-Sep-2024.)
((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
 
20-Sep-2024fimacnv 6556 The preimage of the codomain of a function is the function's domain. (Contributed by FL, 25-Jan-2007.) (Proof shortened by AV, 20-Sep-2024.)
(𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴)
 
20-Sep-2024ffrnbd 6550 A function maps to its range iff the the range is a subset of its codomain. Generalization of ffrn 6548. (Contributed by AV, 20-Sep-2024.)
(𝜑 → ran 𝐹 ⊆ 𝐵)    ⇒   (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶ran 𝐹))
 
20-Sep-2024fnco 6483 Composition of two functions with domains as a function with domain. (Contributed by NM, 22-May-2006.) (Proof shortened by AV, 20-Sep-2024.)
((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)
 
20-Sep-2024ineqcomi 4108 Two ways of expressing that two classes have a given intersection. Inference form of ineqcom 4107. Disjointness inference when 𝐶 = ∅. (Contributed by Peter Mazsa, 26-Mar-2017.) (Proof shortened by SN, 20-Sep-2024.)
(𝐴 ∩ 𝐵) = 𝐶    ⇒   (𝐵 ∩ 𝐴) = 𝐶
 
20-Sep-2024ecase3ad 1036 Deduction for elimination by cases. (Contributed by NM, 24-May-2013.) (Proof shortened by Wolf Lammen, 20-Sep-2024.)
(𝜑 → (𝜓 → 𝜃))    &   (𝜑 → (𝜒 → 𝜃))    &   (𝜑 → ((¬ 𝜓 ∧ ¬ 𝜒) → 𝜃))    ⇒   (𝜑 → 𝜃)
 
19-Sep-2024indthinc 45960 An indiscrete category in which all hom-sets have exactly one morphism is a thin category. Constructed here is an indiscrete category where all morphisms are ∅. This is a special case of prsthinc 45962, where ≤ = (𝐵 × 𝐵). This theorem also implies a functor from the category of sets to the category of small categories. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof shortened by Zhi Wang, 19-Sep-2024.)
(𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶))    &   (𝜑 → ∅ = (comp‘𝐶))    &   (𝜑 → 𝐶 ∈ 𝑉)    ⇒   (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))
 
19-Sep-2024f1omo 45815 There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Proof could be significantly shortened by fvconstdomi 45814 assuming ax-un 7512 (see f1omoALT 45816). (Contributed by Zhi Wang, 19-Sep-2024.)
(𝜑 → 𝐹 = (𝐴 × {1o}))    ⇒   (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))
 
19-Sep-2024mofmo 45801 There is at most one function into a class containing at most one element. (Contributed by Zhi Wang, 19-Sep-2024.)
(∃*𝑥 𝑥 ∈ 𝐵 → ∃*𝑓 𝑓:𝐴⟶𝐵)
 
19-Sep-2024mofsn2 45799 There is at most one function into a singleton. An unconditional variant of mofsn 45798, i.e., the singleton could be empty if 𝑌 is a proper class. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝐵 = {𝑌} → ∃*𝑓 𝑓:𝐴⟶𝐵)
 
19-Sep-2024mofsn 45798 There is at most one function into a singleton, with fewer axioms than eufsn 45796 and eufsn2 45797. See also mofsn2 45799. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝐵 ∈ 𝑉 → ∃*𝑓 𝑓:𝐴⟶{𝐵})
 
19-Sep-2024eufsn2 45797 There is exactly one function into a singleton, assuming ax-pow 5247 and ax-un 7512. Variant of eufsn 45796. If existence is not needed, use mofsn 45798 or mofsn2 45799 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
 
19-Sep-2024eufsn 45796 There is exactly one function into a singleton, assuming ax-rep 5168. See eufsn2 45797 for different axiom requirements. If existence is not needed, use mofsn 45798 or mofsn2 45799 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
 
19-Sep-2024eufsnlem 45795 There is exactly one function into a singleton. For a simpler hypothesis, see eufsn 45796 assuming ax-rep 5168, or eufsn2 45797 assuming ax-pow 5247 and ax-un 7512. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → (𝐴 × {𝐵}) ∈ 𝑉)    ⇒   (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
 
19-Sep-2024mof0ALT 45794 Alternate proof for mof0 45792 with stronger requirements on distinct variables. Uses mo4 2563. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
∃*𝑓 𝑓:𝐴⟶∅
 
19-Sep-2024mof0 45792 There is at most one function into the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
∃*𝑓 𝑓:𝐴⟶∅
 
19-Sep-2024mo0sn 45788 Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 19-Sep-2024.)
(∃*𝑥 𝑥 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
 
19-Sep-2024mo0 45786 "At most one" element in an empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)
 
19-Sep-2024mosn 45785 "At most one" element in a singleton. (Contributed by Zhi Wang, 19-Sep-2024.)
(𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
 
19-Sep-2024vsn 45784 The singleton of the universal class is the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
{V} = ∅
 
19-Sep-2024f1cof1b 44195 If the range of 𝐹 equals the domain of 𝐺, then the composition (𝐺 ∘ 𝐹) is injective iff 𝐹 and 𝐺 are both injective. (Contributed by GL and AV, 19-Sep-2024.)
((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐶⟶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺 ∘ 𝐹):𝐴–1-1→𝐷 ↔ (𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷)))
 
19-Sep-20242oex 8208 2o is a set. (Contributed by BJ, 6-Apr-2019.) Remove dependency on ax-10 2141, ax-11 2158, ax-12 2175, ax-un 7512. (Proof shortened by Zhi Wang, 19-Sep-2024.)
2o ∈ V
 
19-Sep-20241oex 8204 Ordinal 1 is a set. (Contributed by BJ, 6-Apr-2019.) (Proof shortened by AV, 1-Jul-2022.) Remove dependency on ax-10 2141, ax-11 2158, ax-12 2175, ax-un 7512. (Revised by Zhi Wang, 19-Sep-2024.)
1o ∈ V
 
19-Sep-2024ecase2d 1030 Deduction for elimination by cases. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Sep-2024.)
(𝜑 → 𝜓)    &   (𝜑 → ¬ (𝜓 ∧ 𝜒))    &   (𝜑 → ¬ (𝜓 ∧ 𝜃))    &   (𝜑 → (𝜏 ∨ (𝜒 ∨ 𝜃)))    ⇒   (𝜑 → 𝜏)
 
18-Sep-2024prsthinc 45962 Preordered sets as categories. Similar to example 3.3(4.d) of [Adamek] p. 24, but the hom-sets are not pairwise disjoint. One can define a functor from the category of prosets to the category of small thin categories. See catprs 45919 and catprs2 45920 for inducing a preorder from a category. Example 3.26(2) of [Adamek] p. 33 indicates that it induces a bijection from the equivalence class of isomorphic small thin categories to the equivalence class of order-isomorphic preordered sets. (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → ( ≤ × {1o}) = (Hom ‘𝐶))    &   (𝜑 → ∅ = (comp‘𝐶))    &   (𝜑 → ≤ = (le‘𝐶))    &   (𝜑 → 𝐶 ∈ Proset )    ⇒   (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))
 
18-Sep-2024catprsc 45921 A construction of the preorder induced by a category. See catprs2 45920 for details. See also catprsc2 45922 for an alternate construction. (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑 → ≤ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ (𝑥𝐻𝑦) ≠ ∅)})    ⇒   (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 ≤ 𝑤 ↔ (𝑧𝐻𝑤) ≠ ∅))
 
18-Sep-2024catprs2 45920 A category equipped with the induced preorder, where an object 𝑥 is defined to be "less than or equal to" 𝑦 iff there is a morphism from 𝑥 to 𝑦, is a preordered set, or a proset. The category might not be thin. See catprsc 45921 and catprsc2 45922 for constructions satisfying the hypothesis "catprs.1". See catprs 45919 for a more primitive version. See prsthinc 45962 for constructing a thin category from a proset. (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → ≤ = (le‘𝐶))    ⇒   (𝜑 → 𝐶 ∈ Proset )
 
18-Sep-2024catprs 45919 A preorder can be extracted from a category. See catprs2 45920 for more details. (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))    &   (𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   (𝜑 → 𝐶 ∈ Cat)    ⇒   ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑋 ∧ ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)))
 
18-Sep-2024catprslem 45918 Lemma for catprs 45919. (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝑥𝐻𝑦) ≠ ∅))    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅))
 
18-Sep-2024isprsd 45876 Property of being a preordered set (deduction form). (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → ≤ = (le‘𝐾))    &   (𝜑 → 𝐾 ∈ 𝑉)    ⇒   (𝜑 → (𝐾 ∈ Proset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
 
18-Sep-2024f1omoALT 45816 There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Use f1omo 45815 without assuming ax-un 7512. (Contributed by Zhi Wang, 18-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝐹 = (𝐴 × {1o}))    ⇒   (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))
 
18-Sep-2024fvconstdomi 45814 A constant function's value is dominated by the constant. (An artifact of our function value definition.) (Contributed by Zhi Wang, 18-Sep-2024.)
𝐵 ∈ V    ⇒   ((𝐴 × {𝐵})‘𝑋) ≼ 𝐵
 
18-Sep-2024fvconst0ci 45813 A constant function's value is either the constant or the empty set. (An artifact of our function value definition.) (Contributed by Zhi Wang, 18-Sep-2024.)
𝐵 ∈ V    &   𝑌 = ((𝐴 × {𝐵})‘𝑋)    ⇒   (𝑌 = ∅ ∨ 𝑌 = 𝐵)
 
18-Sep-2024fvconstr2 45812 Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑 → 𝐹 = (𝑅 × {𝑌}))    &   (𝜑 → 𝑋 ∈ (𝐴𝐹𝐵))    ⇒   (𝜑 → 𝐴𝑅𝐵)
 
18-Sep-2024fvconstr 45810 Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 18-Sep-2024.)
(𝜑 → 𝐹 = (𝑅 × {𝑌}))    &   (𝜑 → 𝑌 ∈ 𝑉)    &   (𝜑 → 𝑌 ≠ ∅)    ⇒   (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) = 𝑌))
 
18-Sep-2024f1cof1blem 44194 Lemma for f1cof1b 44195 and focofob 44198. (Contributed by AV, 18-Sep-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    &   (𝜑 → 𝐺:𝐶⟶𝐷)    &   𝑌 = (𝐺 ↾ 𝐸)    &   (𝜑 → ran 𝐹 = 𝐶)    ⇒   (𝜑 → ((𝑃 = 𝐴 ∧ 𝐸 = 𝐶) ∧ (𝑋 = 𝐹 ∧ 𝑌 = 𝐺)))
 
18-Sep-2024fcoresf1lem 44188 Lemma for fcoresf1 44189. (Contributed by AV, 18-Sep-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    &   (𝜑 → 𝐺:𝐶⟶𝐷)    &   𝑌 = (𝐺 ↾ 𝐸)    ⇒   ((𝜑 ∧ 𝑍 ∈ 𝑃) → ((𝐺 ∘ 𝐹)‘𝑍) = (𝑌‘(𝑋‘𝑍)))
 
18-Sep-2024sbn1ALT 34736 Alternate proof of sbn1 2109, not using the false constant. (Contributed by BJ, 18-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)
 
18-Sep-2024ssltdisj 33709 If 𝐴 preceeds 𝐵, then 𝐴 and 𝐵 are disjoint. (Contributed by Scott Fenton, 18-Sep-2024.)
(𝐴 <<s 𝐵 → (𝐴 ∩ 𝐵) = ∅)
 
18-Sep-2024catcone0 17162 Composition of non-empty hom-sets is non-empty. (Contributed by Zhi Wang, 18-Sep-2024.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &    · = (comp‘𝐶)    &   (𝜑 → 𝐶 ∈ Cat)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → (𝑋𝐻𝑌) ≠ ∅)    &   (𝜑 → (𝑌𝐻𝑍) ≠ ∅)    ⇒   (𝜑 → (𝑋𝐻𝑍) ≠ ∅)
 
18-Sep-2024f1cof1 6615 Composition of two one-to-one functions. Generalization of f1co 6616. (Contributed by AV, 18-Sep-2024.)
((𝐹:𝐶–1-1→𝐷 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)–1-1→𝐷)
 
18-Sep-2024fcof 6557 Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 6558. (Contributed by AV, 18-Sep-2024.)
((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
 
18-Sep-2024cnvimassrndm 6004 The preimage of a superset of the range of a class is the domain of the class. Generalization of cnvimarndm 5939 for subsets. (Contributed by AV, 18-Sep-2024.)
(ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹)
 
18-Sep-2024abeq2w 2811 Version of abeq2 2865 using implicit substitution, which requires fewer axioms. (Contributed by GG and AV, 18-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑦(𝑦 ∈ 𝐴 ↔ 𝜓))
 
17-Sep-2024indthincALT 45961 An alternate proof for indthinc 45960 assuming more axioms including ax-pow 5247 and ax-un 7512. (Contributed by Zhi Wang, 17-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → ((𝐵 × 𝐵) × {1o}) = (Hom ‘𝐶))    &   (𝜑 → ∅ = (comp‘𝐶))    &   (𝜑 → 𝐶 ∈ 𝑉)    ⇒   (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ ∅)))
 
17-Sep-20240thinc 45959 The empty category (see 0cat 17164) is thin. (Contributed by Zhi Wang, 17-Sep-2024.)
∅ ∈ ThinCat
 
17-Sep-20240thincg 45958 Any structure with an empty set of objects is a thin category. (Contributed by Zhi Wang, 17-Sep-2024.)
((𝐶 ∈ 𝑉 ∧ ∅ = (Base‘𝐶)) → 𝐶 ∈ ThinCat)
 
17-Sep-2024isthincd2 45946 The predicate "𝐶 is a thin category" without knowing 𝐶 is a category (deduction form). The identity arrow operator is also provided as a byproduct. (Contributed by Zhi Wang, 17-Sep-2024.)
(𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))    &   (𝜑 → · = (comp‘𝐶))    &   (𝜑 → 𝐶 ∈ 𝑉)    &   (𝜓 ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))))    &   ((𝜑 ∧ 𝑦 ∈ 𝐵) → 1 ∈ (𝑦𝐻𝑦))    &   ((𝜑 ∧ 𝜓) → (𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))    ⇒   (𝜑 → (𝐶 ∈ ThinCat ∧ (Id‘𝐶) = (𝑦 ∈ 𝐵 ↦ 1 )))
 
17-Sep-2024isthincd 45945 The predicate "is a thin category" (deduction form). (Contributed by Zhi Wang, 17-Sep-2024.)
(𝜑 → 𝐵 = (Base‘𝐶))    &   (𝜑 → 𝐻 = (Hom ‘𝐶))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))    &   (𝜑 → 𝐶 ∈ Cat)    ⇒   (𝜑 → 𝐶 ∈ ThinCat)
 
17-Sep-2024isthincd2lem2 45944 Lemma for isthincd2 45946. (Contributed by Zhi Wang, 17-Sep-2024.)
(𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))    &   (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩ · 𝑧)𝑓) ∈ (𝑥𝐻𝑧))    ⇒   (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) ∈ (𝑋𝐻𝑍))
 
17-Sep-2024thincmo2 45936 Morphisms in the same hom-set are identical. (Contributed by Zhi Wang, 17-Sep-2024.)
(𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → 𝐶 ∈ ThinCat)    ⇒   (𝜑 → 𝐹 = 𝐺)
 
17-Sep-2024isthincd2lem1 45935 Lemma for isthincd2 45946 and thincmo2 45936. (Contributed by Zhi Wang, 17-Sep-2024.)
(𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))    &   (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))    &   (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))    ⇒   (𝜑 → 𝐹 = 𝐺)
 
17-Sep-2024thincssc 45934 A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.)
ThinCat ⊆ Cat
 
17-Sep-2024thincc 45932 A thin category is a category. (Contributed by Zhi Wang, 17-Sep-2024.)
(𝐶 ∈ ThinCat → 𝐶 ∈ Cat)
 
17-Sep-2024isthinc3 45931 A thin category is a category in which, given a pair of objects 𝑥 and 𝑦 and any two morphisms 𝑓, 𝑔 from 𝑥 to 𝑦, the morphisms are equal. (Contributed by Zhi Wang, 17-Sep-2024.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    ⇒   (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑥𝐻𝑦)𝑓 = 𝑔))
 
17-Sep-2024isthinc2 45930 A thin category is a category in which all hom-sets have cardinality less than or equal to the cardinality of 1o. (Contributed by Zhi Wang, 17-Sep-2024.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    ⇒   (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ≼ 1o))
 
17-Sep-2024isthinc 45929 The predicate "is a thin category". (Contributed by Zhi Wang, 17-Sep-2024.)
𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    ⇒   (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
 
17-Sep-2024df-thinc 45928 Definition of the class of thin categories, or posetal categories, whose hom-sets each contain at most one morphism. Example 3.26(2) of [Adamek] p. 33. "ThinCat" was taken instead of "PosCat" because the latter might mean the category of posets. (Contributed by Zhi Wang, 17-Sep-2024.)
ThinCat = {𝑐 ∈ Cat ∣ [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃*𝑓 𝑓 ∈ (𝑥ℎ𝑦)}
 
17-Sep-2024fcoresfo 44191 If a composition is surjective, then the restriction of its first component to the minimum domain is surjective. (Contributed by AV, 17-Sep-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    &   (𝜑 → 𝐺:𝐶⟶𝐷)    &   𝑌 = (𝐺 ↾ 𝐸)    &   (𝜑 → (𝐺 ∘ 𝐹):𝑃–onto→𝐷)    ⇒   (𝜑 → 𝑌:𝐸–onto→𝐷)
 
17-Sep-2024fcores 44187 Every composite function (𝐺 ∘ 𝐹) can be written as composition of restrictions of the composed functions (to their minimum domains). (Contributed by GL and AV, 17-Sep-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    &   (𝜑 → 𝐺:𝐶⟶𝐷)    &   𝑌 = (𝐺 ↾ 𝐸)    ⇒   (𝜑 → (𝐺 ∘ 𝐹) = (𝑌 ∘ 𝑋))
 
17-Sep-2024fcoreslem4 44186 Lemma 4 for fcores 44187. (Contributed by AV, 17-Sep-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    &   (𝜑 → 𝐺:𝐶⟶𝐷)    &   𝑌 = (𝐺 ↾ 𝐸)    ⇒   (𝜑 → (𝑌 ∘ 𝑋) Fn 𝑃)
 
17-Sep-2024fcoreslem2 44184 Lemma 2 for fcores 44187. (Contributed by AV, 17-Sep-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    ⇒   (𝜑 → ran 𝑋 = 𝐸)
 
17-Sep-2024fcoreslem1 44183 Lemma 1 for fcores 44187. (Contributed by AV, 17-Sep-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    ⇒   (𝜑 → 𝑃 = (◡𝐹 “ 𝐸))
 
17-Sep-2024sltlpss 33781 If two surreals share a birthday, then 𝑋 <s 𝑌 iff the left set of 𝑋 is a proper subset of the left set of 𝑌. (Contributed by Scott Fenton, 17-Sep-2024.)
((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (𝑋 <s 𝑌 ↔ ( L ‘𝑋) ⊊ ( L ‘𝑌)))
 
17-Sep-2024lruneq 33780 If two surreals share a birthday, then the union of their left and right sets are equal. (Contributed by Scott Fenton, 17-Sep-2024.)
((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ ( bday ‘𝑋) = ( bday ‘𝑌)) → (( L ‘𝑋) ∪ ( R ‘𝑋)) = (( L ‘𝑌) ∪ ( R ‘𝑌)))
 
17-Sep-2024cnvimainrn 6876 The preimage of the intersection of the range of a class and a class 𝐴 is the preimage of the class 𝐴. (Contributed by AV, 17-Sep-2024.)
(Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐴)) = (◡𝐹 “ 𝐴))
 
17-Sep-2024fncofn 6482 Composition of a function with domain and a function as a function with domain. Generalization of fnco 6483. (Contributed by AV, 17-Sep-2024.)
((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))
 
16-Sep-2024neircl 45825 Reverse closure of the neighborhood operation. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by Zhi Wang, 16-Sep-2024.)
(𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝐽 ∈ Top)
 
16-Sep-2024elfvne0 45803 If a function value has a member, then the function is not an empty set (An artifact of our function value definition.) (Contributed by Zhi Wang, 16-Sep-2024.)
(𝐴 ∈ (𝐹‘𝐵) → 𝐹 ≠ ∅)
 
16-Sep-2024isdomn5 39845 The right conjunct in the right hand side of the equivalence of isdomn 20304 is logically equivalent to a less symmetric version where one of the variables is restricted to be nonzero. (Contributed by SN, 16-Sep-2024.)
(∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 )) ↔ ∀𝑎 ∈ (𝐵 ∖ { 0 })∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 ))
 
15-Sep-2024dvdsexpb 40002 dvdssq 16105 generalized to positive integer exponents. (Contributed by SN, 15-Sep-2024.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 ∥ 𝐵 ↔ (𝐴↑𝑁) ∥ (𝐵↑𝑁)))
 
15-Sep-2024dvdsexpnn0 40001 dvdsexpnn 40000 generalized to include zero bases. (Contributed by SN, 15-Sep-2024.)
((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝐴 ∥ 𝐵 ↔ (𝐴↑𝑁) ∥ (𝐵↑𝑁)))
 
15-Sep-2024absdvdsabsb 39987 Divisibility is invariant under taking the absolute value on both sides. (Contributed by SN, 15-Sep-2024.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (abs‘𝑀) ∥ (abs‘𝑁)))
 
15-Sep-20240dvds0 39986 0 divides 0. (Contributed by SN, 15-Sep-2024.)
0 ∥ 0
 
15-Sep-2024zexpcld 39981 Closure of exponentiation of integers, deductive form. (Contributed by SN, 15-Sep-2024.)
(𝜑 → 𝐴 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℕ0)    ⇒   (𝜑 → (𝐴↑𝑁) ∈ ℤ)
 
15-Sep-2024syl3an12 39849 A double syllogism inference. (Contributed by SN, 15-Sep-2024.)
(𝜑 → 𝜓)    &   (𝜒 → 𝜃)    &   ((𝜓 ∧ 𝜃 ∧ 𝜏) → 𝜂)    ⇒   ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂)
 
15-Sep-2024isdomn4 39846 A ring is a domain iff it is nonzero and the cancellation law for multiplication holds. (Contributed by SN, 15-Sep-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    ⇒   (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑎 ∈ (𝐵 ∖ { 0 })∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ((𝑎 · 𝑏) = (𝑎 · 𝑐) → 𝑏 = 𝑐)))
 
15-Sep-2024cat1 17575 The definition of category df-cat 17143 does not impose pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. See setc2obas 17572 and setc2ohom 17573 for a counterexample. For a version with pairwise disjoint hom-sets, see df-homa 17504 and its subsection. (Contributed by Zhi Wang, 15-Sep-2024.)
∃𝑐 ∈ Cat [(Base‘𝑐) / 𝑏][(Hom ‘𝑐) / ℎ] ¬ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ∀𝑤 ∈ 𝑏 (((𝑥ℎ𝑦) ∩ (𝑧ℎ𝑤)) ≠ ∅ → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))
 
15-Sep-2024cat1lem 17574 The category of sets in a "universe" containing the empty set and another set does not have pairwise disjoint hom-sets as required in Axiom CAT 1 in [Lang] p. 53. Lemma for cat1 17575. (Contributed by Zhi Wang, 15-Sep-2024.)
𝐶 = (SetCat‘𝑈)    &   (𝜑 → 𝑈 ∈ 𝑉)    &   𝐵 = (Base‘𝐶)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑 → ∅ ∈ 𝑈)    &   (𝜑 → 𝑌 ∈ 𝑈)    &   (𝜑 → ∅ ≠ 𝑌)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐵 (((𝑥𝐻𝑦) ∩ (𝑧𝐻𝑤)) ≠ ∅ ∧ ¬ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
 
15-Sep-2024gcdabs 16071 The gcd of two integers is the same as that of their absolute values. (Contributed by Paul Chapman, 31-Mar-2011.) (Proof shortened by SN, 15-Sep-2024.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁))
 
15-Sep-2024fsetexb 8534 The class of all functions from a class 𝐴 into a class 𝐵 is a set iff 𝐵 is a set or 𝐴 is not a set or 𝐴 is empty. (Contributed by AV, 15-Sep-2024.)
({𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V ↔ (𝐴 ∉ V ∨ 𝐴 = ∅ ∨ 𝐵 ∈ V))
 
15-Sep-2024fsetcdmex 8533 The class of all functions from a nonempty set 𝐴 into a class 𝐵 is a set iff 𝐵 is a set . (Contributed by AV, 15-Sep-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) → (𝐵 ∈ V ↔ {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V))
 
15-Sep-2024fsetprcnex 8532 The class of all functions from a nonempty set 𝐴 into a proper class 𝐵 is not a set. If one of the preconditions is not fufilled, then {𝑓 ∣ 𝑓:𝐴⟶𝐵} is a set, see fsetdmprc0 8525 for 𝐴 ∉ V, fset0 8524 for 𝐴 = ∅, and fsetex 8526 for 𝐵 ∈ V, see also fsetexb 8534. (Contributed by AV, 14-Sep-2024.) (Proof shortened by BJ, 15-Sep-2024.)
(((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)
 
15-Sep-2024fsetfocdm 8531 The class of functions with a given domain that is a set and a given codomain is mapped, through evaluation at a point of the domain, onto the codomain. (Contributed by AV, 15-Sep-2024.)
𝐹 = {𝑓 ∣ 𝑓:𝐴⟶𝐵}    &   𝑆 = (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴) → 𝑆:𝐹–onto→𝐵)
 
15-Sep-2024fsetfcdm 8530 The class of functions with a given domain and a given codomain is mapped, through evaluation at a point of the domain, into the codomain. (Contributed by AV, 15-Sep-2024.)
𝐹 = {𝑓 ∣ 𝑓:𝐴⟶𝐵}    &   𝑆 = (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋))    ⇒   (𝑋 ∈ 𝐴 → 𝑆:𝐹⟶𝐵)
 
14-Sep-2024fsetprcnexALT 44182 First version of proof for fsetprcnex 8532, which was much more complicated. (Contributed by AV, 14-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝐴 ∈ 𝑉 ∧ 𝐴 ≠ ∅) ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∉ V)
 
14-Sep-2024cfsetsnfsetf1o 44181 The mapping of the class of singleton functions into the class of constant functions is a bijection. (Contributed by AV, 14-Sep-2024.)
𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–1-1-onto→𝐹)
 
14-Sep-2024cfsetsnfsetfo 44180 The mapping of the class of singleton functions into the class of constant functions is a surjection. (Contributed by AV, 14-Sep-2024.)
𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–onto→𝐹)
 
14-Sep-2024cfsetsnfsetf1 44179 The mapping of the class of singleton functions into the class of constant functions is an injection. (Contributed by AV, 14-Sep-2024.)
𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–1-1→𝐹)
 
14-Sep-2024cfsetsnfsetf 44178 The mapping of the class of singleton functions into the class of constant functions is a function. (Contributed by AV, 14-Sep-2024.)
𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺⟶𝐹)
 
13-Sep-2024fcoreslem3 44185 Lemma 3 for fcores 44187. (Contributed by AV, 13-Sep-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   𝐸 = (ran 𝐹 ∩ 𝐶)    &   𝑃 = (◡𝐹 “ 𝐶)    &   𝑋 = (𝐹 ↾ 𝑃)    ⇒   (𝜑 → 𝑋:𝑃–onto→𝐸)
 
13-Sep-2024cfsetsnfsetfv 44177 The function value of the mapping of the class of singleton functions into the class of constant functions. (Contributed by AV, 13-Sep-2024.)
𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    &   𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵}    &   𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌)))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐺) → (𝐻‘𝑋) = (𝑎 ∈ 𝐴 ↦ (𝑋‘𝑌)))
 
13-Sep-2024cfsetssfset 44176 The class of constant functions is a subclass of the class of functions. (Contributed by AV, 13-Sep-2024.)
𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)}    ⇒   𝐹 ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}
 
13-Sep-2024fsetsnprcnex 44175 The class of all functions from a (proper) singleton into a proper class 𝐵 is not a set. (Contributed by AV, 13-Sep-2024.)
((𝑆 ∈ 𝑉 ∧ 𝐵 ∉ V) → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} ∉ V)
 
13-Sep-2024fsetsnf1o 44174 The mapping of an element of a class to a singleton function is a bijection. (Contributed by AV, 13-Sep-2024.)
𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}    &   𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})    ⇒   (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1-onto→𝐴)
 
13-Sep-2024fsetsnfo 44173 The mapping of an element of a class to a singleton function is a surjection. (Contributed by AV, 13-Sep-2024.)
𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}    &   𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})    ⇒   (𝑆 ∈ 𝑉 → 𝐹:𝐵–onto→𝐴)
 
13-Sep-2024fsetsnf1 44172 The mapping of an element of a class to a singleton function is an injection. (Contributed by AV, 13-Sep-2024.)
𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}    &   𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})    ⇒   (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1→𝐴)
 
13-Sep-2024fsetsnf 44171 The mapping of an element of a class to a singleton function is a function. (Contributed by AV, 13-Sep-2024.)
𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}}    &   𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩})    ⇒   (𝑆 ∈ 𝑉 → 𝐹:𝐵⟶𝐴)
 
13-Sep-2024fsetabsnop 44170 The class of all functions from a (proper) singleton into 𝐵 is the class of all the singletons of (proper) ordered pairs over the elements of 𝐵 as second component. (Contributed by AV, 13-Sep-2024.)
(𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}})
 
13-Sep-2024fsetsniunop 44169 The class of all functions from a (proper) singleton into 𝐵 is the union of all the singletons of (proper) ordered pairs over the elements of 𝐵 as second component. (Contributed by AV, 13-Sep-2024.)
(𝑆 ∈ 𝑉 → {𝑓 ∣ 𝑓:{𝑆}⟶𝐵} = ∪ 𝑏 ∈ 𝐵 {{⟨𝑆, 𝑏⟩}})
 
13-Sep-2024fset0 8524 The set of functions from the empty set is the singleton containing the empty set. (Contributed by AV, 13-Sep-2024.)
{𝑓 ∣ 𝑓:∅⟶𝐵} = {∅}
 
13-Sep-2024fsetsspwxp 8523 The class of all functions from 𝐴 into 𝐵 is a subclass of the power class of the cartesion product of 𝐴 and 𝐵. (Contributed by AV, 13-Sep-2024.)
{𝑓 ∣ 𝑓:𝐴⟶𝐵} ⊆ 𝒫 (𝐴 × 𝐵)
 
12-Sep-2024fineqvpow 32750 If the Axiom of Infinity is negated, then the Axiom of Power Sets becomes redundant. (Contributed by BTernaryTau, 12-Sep-2024.)
(Fin = V → ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
 
12-Sep-2024fineqvrep 32749 If the Axiom of Infinity is negated, then the Axiom of Replacement becomes redundant. (Contributed by BTernaryTau, 12-Sep-2024.)
(Fin = V → (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
 
10-Sep-2024iscnrm3v 45874 A topology is completely normal iff two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 10-Sep-2024.)
(𝐽 ∈ Top → (𝐽 ∈ CNrm ↔ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
 
10-Sep-2024onunel 33377 The union of two ordinals is in a third iff both of the first two are. (Contributed by Scott Fenton, 10-Sep-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∪ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶)))
 
10-Sep-2024entrfil 8852 Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 8669). (Contributed by BTernaryTau, 10-Sep-2024.)
((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)
 
9-Sep-2024seppcld 45850 If two sets are precisely separated by a continuous function, then they are closed. An alternate proof involves II ∈ Fre. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1})))    ⇒   (𝜑 → (𝑆 ∈ (Clsd‘𝐽) ∧ 𝑇 ∈ (Clsd‘𝐽)))
 
9-Sep-2024seppsepf 45849 If two sets are precisely separated by a continuous function, then they are separated by the continuous function. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1})))    ⇒   (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (◡𝑓 “ {0}) ∧ 𝑇 ⊆ (◡𝑓 “ {1})))
 
9-Sep-2024sepfsepc 45848 If two sets are separated by a continuous function, then they are separated by closed neighborhoods. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (◡𝑓 “ {0}) ∧ 𝑇 ⊆ (◡𝑓 “ {1})))    ⇒   (𝜑 → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅))
 
9-Sep-2024io1ii 45841 (𝐴(,]1) is open in II. (Contributed by Zhi Wang, 9-Sep-2024.)
(0 ≤ 𝐴 → (𝐴(,]1) ∈ II)
 
9-Sep-2024i0oii 45840 (0[,)𝐴) is open in II. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝐴 ≤ 1 → (0[,)𝐴) ∈ II)
 
9-Sep-2024iooii 45838 Open intervals are open sets of II. (Contributed by Zhi Wang, 9-Sep-2024.)
((0 ≤ 𝐴 ∧ 𝐵 ≤ 1) → (𝐴(,)𝐵) ∈ II)
 
9-Sep-2024cnneiima 45837 Given a continuous function, the preimage of a neighborhood is a neighborhood. To be precise, the preimage of a neighborhood of a subset 𝑇 of the codomain of a continuous function is a neighborhood of any subset of the preimage of 𝑇. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   (𝜑 → 𝑁 ∈ ((nei‘𝐾)‘𝑇))    &   (𝜑 → 𝑆 ⊆ (◡𝐹 “ 𝑇))    ⇒   (𝜑 → (◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆))
 
9-Sep-2024iccdisj 45819 If the upper bound of one closed interval is less than the lower bound of the other, the intervals are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)
((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝐵 < 𝐶) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = ∅)
 
9-Sep-2024iccdisj2 45818 If the upper bound of one closed interval is less than the lower bound of the other, the intervals are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)
((𝐴 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐵 < 𝐶) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = ∅)
 
9-Sep-2024iccin 45817 Intersection of two closed intervals of extended reals. (Contributed by Zhi Wang, 9-Sep-2024.)
(((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)[,]if(𝐵 ≤ 𝐷, 𝐵, 𝐷)))
 
9-Sep-2024predisj 45783 Preimages of disjoint sets are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)
(𝜑 → Fun 𝐹)    &   (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   (𝜑 → 𝑆 ⊆ (◡𝐹 “ 𝐴))    &   (𝜑 → 𝑇 ⊆ (◡𝐹 “ 𝐵))    ⇒   (𝜑 → (𝑆 ∩ 𝑇) = ∅)
 
9-Sep-2024naddss2 33536 Ordinal less-than-or-equal is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +no 𝐴) ⊆ (𝐶 +no 𝐵)))
 
9-Sep-2024naddss1 33535 Ordinal less-than-or-equal is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))
 
9-Sep-2024naddel2 33534 Ordinal less-than is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 ↔ (𝐶 +no 𝐴) ∈ (𝐶 +no 𝐵)))
 
9-Sep-2024naddel1 33533 Ordinal less-than is not affected by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 ↔ (𝐴 +no 𝐶) ∈ (𝐵 +no 𝐶)))
 
9-Sep-2024naddelim 33532 Ordinal less-than is preserved by natural addition. (Contributed by Scott Fenton, 9-Sep-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 +no 𝐶) ∈ (𝐵 +no 𝐶)))
 
9-Sep-2024ensymfib 8851 Symmetry of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike ensymb 8665). (Contributed by BTernaryTau, 9-Sep-2024.)
(𝐴 ∈ Fin → (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴))
 
9-Sep-2024f1oenfirn 8849 If the range of a one-to-one, onto function is finite, then the domain and range of the function are equinumerous. (Contributed by BTernaryTau, 9-Sep-2024.)
((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
 
9-Sep-2024cnvfi 8846 If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.) Avoid ax-pow 5247. (Revised by BTernaryTau, 9-Sep-2024.)
(𝐴 ∈ Fin → ◡𝐴 ∈ Fin)
 
8-Sep-2024sepcsepo 45847 If two sets are separated by closed neighborhoods, then they are separated by (open) neighborhoods. See sepnsepo 45844 for the equivalence between separatedness by open neighborhoods and separatedness by neighborhoods. Although 𝐽 ∈ Top might be redundant here, it is listed for explicitness. 𝐽 ∈ Top can be obtained from neircl 45825, adantr 484, and rexlimiva 3193. (Contributed by Zhi Wang, 8-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → ∃𝑛 ∈ ((nei‘𝐽)‘𝑆)∃𝑚 ∈ ((nei‘𝐽)‘𝑇)(𝑛 ∈ (Clsd‘𝐽) ∧ 𝑚 ∈ (Clsd‘𝐽) ∧ (𝑛 ∩ 𝑚) = ∅))    ⇒   (𝜑 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))
 
8-Sep-2024icccldii 45839 Closed intervals are closed sets of II. Note that iccss 12986, iccordt 22083, and ordtresticc 22092 are proved from ixxss12 12938, ordtcld3 22068, and ordtrest2 22073, respectively. An alternate proof uses restcldi 22042, dfii2 23751, and icccld 23636. (Contributed by Zhi Wang, 8-Sep-2024.)
((0 ≤ 𝐴 ∧ 𝐵 ≤ 1) → (𝐴[,]𝐵) ∈ (Clsd‘II))
 
8-Sep-2024enreffi 8850 Equinumerosity is reflexive for finite sets, proved without using the Axiom of Power Sets (unlike enrefg 8649). (Contributed by BTernaryTau, 8-Sep-2024.)
(𝐴 ∈ Fin → 𝐴 ≈ 𝐴)
 
8-Sep-2024f1oenfi 8848 If the domain of a one-to-one, onto function is finite, then the domain and range of the function are equinumerous. This theorem is proved without using the Axiom of Replacement or the Axiom of Power Sets (unlike f1oeng 8636). (Contributed by BTernaryTau, 8-Sep-2024.)
((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
 
8-Sep-2024relopabv 5680 A class of ordered pairs is a relation. For a version without a disjoint variable condition, but using ax-11 2158 and ax-12 2175, see relopab 5683. (Contributed by SN, 8-Sep-2024.)
Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
 
8-Sep-2024ab0 4279 The class of sets verifying a property is the empty class if and only if that property is a contradiction. See also abn0 4285 (from which it could be proved using as many essential proof steps but one fewer syntactic step, at the cost of depending on df-ne 2936). (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2812, ax-8 2112. (Revised by Gino Giotto, 30-Aug-2024.) (Proof shortened by SN, 8-Sep-2024.)
({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
 
8-Sep-2024ceqsralv 3438 Restricted quantifier version of ceqsalv 3436. (Contributed by NM, 21-Jun-2013.) Avoid ax-9 2120, ax-12 2175, ax-ext 2706. (Revised by SN, 8-Sep-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ 𝜓))
 
8-Sep-2024ceqsalv 3436 A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.) Avoid ax-12 2175. (Revised by SN, 8-Sep-2024.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓)
 
7-Sep-2024seposep 45846 If two sets are separated by (open) neighborhoods, then they are separated subsets of the underlying set. Note that separatedness by open neighborhoods is equivalent to separatedness by neighborhoods. See sepnsepo 45844. The relationship between separatedness and closure is also seen in isnrm 22204, isnrm2 22227, isnrm3 22228. (Contributed by Zhi Wang, 7-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))    ⇒   (𝜑 → ((𝑆 ⊆ ∪ 𝐽 ∧ 𝑇 ⊆ ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)))
 
7-Sep-2024sepdisj 45845 Separated sets are disjoint. Note that in general separatedness also requires 𝑇 ⊆ ∪ 𝐽 and (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ as well but they are unnecessary here. (Contributed by Zhi Wang, 7-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)    ⇒   (𝜑 → (𝑆 ∩ 𝑇) = ∅)
 
7-Sep-2024ssdisjdr 45781 Subset preserves disjointness. Deduction form of ssdisj 4364. Alternatively this could be proved with ineqcom 4107 in tandem with ssdisjd 45780. (Contributed by Zhi Wang, 7-Sep-2024.)
(𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → (𝐶 ∩ 𝐵) = ∅)    ⇒   (𝜑 → (𝐶 ∩ 𝐴) = ∅)
 
7-Sep-2024ssdisjd 45780 Subset preserves disjointness. Deduction form of ssdisj 4364. (Contributed by Zhi Wang, 7-Sep-2024.)
(𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → (𝐵 ∩ 𝐶) = ∅)    ⇒   (𝜑 → (𝐴 ∩ 𝐶) = ∅)
 
7-Sep-2024naddssim 33531 Ordinal less-than-or-equal is preserved by natural addition. (Contributed by Scott Fenton, 7-Sep-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))
 
7-Sep-2024pwfi 8845 The power set of a finite set is finite and vice-versa. Theorem 38 of [Suppes] p. 104 and its converse, Theorem 40 of [Suppes] p. 105. (Contributed by NM, 26-Mar-2007.) Avoid ax-pow 5247. (Revised by BTernaryTau, 7-Sep-2024.)
(𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
 
7-Sep-2024pwfilem 8844 Lemma for pwfi 8845. (Contributed by NM, 26-Mar-2007.) Avoid ax-pow 5247. (Revised by BTernaryTau, 7-Sep-2024.)
𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))    ⇒   (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)
 
7-Sep-2024pwfir 8843 If the power set of a set is finite, then the set itself is finite. (Contributed by BTernaryTau, 7-Sep-2024.)
(𝒫 𝐵 ∈ Fin → 𝐵 ∈ Fin)
 
7-Sep-2024imafi 8842 Images of finite sets are finite. For a shorter proof using ax-pow 5247, see imafiALT 8958. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid ax-pow 5247. (Revised by BTernaryTau, 7-Sep-2024.)
((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin)
 
6-Sep-2024clddisj 45824 Two ways of saying that two closed sets are disjoint, if 𝐽 is a topology and 𝑋 is a closed set. An alternative proof is similar to that of opndisj 45823 with elssuni 4841 replaced by the combination of cldss 21898 and eqid 2734. (Contributed by Zhi Wang, 6-Sep-2024.)
(𝑍 = (∪ 𝐽 ∖ 𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅)))
 
6-Sep-2024opndisj 45823 Two ways of saying that two open sets are disjoint, if 𝐽 is a topology and 𝑋 is an open set. (Contributed by Zhi Wang, 6-Sep-2024.)
(𝑍 = (∪ 𝐽 ∖ 𝑋) → (𝑌 ∈ (𝐽 ∩ 𝒫 𝑍) ↔ (𝑌 ∈ 𝐽 ∧ (𝑋 ∩ 𝑌) = ∅)))
 
6-Sep-2024clduni 45821 The union of closed sets is the underlying set of the topology (the union of open sets). (Contributed by Zhi Wang, 6-Sep-2024.)
(𝐽 ∈ Top → ∪ (Clsd‘𝐽) = ∪ 𝐽)
 
6-Sep-2024r19.41dv 45774 A complex deduction form of r19.41v 3253. (Contributed by Zhi Wang, 6-Sep-2024.)
(𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ((𝜑 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))
 
6-Sep-2024ralbidb 45772 Formula-building rule for restricted universal quantifier and additional condition (deduction form). See ralbidc 45773 for a more generalized form. (Contributed by Zhi Wang, 6-Sep-2024.)
(𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)))    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜒 ↔ 𝜃))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥 ∈ 𝐵 (𝜓 → 𝜃)))
 
6-Sep-2024pm5.32dra 45767 Reverse distribution of implication over biconditional (deduction form). (Contributed by Zhi Wang, 6-Sep-2024.)
(𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃)))    ⇒   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
 
6-Sep-2024eq0rdv 4309 Deduction for equality to the empty set. (Contributed by NM, 11-Jul-2014.) Avoid ax-8 2112, df-clel 2812. (Revised by Gino Giotto, 6-Sep-2024.)
(𝜑 → ¬ 𝑥 ∈ 𝐴)    ⇒   (𝜑 → 𝐴 = ∅)
 
6-Sep-2024eq0 4248 A class is equal to the empty set if and only if it has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by NM, 29-Aug-1993.) Avoid ax-11 2158, ax-12 2175. (Revised by Gino Giotto and Steven Nguyen, 28-Jun-2024.) Avoid ax-8 2112, df-clel 2812. (Revised by Gino Giotto, 6-Sep-2024.)
(𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
 
6-Sep-2024vn0 4243 The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.) Avoid ax-8 2112, df-clel 2812. (Revised by Gino Giotto, 6-Sep-2024.)
V ≠ ∅
 
5-Sep-2024iscnrm4 45875 A completely normal topology is a topology in which two separated sets can be separated by neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ ((nei‘𝐽)‘𝑠)∃𝑚 ∈ ((nei‘𝐽)‘𝑡)(𝑛 ∩ 𝑚) = ∅)))
 
5-Sep-2024iscnrm3 45873 A completely normal topology is a topology in which two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝐽 ∈ CNrm ↔ (𝐽 ∈ Top ∧ ∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅))))
 
5-Sep-2024iscnrm3l 45872 Lemma for iscnrm3 45873. Given a topology 𝐽, if two separated sets can be separated by open neighborhoods, then all subspaces of the topology 𝐽 are normal, i.e., two disjoint closed sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝐽 ∈ Top → (∀𝑠 ∈ 𝒫 ∪ 𝐽∀𝑡 ∈ 𝒫 ∪ 𝐽(((𝑠 ∩ ((cls‘𝐽)‘𝑡)) = ∅ ∧ (((cls‘𝐽)‘𝑠) ∩ 𝑡) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)) → ((𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) → ((𝐶 ∩ 𝐷) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑍)∃𝑘 ∈ (𝐽 ↾t 𝑍)(𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))))
 
5-Sep-2024iscnrm3llem2 45871 Lemma for iscnrm3l 45872. If there exist disjoint open neighborhoods in the orignal topology for two disjoint closed sets in a subspace, then they can be separated by open neighborhoods in the subspace topology. (Could shorten proof with ssin0 42228.) (Contributed by Zhi Wang, 5-Sep-2024.)
((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝐶 ⊆ 𝑛 ∧ 𝐷 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅) → ∃𝑙 ∈ (𝐽 ↾t 𝑍)∃𝑘 ∈ (𝐽 ↾t 𝑍)(𝐶 ⊆ 𝑙 ∧ 𝐷 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)))
 
5-Sep-2024iscnrm3r 45869 Lemma for iscnrm3 45873. If all subspaces of a topology are normal, i.e., two disjoint closed sets can be separated by open neighborhoods, then in the original topology two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝐽 ∈ Top → (∀𝑧 ∈ 𝒫 ∪ 𝐽∀𝑐 ∈ (Clsd‘(𝐽 ↾t 𝑧))∀𝑑 ∈ (Clsd‘(𝐽 ↾t 𝑧))((𝑐 ∩ 𝑑) = ∅ → ∃𝑙 ∈ (𝐽 ↾t 𝑧)∃𝑘 ∈ (𝐽 ↾t 𝑧)(𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅)) → ((𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) → (((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))))
 
5-Sep-2024iscnrm3rlem8 45868 Lemma for iscnrm3r 45869. Disjoint open neighborhoods in the subspace topology are disjoint open neighborhoods in the original topology given that the subspace is an open set in the original topology. Therefore, given any two sets separated in the original topology and separated by open neighborhoods in the subspace topology, they must be separated by open neighborhoods in the original topology. (Contributed by Zhi Wang, 5-Sep-2024.)
((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽) ∧ ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) → (∃𝑙 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))∃𝑘 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))((((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ⊆ 𝑙 ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ⊆ 𝑘 ∧ (𝑙 ∩ 𝑘) = ∅) → ∃𝑛 ∈ 𝐽 ∃𝑚 ∈ 𝐽 (𝑆 ⊆ 𝑛 ∧ 𝑇 ⊆ 𝑚 ∧ (𝑛 ∩ 𝑚) = ∅)))
 
5-Sep-2024iscnrm3rlem7 45867 Lemma for iscnrm3rlem8 45868. Open neighborhoods in the subspace topology are open neighborhoods in the original topology given that the subspace is an open set in the original topology. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   (𝜑 → 𝑇 ⊆ ∪ 𝐽)    &   (𝜑 → 𝑂 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))    ⇒   (𝜑 → 𝑂 ∈ 𝐽)
 
5-Sep-2024iscnrm3rlem6 45866 Lemma for iscnrm3rlem7 45867. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   (𝜑 → 𝑇 ⊆ ∪ 𝐽)    &   (𝜑 → 𝑂 ⊆ (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))    ⇒   (𝜑 → (𝑂 ∈ (𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))) ↔ 𝑂 ∈ 𝐽))
 
5-Sep-2024iscnrm3rlem5 45865 Lemma for iscnrm3rlem6 45866. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   (𝜑 → 𝑇 ⊆ ∪ 𝐽)    ⇒   (𝜑 → (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝐽)
 
5-Sep-2024iscnrm3rlem4 45864 Lemma for iscnrm3rlem8 45868. Given two disjoint subsets 𝑆 and 𝑇 of the underlying set of a topology 𝐽, if 𝑁 is a superset of (((cls‘𝐽)‘𝑆) ∖ 𝑇), then it is a superset of 𝑆. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑆 ⊆ ∪ 𝐽)    &   (𝜑 → (𝑆 ∩ 𝑇) = ∅)    &   (𝜑 → (((cls‘𝐽)‘𝑆) ∖ 𝑇) ⊆ 𝑁)    ⇒   (𝜑 → 𝑆 ⊆ 𝑁)
 
5-Sep-2024iscnrm3rlem3 45863 Lemma for iscnrm3r 45869. The designed subspace is a subset of the original set; the two sets are closed sets in the subspace. (Contributed by Zhi Wang, 5-Sep-2024.)
((𝐽 ∈ Top ∧ (𝑆 ∈ 𝒫 ∪ 𝐽 ∧ 𝑇 ∈ 𝒫 ∪ 𝐽)) → ((∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))) ∈ 𝒫 ∪ 𝐽 ∧ (((cls‘𝐽)‘𝑆) ∖ ((cls‘𝐽)‘𝑇)) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇))))) ∧ (((cls‘𝐽)‘𝑇) ∖ ((cls‘𝐽)‘𝑆)) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ ((cls‘𝐽)‘𝑇)))))))
 
5-Sep-2024iscnrm3rlem2 45862 Lemma for iscnrm3rlem3 45863. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑆 ⊆ ∪ 𝐽)    ⇒   (𝜑 → (((cls‘𝐽)‘𝑆) ∖ 𝑇) ∈ (Clsd‘(𝐽 ↾t (∪ 𝐽 ∖ (((cls‘𝐽)‘𝑆) ∩ 𝑇)))))
 
5-Sep-2024iscnrm3rlem1 45861 Lemma for iscnrm3rlem2 45862. The hypothesis could be generalized to (𝜑 → (𝑆 ∖ 𝑇) ⊆ 𝑋). (Contributed by Zhi Wang, 5-Sep-2024.)
(𝜑 → 𝑆 ⊆ 𝑋)    ⇒   (𝜑 → (𝑆 ∖ 𝑇) = (𝑆 ∩ (𝑋 ∖ (𝑆 ∩ 𝑇))))
 
5-Sep-2024iscnrm3lem7 45860 Lemma for iscnrm3rlem8 45868 and iscnrm3llem2 45871 involving restricted existential quantifications. (Contributed by Zhi Wang, 5-Sep-2024.)
(𝑧 = 𝑍 → (𝜒 ↔ 𝜃))    &   (𝑤 = 𝑊 → (𝜃 ↔ 𝜏))    &   ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → (𝑍 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝜏))    ⇒   (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝐷 𝜒))
 
5-Sep-2024iscnrm3lem6 45859 Lemma for iscnrm3lem7 45860. (Contributed by Zhi Wang, 5-Sep-2024.)
((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑊) ∧ 𝜓) → 𝜒)    ⇒   (𝜑 → (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑊 𝜓 → 𝜒))
 
5-Sep-2024disjdifb 45782 Relative complement is anticommutative regarding intersection. (Contributed by Zhi Wang, 5-Sep-2024.)
((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅
 
4-Sep-2024iscnrm3llem1 45870 Lemma for iscnrm3l 45872. Closed sets in the subspace are subsets of the underlying set of the original topology. (Contributed by Zhi Wang, 4-Sep-2024.)
((𝐽 ∈ Top ∧ (𝑍 ∈ 𝒫 ∪ 𝐽 ∧ 𝐶 ∈ (Clsd‘(𝐽 ↾t 𝑍)) ∧ 𝐷 ∈ (Clsd‘(𝐽 ↾t 𝑍))) ∧ (𝐶 ∩ 𝐷) = ∅) → (𝐶 ∈ 𝒫 ∪ 𝐽 ∧ 𝐷 ∈ 𝒫 ∪ 𝐽))
 
4-Sep-2024iscnrm3lem4 45857 Lemma for iscnrm3lem5 45858 and iscnrm3r 45869. (Contributed by Zhi Wang, 4-Sep-2024.)
(𝜂 → (𝜓 → 𝜁))    &   ((𝜑 ∧ 𝜒 ∧ 𝜃) → 𝜂)    &   ((𝜑 ∧ 𝜒 ∧ 𝜃) → (𝜁 → 𝜏))    ⇒   (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏))))
 
4-Sep-2024iscnrm3lem3 45856 Lemma for iscnrm3lem4 45857. (Contributed by Zhi Wang, 4-Sep-2024.)
(((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜓))
 
4-Sep-2024on3ind 33523 Triple induction over ordinals. (Contributed by Scott Fenton, 4-Sep-2024.)
(𝑎 = 𝑑 → (𝜑 ↔ 𝜓))    &   (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))    &   (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))    &   (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))    &   (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))    &   (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))    &   (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))    &   (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))    &   (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))    &   (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))    &   ((𝑎 ∈ On ∧ 𝑏 ∈ On ∧ 𝑐 ∈ On) → (((∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜃 ∧ ∀𝑑 ∈ 𝑎 ∀𝑒 ∈ 𝑏 𝜒 ∧ ∀𝑑 ∈ 𝑎 ∀𝑓 ∈ 𝑐 𝜁) ∧ (∀𝑑 ∈ 𝑎 𝜓 ∧ ∀𝑒 ∈ 𝑏 ∀𝑓 ∈ 𝑐 𝜏 ∧ ∀𝑒 ∈ 𝑏 𝜎) ∧ ∀𝑓 ∈ 𝑐 𝜂) → 𝜑))    ⇒   ((𝑋 ∈ On ∧ 𝑌 ∈ On ∧ 𝑍 ∈ On) → 𝜆)
 
4-Sep-2024xpord3ind 33488 Induction over the triple cross product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 4-Sep-2024.)
𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    &   𝑅 Fr 𝐴    &   𝑅 Po 𝐴    &   𝑅 Se 𝐴    &   𝑆 Fr 𝐵    &   𝑆 Po 𝐵    &   𝑆 Se 𝐵    &   𝑇 Fr 𝐶    &   𝑇 Po 𝐶    &   𝑇 Se 𝐶    &   (𝑎 = 𝑑 → (𝜑 ↔ 𝜓))    &   (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))    &   (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))    &   (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))    &   (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))    &   (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))    &   (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))    &   (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))    &   (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))    &   (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))    &   ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶) → (((∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜂) → 𝜑))    ⇒   ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝜆)
 
4-Sep-2024vex 3405 All setvar variables are sets (see isset 3414). Theorem 6.8 of [Quine] p. 43. A shorter proof is possible from eleq2i 2825 but it uses more axioms. (Contributed by NM, 26-May-1993.) Remove use of ax-12 2175. (Revised by SN, 28-Aug-2023.) (Proof shortened by BJ, 4-Sep-2024.)
𝑥 ∈ V
 
3-Sep-2024iscnrm3lem5 45858 Lemma for iscnrm3l 45872. (Contributed by Zhi Wang, 3-Sep-2024.)
((𝑥 = 𝑆 ∧ 𝑦 = 𝑇) → (𝜑 ↔ 𝜓))    &   ((𝑥 = 𝑆 ∧ 𝑦 = 𝑇) → (𝜒 ↔ 𝜃))    &   ((𝜏 ∧ 𝜂 ∧ 𝜁) → (𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊))    &   ((𝜏 ∧ 𝜂 ∧ 𝜁) → ((𝜓 → 𝜃) → 𝜎))    ⇒   (𝜏 → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 (𝜑 → 𝜒) → (𝜂 → (𝜁 → 𝜎))))
 
3-Sep-2024iscnrm3lem2 45855 Lemma for iscnrm3 45873 proving a biconditional on restricted universal quantifications. (Contributed by Zhi Wang, 3-Sep-2024.)
(𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓 → ((𝑤 ∈ 𝐷 ∧ 𝑣 ∈ 𝐸) → 𝜒)))    &   (𝜑 → (∀𝑤 ∈ 𝐷 ∀𝑣 ∈ 𝐸 𝜒 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜓)))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓 ↔ ∀𝑤 ∈ 𝐷 ∀𝑣 ∈ 𝐸 𝜒))
 
3-Sep-2024iscnrm3lem1 45854 Lemma for iscnrm3 45873. Subspace topology is a topology. (Contributed by Zhi Wang, 3-Sep-2024.)
(𝐽 ∈ Top → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ((𝐽 ↾t 𝑥) ∈ Top ∧ 𝜑)))
 
3-Sep-2024exp12bd 45768 The import-export theorem (impexp 454) for biconditionals (deduction form). (Contributed by Zhi Wang, 3-Sep-2024.)
(𝜑 → (((𝜓 ∧ 𝜒) → 𝜃) ↔ ((𝜏 ∧ 𝜂) → 𝜁)))    ⇒   (𝜑 → ((𝜓 → (𝜒 → 𝜃)) ↔ (𝜏 → (𝜂 → 𝜁))))
 
3-Sep-2024on2recsov 33521 Calculate the value of the double ordinal recursion operator. (Contributed by Scott Fenton, 3-Sep-2024.)
𝐹 = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st ‘𝑥) E (1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥) E (2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), 𝐺)    ⇒   ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐹𝐵) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ ((suc 𝐴 × suc 𝐵) ∖ {⟨𝐴, 𝐵⟩}))))
 
3-Sep-2024on2recsfn 33520 Show that double recursion over ordinals yields a function over pairs of ordinals. (Contributed by Scott Fenton, 3-Sep-2024.)
𝐹 = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st ‘𝑥) E (1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥) E (2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), 𝐺)    ⇒   𝐹 Fn (On × On)
 
2-Sep-2024dfnrm3 45853 A topological space is normal if any disjoint closed sets can be separated by neighborhoods. An alternate definition of df-nrm 22186. (Contributed by Zhi Wang, 2-Sep-2024.)
Nrm = {𝑗 ∈ Top ∣ ∀𝑐 ∈ (Clsd‘𝑗)∀𝑑 ∈ (Clsd‘𝑗)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝑗)‘𝑐)∃𝑦 ∈ ((nei‘𝑗)‘𝑑)(𝑥 ∩ 𝑦) = ∅)}
 
2-Sep-2024restclssep 45836 Two disjoint closed sets in a subspace topology are separated in the original topology. (Contributed by Zhi Wang, 2-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑋 = ∪ 𝐽)    &   (𝜑 → 𝑌 ⊆ 𝑋)    &   (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))    &   (𝜑 → 𝑆 ∈ (Clsd‘𝐾))    &   (𝜑 → (𝑆 ∩ 𝑇) = ∅)    &   (𝜑 → 𝑇 ∈ (Clsd‘𝐾))    ⇒   (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅))
 
2-Sep-2024restclsseplem 45835 Lemma for restclssep 45836. (Contributed by Zhi Wang, 2-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑋 = ∪ 𝐽)    &   (𝜑 → 𝑌 ⊆ 𝑋)    &   (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))    &   (𝜑 → 𝑆 ∈ (Clsd‘𝐾))    &   (𝜑 → (𝑆 ∩ 𝑇) = ∅)    &   (𝜑 → 𝑇 ⊆ 𝑌)    ⇒   (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)
 
2-Sep-2024restcls2 45834 A closed set in a subspace topology is the closure in the original topology intersecting with the subspace. (Contributed by Zhi Wang, 2-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑋 = ∪ 𝐽)    &   (𝜑 → 𝑌 ⊆ 𝑋)    &   (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))    &   (𝜑 → 𝑆 ∈ (Clsd‘𝐾))    ⇒   (𝜑 → 𝑆 = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
 
2-Sep-2024restcls2lem 45833 A closed set in a subspace topology is a subset of the subspace. (Contributed by Zhi Wang, 2-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    &   (𝜑 → 𝑋 = ∪ 𝐽)    &   (𝜑 → 𝑌 ⊆ 𝑋)    &   (𝜑 → 𝐾 = (𝐽 ↾t 𝑌))    &   (𝜑 → 𝑆 ∈ (Clsd‘𝐾))    ⇒   (𝜑 → 𝑆 ⊆ 𝑌)
 
2-Sep-2024elrab2w 39841 Membership in a restricted class abstraction. This is to elrab2 3598 what elab2gw 39840 is to elab2g 3582. (Contributed by SN, 2-Sep-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))    &   𝐶 = {𝑥 ∈ 𝐵 ∣ 𝜑}    ⇒   (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ 𝐵 ∧ 𝜒))
 
2-Sep-2024ralf0 4415 The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.) Avoid df-clel 2812, ax-8 2112. (Revised by Gino Giotto, 2-Sep-2024.)
¬ 𝜑    ⇒   (∀𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 = ∅)
 
2-Sep-2024ral0 4414 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.) Avoid df-clel 2812, ax-8 2112. (Revised by Gino Giotto, 2-Sep-2024.)
∀𝑥 ∈ ∅ 𝜑
 
2-Sep-2024ralidm 4413 Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.) Reduce axiom usage. (Revised by Gino Giotto, 2-Sep-2024.)
(∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)
 
2-Sep-2024rexn0 4412 Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) Avoid df-clel 2812, ax-8 2112. (Revised by Gino Giotto, 2-Sep-2024.)
(∃𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅)
 
2-Sep-2024rzal 4410 Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) Avoid df-clel 2812, ax-8 2112. (Revised by Gino Giotto, 2-Sep-2024.)
(𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
 
2-Sep-2024sbc5 3715 An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by SN, 2-Sep-2024.)
([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
 
2-Sep-2024vtocld 3463 Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-10 2141, ax-11 2158, ax-12 2175. (Revised by SN, 2-Sep-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))    &   (𝜑 → 𝜓)    ⇒   (𝜑 → 𝜒)
 
2-Sep-2024clelab 2876 Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) Avoid ax-11 2158, see sbc5ALT 3716 for more details. (Revised by SN, 2-Sep-2024.)
(𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
 
1-Sep-2024isnrm4 45851 A topological space is normal iff any two disjoint closed sets are separated by neighborhoods. (Contributed by Zhi Wang, 1-Sep-2024.)
(𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑑 ∈ (Clsd‘𝐽)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ ((nei‘𝐽)‘𝑐)∃𝑦 ∈ ((nei‘𝐽)‘𝑑)(𝑥 ∩ 𝑦) = ∅)))
 
1-Sep-2024sepnsepo 45844 Open neighborhood and neighborhood is equivalent regarding disjointness for both sides. Namely, separatedness by open neighborhoods is equivalent to separatedness by neighborhoods. (Contributed by Zhi Wang, 1-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    ⇒   (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝐶)∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝐶 ⊆ 𝑥 ∧ 𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
 
1-Sep-2024sepnsepolem2 45843 Open neighborhood and neighborhood is equivalent regarding disjointness. Lemma for sepnsepo 45844. Proof could be shortened by 1 step using ssdisjdr 45781. (Contributed by Zhi Wang, 1-Sep-2024.)
(𝜑 → 𝐽 ∈ Top)    ⇒   (𝜑 → (∃𝑦 ∈ ((nei‘𝐽)‘𝐷)(𝑥 ∩ 𝑦) = ∅ ↔ ∃𝑦 ∈ 𝐽 (𝐷 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅)))
 
1-Sep-2024sepnsepolem1 45842 Lemma for sepnsepo 45844. (Contributed by Zhi Wang, 1-Sep-2024.)
(∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐽 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥 ∈ 𝐽 (𝜑 ∧ ∃𝑦 ∈ 𝐽 (𝜓 ∧ 𝜒)))
 
1-Sep-2024ruvALT 39842 Alternate proof of ruv 9207 with one fewer syntax step thanks to using elirrv 9201 instead of elirr 9202. However, it does not change the compressed proof size or the number of symbols in the generated display, so it is not considered a shortening according to conventions 28455. (Contributed by SN, 1-Sep-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
{𝑥 ∣ 𝑥 ∉ 𝑥} = V
 
1-Sep-2024bj-clel3gALT 34915 Alternate proof of clel3g 3562. (Contributed by BJ, 1-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥)))
 
1-Sep-2024vopelopabsb 5399 The law of concretion in terms of substitutions. Version of opelopabsb 5400 with set variables. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) Remove unnecessary commutation. (Revised by SN, 1-Sep-2024.)
(⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
 
1-Sep-2024copsex2g 5365 Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.) Use a similar proof to copsex4g 5367 to reduce axiom usage. (Revised by SN, 1-Sep-2024.)
((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥∃𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
 
1-Sep-2024intpr 4883 The intersection of a pair is the intersection of its members. Theorem 71 of [Suppes] p. 42. (Contributed by NM, 14-Oct-1999.) Prove from intprg 4882. (Revised by BJ, 1-Sep-2024.)
𝐴 ∈ V    &   𝐵 ∈ V    ⇒   ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵)
 
1-Sep-2024intprg 4882 The intersection of a pair is the intersection of its members. Closed form of intpr 4883. Theorem 71 of [Suppes] p. 42. (Contributed by FL, 27-Apr-2008.) (Proof shortened by BJ, 1-Sep-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {𝐴, 𝐵} = (𝐴 ∩ 𝐵))
 
1-Sep-2024unipr 4827 The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.) (Proof shortened by BJ, 1-Sep-2024.)
𝐴 ∈ V    &   𝐵 ∈ V    ⇒   ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)
 
1-Sep-2024uniprg 4826 The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.) Avoid using unipr 4827 to prove it from uniprg 4826. (Revised by BJ, 1-Sep-2024.)
((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
 
1-Sep-2024clel4g 3564 Alternate definition of membership in a set. (Contributed by NM, 18-Aug-1993.) Strengthen from sethood hypothesis to sethood antecedent and avoid ax-12 2175. (Revised by BJ, 1-Sep-2024.)
(𝐵 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐵 → 𝐴 ∈ 𝑥)))
 
1-Sep-2024clel2g 3559 Alternate definition of membership when the member is a set. (Contributed by NM, 18-Aug-1993.) Strengthen from sethood hypothesis to sethood antecedent. (Revised by BJ, 12-Feb-2022.) Avoid ax-12 2175. (Revised by BJ, 1-Sep-2024.)
(𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵)))
 
31-Aug-2024opnneieqvv 45832 The equivalence between neighborhood and open neighborhood. A variant of opnneieqv 45831 with two dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
(𝜑 → 𝐽 ∈ Top)    &   ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    &   ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
 
31-Aug-2024opnneieqv 45831 The equivalence between neighborhood and open neighborhood. See opnneieqvv 45832 for different dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
(𝜑 → 𝐽 ∈ Top)    &   ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    &   ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓)))
 
31-Aug-2024opnneil 45830 A variant of opnneilv 45829. (Contributed by Zhi Wang, 31-Aug-2024.)
(𝜑 → 𝐽 ∈ Top)    &   ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    &   ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓)))
 
31-Aug-2024opnneilv 45829 The converse of opnneir 45827 with different dummy variables. Note that the second hypothesis could be generalized by adding 𝑦 ∈ 𝐽 to the antecedent. See the proof for details. Although 𝐽 ∈ Top might be redundant here (see neircl 45825), it is listed for explicitness. (Contributed by Zhi Wang, 31-Aug-2024.)
(𝜑 → 𝐽 ∈ Top)    &   ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    ⇒   (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
 
31-Aug-2024opnneirv 45828 A variant of opnneir 45827 with different dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
(𝜑 → 𝐽 ∈ Top)    &   ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) → ∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝜒))
 
31-Aug-2024opnneir 45827 If something is true for an open neighborhood, it must be true for a neighborhood. (Contributed by Zhi Wang, 31-Aug-2024.)
(𝜑 → 𝐽 ∈ Top)    ⇒   (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) → ∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓))
 
31-Aug-2024opnneilem 45826 Lemma factoring out common proof steps of opnneil 45830 and opnneirv 45828. (Contributed by Zhi Wang, 31-Aug-2024.)
((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
 
31-Aug-2024dfaiota3 44210 Alternate definition of ℩': this is to df-aiota 44203 what dfiota4 6361 is to df-iota 6327. operation using the if operator. It is simpler than df-aiota 44203 and uses no dummy variables, so it would be the preferred definition if ℩' becomes the description binder used in set.mm. (Contributed by BJ, 31-Aug-2024.)
(℩'𝑥𝜑) = if(∃!𝑥𝜑, ∩ {𝑥 ∣ 𝜑}, V)
 
31-Aug-2024aiotaint 44209 This is to df-aiota 44203 what iotauni 6344 is to df-iota 6327 (it uses intersection like df-aiota 44203, similar to iotauni 6344 using union like df-iota 6327; we could also prove an analogous result using union here too, in the same way that we have iotaint 6345). (Contributed by BJ, 31-Aug-2024.)
(∃!𝑥𝜑 → (℩'𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})
 
31-Aug-2024acos1half 39844 The arccosine of 1 / 2 is π / 3. (Contributed by SN, 31-Aug-2024.)
(arccos‘(1 / 2)) = (π / 3)
 
31-Aug-2024f1ofvswap 7105 Swapping two values in a bijection between two classes yields another bijection between those classes. (Contributed by BTernaryTau, 31-Aug-2024.)
((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, (𝐹‘𝑌)⟩, ⟨𝑌, (𝐹‘𝑋)⟩}):𝐴–1-1-onto→𝐵)
 
30-Aug-2024dfnrm2 45852 A topological space is normal if any disjoint closed sets can be separated by open neighborhoods. An alternate definition of df-nrm 22186. (Contributed by Zhi Wang, 30-Aug-2024.)
Nrm = {𝑗 ∈ Top ∣ ∀𝑐 ∈ (Clsd‘𝑗)∀𝑑 ∈ (Clsd‘𝑗)((𝑐 ∩ 𝑑) = ∅ → ∃𝑥 ∈ 𝑗 ∃𝑦 ∈ 𝑗 (𝑐 ⊆ 𝑥 ∧ 𝑑 ⊆ 𝑦 ∧ (𝑥 ∩ 𝑦) = ∅))}
 
30-Aug-2024opncldeqv 45822 Conditions on open sets are equivalent to conditions on closed sets. (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → 𝐽 ∈ Top)    &   ((𝜑 ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦)) → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐽 𝜓 ↔ ∀𝑦 ∈ (Clsd‘𝐽)𝜒))
 
30-Aug-2024ralbidc 45773 Formula-building rule for restricted universal quantifier and additional condition (deduction form). A variant of ralbidb 45772. (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)))    &   (𝜑 → ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → (𝜒 ↔ 𝜃)))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥 ∈ 𝐵 (𝜓 → 𝜃)))
 
30-Aug-2024pm5.32dav 45766 Distribution of implication over biconditional (deduction form). Variant of pm5.32da 582. (Contributed by Zhi Wang, 30-Aug-2024.)
((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ⇒   (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜓)))
 
30-Aug-2024logic2 45765 Variant of logic1 45763. (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → (𝜓 ↔ 𝜒))    &   (𝜑 → ((𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)))    ⇒   (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))
 
30-Aug-2024logic1a 45764 Variant of logic1 45763. (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → (𝜓 ↔ 𝜒))    &   ((𝜑 ∧ 𝜓) → (𝜃 ↔ 𝜏))    ⇒   (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))
 
30-Aug-2024logic1 45763 Distribution of implication over biconditional with replacement (deduction form). (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → (𝜓 ↔ 𝜒))    &   (𝜑 → (𝜓 → (𝜃 ↔ 𝜏)))    ⇒   (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))
 
30-Aug-2024pm4.71da 45762 Deduction converting a biconditional to a biconditional with conjunction. Variant of pm4.71d 565. (Contributed by Zhi Wang, 30-Aug-2024.)
(𝜑 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒)))
 
30-Aug-2024abn0 4285 Nonempty class abstraction. See also ab0 4279. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) Avoid df-clel 2812, ax-8 2112. (Revised by Gino Giotto, 30-Aug-2024.)
({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
 
30-Aug-2024ab0orv 4283 The class abstraction defined by a formula not containing the abstraction variable is either the empty set or the universal class. (Contributed by Mario Carneiro, 29-Aug-2013.) (Revised by BJ, 22-Mar-2020.) Reduce axiom usage. (Revised by Gino Giotto, 30-Aug-2024.)
({𝑥 ∣ 𝜑} = V ∨ {𝑥 ∣ 𝜑} = ∅)
 
30-Aug-2024ab0OLD 4280 Obsolete version of ab0 4279 as of 8-Sep-2024. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2812, ax-8 2112. (Revised by Gino Giotto, 30-Aug-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
 
30-Aug-2024rru 3685 Relative version of Russell's paradox ru 3686 (which corresponds to the case 𝐴 = V).

Originally a subproof in pwnss 5230. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid df-nel 3040. (Revised by Steven Nguyen, 23-Nov-2022.) Reduce axiom usage. (Revised by Gino Giotto, 30-Aug-2024.)

¬ {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝑥} ∈ 𝐴
 
30-Aug-2024abv 3412 The class of sets verifying a property is the universal class if and only if that property is a tautology. The reverse implication (bj-abv 34785) requires fewer axioms. (Contributed by BJ, 19-Mar-2021.) Avoid df-clel 2812, ax-8 2112. (Revised by Gino Giotto, 30-Aug-2024.) (Proof shortened by BJ, 30-Aug-2024.)
({𝑥 ∣ 𝜑} = V ↔ ∀𝑥𝜑)
 
29-Aug-2024dftermo3 17484 An alternate definition of df-termo 17463 depending on df-inito 17462, without dummy variables. (Contributed by Zhi Wang, 29-Aug-2024.)
TermO = (InitO ∘ (oppCat ↾ Cat))
 
29-Aug-2024dfinito3 17483 An alternate definition of df-inito 17462 depending on df-termo 17463, without dummy variables. (Contributed by Zhi Wang, 29-Aug-2024.)
InitO = (TermO ∘ (oppCat ↾ Cat))
 
29-Aug-2024dftermo2 17482 A terminal object is an initial object in the opposite category. An alternate definition of df-termo 17463 depending on df-inito 17462. (Contributed by Zhi Wang, 29-Aug-2024.)
TermO = (𝑐 ∈ Cat ↦ (InitO‘(oppCat‘𝑐)))
 
29-Aug-2024dfinito2 17481 An initial object is a terminal object in the opposite category. An alternate definition of df-inito 17462 depending on df-termo 17463. (Contributed by Zhi Wang, 29-Aug-2024.)
InitO = (𝑐 ∈ Cat ↦ (TermO‘(oppCat‘𝑐)))
 
29-Aug-2024zeroofn 17467 ZeroO is a function on Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
ZeroO Fn Cat
 
29-Aug-2024termofn 17466 TermO is a function on Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
TermO Fn Cat
 
29-Aug-2024initofn 17465 InitO is a function on Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
InitO Fn Cat
 
29-Aug-2024oppccatf 17204 oppCat restricted to Cat is a function from Cat to Cat. (Contributed by Zhi Wang, 29-Aug-2024.)
(oppCat ↾ Cat):Cat⟶Cat
 
27-Aug-2024nmfval0 23460 The value of the norm function on a structure containing a zero as the distance restricted to the elements of the base set to zero. Examples of structures containing a "zero" are groups (see nmfval2 23461 proved from this theorem and grpidcl 18367) or more generally monoids (see mndidcl 18160), or pointed sets). (Contributed by Mario Carneiro, 2-Oct-2015.) Extract this result from the proof of nmfval2 23461. (Revised by BJ, 27-Aug-2024.)
𝑁 = (norm‘𝑊)    &   𝑋 = (Base‘𝑊)    &    0 = (0g‘𝑊)    &   𝐷 = (dist‘𝑊)    &   𝐸 = (𝐷 ↾ (𝑋 × 𝑋))    ⇒   ( 0 ∈ 𝑋 → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))
 
26-Aug-2024naddid1 33530 Ordinal zero is the additive identity for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
(𝐴 ∈ On → (𝐴 +no ∅) = 𝐴)
 
26-Aug-2024naddcom 33529 Natural addition commutes. (Contributed by Scott Fenton, 26-Aug-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = (𝐵 +no 𝐴))
 
26-Aug-2024naddov2 33528 Alternate expression for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝐵) ∈ 𝑥)})
 
26-Aug-2024naddov 33527 The value of natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)})
 
26-Aug-2024naddcl 33526 Closure law for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) ∈ On)
 
26-Aug-2024naddcllem 33525 Lemma for ordinal addition closure. (Contributed by Scott Fenton, 26-Aug-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +no 𝐵) ∈ On ∧ (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)}))
 
26-Aug-2024naddfn 33524 Natural addition is a function over pairs of ordinals. (Contributed by Scott Fenton, 26-Aug-2024.)
+no Fn (On × On)
 
26-Aug-2024on2ind 33522 Double induction over ordinal numbers. (Contributed by Scott Fenton, 26-Aug-2024.)
(𝑎 = 𝑐 → (𝜑 ↔ 𝜓))    &   (𝑏 = 𝑑 → (𝜓 ↔ 𝜒))    &   (𝑎 = 𝑐 → (𝜃 ↔ 𝜒))    &   (𝑎 = 𝑋 → (𝜑 ↔ 𝜏))    &   (𝑏 = 𝑌 → (𝜏 ↔ 𝜂))    &   ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 𝜒 ∧ ∀𝑐 ∈ 𝑎 𝜓 ∧ ∀𝑑 ∈ 𝑏 𝜃) → 𝜑))    ⇒   ((𝑋 ∈ On ∧ 𝑌 ∈ On) → 𝜂)
 
26-Aug-2024df-nadd 33519 Define natural ordinal addition. This is a commutative form of addition over the ordinals. (Contributed by Scott Fenton, 26-Aug-2024.)
+no = frecs({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On) ∧ (((1st ‘𝑥) E (1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥) E (2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}, (On × On), (𝑧 ∈ V, 𝑎 ∈ V ↦ ∩ {𝑤 ∈ On ∣ ((𝑎 “ ({(1st ‘𝑧)} × (2nd ‘𝑧))) ⊆ 𝑤 ∧ (𝑎 “ ((1st ‘𝑧) × {(2nd ‘𝑧)})) ⊆ 𝑤)}))
 
26-Aug-2024findcard2 8831 Schema for induction on the cardinality of a finite set. The inductive step shows that the result is true if one more element is added to the set. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 8-Jul-2010.) Avoid ax-pow 5247. (Revised by BTernaryTau, 26-Aug-2024.)
(𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃))    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   𝜓    &   (𝑦 ∈ Fin → (𝜒 → 𝜃))    ⇒   (𝐴 ∈ Fin → 𝜏)
 
26-Aug-2024dif1en 8829 If a set 𝐴 is equinumerous to the successor of a natural number 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. For a proof with fewer symbols using ax-pow 5247, see dif1enALT 8896. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) Avoid ax-pow 5247. (Revised by BTernaryTau, 26-Aug-2024.)
((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
 
26-Aug-2024rexdif1en 8828 If a set is equinumerous to a nonzero finite ordinal, then there exists an element in that set such that removing it leaves the set equinumerous to the predecessor of that ordinal. (Contributed by BTernaryTau, 26-Aug-2024.)
((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
 
25-Aug-2024flt4lem7 40151 Convert flt4lem5f 40149 into a convenient form for nna4b4nsq 40152. TODO-SN: The change to (𝐴 gcd 𝐵) = 1 points at some inefficiency in the lemmas. EDITORIAL: This is not minimized! (Contributed by SN, 25-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐵) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   (𝜑 → ∃𝑙 ∈ ℕ (∃𝑔 ∈ ℕ ∃ℎ ∈ ℕ (¬ 2 ∥ 𝑔 ∧ ((𝑔 gcd ℎ) = 1 ∧ ((𝑔↑4) + (ℎ↑4)) = (𝑙↑2))) ∧ 𝑙 < 𝐶))
 
25-Aug-2024gcdle2d 39991 The greatest common divisor of a positive integer and another integer is less than or equal to the positive integer. (Contributed by SN, 25-Aug-2024.)
(𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℕ)    ⇒   (𝜑 → (𝑀 gcd 𝑁) ≤ 𝑁)
 
25-Aug-2024gcdle1d 39990 The greatest common divisor of a positive integer and another integer is less than or equal to the positive integer. (Contributed by SN, 25-Aug-2024.)
(𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℤ)    ⇒   (𝜑 → (𝑀 gcd 𝑁) ≤ 𝑀)
 
25-Aug-2024gcdnn0id 39989 The gcd of a nonnegative integer and itself is the integer. (Contributed by SN, 25-Aug-2024.)
(𝑁 ∈ ℕ0 → (𝑁 gcd 𝑁) = 𝑁)
 
24-Aug-2024flt4lem5f 40149 Final equation of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. Given 𝐴↑4 + 𝐵↑4 = 𝐶↑2, provide a smaller solution. This satisfies the infinite descent condition. (Contributed by SN, 24-Aug-2024.)
𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀 − 𝑁))) / 2)    &   𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   (𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   (𝜑 → ((𝑀 gcd (𝐵 / 2))↑2) = (((𝑅 gcd (𝐵 / 2))↑4) + ((𝑆 gcd (𝐵 / 2))↑4)))
 
24-Aug-2024flt4lem5elem 40143 Version of fltaccoprm 40132 and fltbccoprm 40133 where 𝑀 is not squared. This can be proved in general for any polynomial in three variables: using prmdvdsncoprmbd 16264, dvds2addd 15834, and prmdvdsexp 16253, we can show that if two variables are coprime, the third is also coprime to the two. (Contributed by SN, 24-Aug-2024.)
(𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑅 ∈ ℕ)    &   (𝜑 → 𝑆 ∈ ℕ)    &   (𝜑 → 𝑀 = ((𝑅↑2) + (𝑆↑2)))    &   (𝜑 → (𝑅 gcd 𝑆) = 1)    ⇒   (𝜑 → ((𝑅 gcd 𝑀) = 1 ∧ (𝑆 gcd 𝑀) = 1))
 
24-Aug-20242rspcedvdw 39855 Double application of rspcedvdw 39854. (Contributed by SN, 24-Aug-2024.)
(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    &   (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))    &   (𝜑 → 𝐴 ∈ 𝑋)    &   (𝜑 → 𝐵 ∈ 𝑌)    &   (𝜑 → 𝜃)    ⇒   (𝜑 → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 𝜓)
 
24-Aug-2024prmdvdsncoprmbd 16264 Two positive integers are not coprime iff a prime divides both integers. Deduction version of ncoprmgcdne1b 16188 with the existential quantifier over the primes instead of integers greater than or equal to 2. (Contributed by SN, 24-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    ⇒   (𝜑 → (∃𝑝 ∈ ℙ (𝑝 ∥ 𝐴 ∧ 𝑝 ∥ 𝐵) ↔ (𝐴 gcd 𝐵) ≠ 1))
 
24-Aug-2024gcdcomd 16054 The gcd operator is commutative, deduction version. (Contributed by SN, 24-Aug-2024.)
(𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    ⇒   (𝜑 → (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀))
 
23-Aug-2024nna4b4nsq 40152 Strengthening of Fermat's last theorem for exponent 4, where the sum is only assumed to be a square. (Contributed by SN, 23-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    ⇒   (𝜑 → ((𝐴↑4) + (𝐵↑4)) ≠ (𝐶↑2))
 
23-Aug-2024flt4lem5e 40148 Satisfy the hypotheses of flt4lem4 40141. EDITORIAL: This is not minimized! (Contributed by SN, 23-Aug-2024.)
𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀 − 𝑁))) / 2)    &   𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   (𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   (𝜑 → (((𝑅 gcd 𝑆) = 1 ∧ (𝑅 gcd 𝑀) = 1 ∧ (𝑆 gcd 𝑀) = 1) ∧ (𝑅 ∈ ℕ ∧ 𝑆 ∈ ℕ ∧ 𝑀 ∈ ℕ) ∧ ((𝑀 · (𝑅 · 𝑆)) = ((𝐵 / 2)↑2) ∧ (𝐵 / 2) ∈ ℕ)))
 
23-Aug-2024flt4lem5d 40147 Part 3 of Equation 2 of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. (Contributed by SN, 23-Aug-2024.)
𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀 − 𝑁))) / 2)    &   𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   (𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   (𝜑 → 𝑀 = ((𝑅↑2) + (𝑆↑2)))
 
23-Aug-2024flt4lem5 40142 In the context of the lemmas of pythagtrip 16368, 𝑀 and 𝑁 are coprime. (Contributed by SN, 23-Aug-2024.)
𝑀 = (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)    &   𝑁 = (((√‘(𝐶 + 𝐵)) − (√‘(𝐶 − 𝐵))) / 2)    ⇒   (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝑀 gcd 𝑁) = 1)
 
23-Aug-2024posqsqznn 40003 When a positive rational squared is an integer, the rational is a positive integer. zsqrtelqelz 16295 with all terms squared and positive. (Contributed by SN, 23-Aug-2024.)
(𝜑 → (𝐴↑2) ∈ ℤ)    &   (𝜑 → 𝐴 ∈ ℚ)    &   (𝜑 → 0 < 𝐴)    ⇒   (𝜑 → 𝐴 ∈ ℕ)
 
23-Aug-2024lsubcom23d 39966 Swap the second and third variables in an equation with subtraction on the left, converting it into an addition. (Contributed by SN, 23-Aug-2024.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 − 𝐵) = 𝐶)    ⇒   (𝜑 → (𝐴 − 𝐶) = 𝐵)
 
23-Aug-2024addscllem1 33825 Lemma for addscl (future) Alternate expression for surreal addition. (Contributed by Scott Fenton, 23-Aug-2024.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = ((( +s “ (( L ‘𝐴) × {𝐵})) ∪ ( +s “ ({𝐴} × ( L ‘𝐵)))) |s (( +s “ (( R ‘𝐴) × {𝐵})) ∪ ( +s “ ({𝐴} × ( R ‘𝐵))))))
 
23-Aug-2024scutfo 33778 The surreal cut function is onto. (Contributed by Scott Fenton, 23-Aug-2024.)
|s : <<s –onto→ No
 
23-Aug-2024scutcld 33691 Closure law for surreal cuts. (Contributed by Scott Fenton, 23-Aug-2024.)
(𝜑 → 𝐴 <<s 𝐵)    ⇒   (𝜑 → (𝐴 |s 𝐵) ∈ No )
 
23-Aug-2024scutcl 33690 Closure law for surreal cuts. (Contributed by Scott Fenton, 23-Aug-2024.)
(𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
 
23-Aug-2024dvdsmultr2d 15841 Deduction form of dvdsmultr2 15840. (Contributed by SN, 23-Aug-2024.)
(𝜑 → 𝐾 ∈ ℤ)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐾 ∥ 𝑁)    ⇒   (𝜑 → 𝐾 ∥ (𝑀 · 𝑁))
 
23-Aug-2024dmfexALT 7677 Alternate proof of dmfex 7674: shorter but using ax-rep 5168. (Contributed by NM, 27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Proof shortened by AV, 23-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐹 ∈ 𝐶 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V)
 
23-Aug-2024nfceqdf 2895 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.) Avoid ax-8 2112 and df-clel 2812. (Revised by WL and SN, 23-Aug-2024.)
Ⅎ𝑥𝜑    &   (𝜑 → 𝐴 = 𝐵)    ⇒   (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵))
 
23-Aug-2024eleq2w2 2730 A weaker version of eleq2 2822 (but stronger than ax-9 2120 and elequ2 2125) that uses ax-12 2175 to avoid ax-8 2112 and df-clel 2812. Compare eleq2w 2817, whose setvars appear where the class variables are in this theorem, and vice versa. (Contributed by BJ, 24-Jun-2019.) Strengthen from setvar variables to class variables. (Revised by WL and SN, 23-Aug-2024.)
(𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
 
22-Aug-2024flt4lem5c 40146 Part 2 of Equation 2 of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. (Contributed by SN, 22-Aug-2024.)
𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀 − 𝑁))) / 2)    &   𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   (𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   (𝜑 → 𝑁 = (2 · (𝑅 · 𝑆)))
 
22-Aug-2024flt4lem5b 40145 Part 2 of Equation 1 of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. (Contributed by SN, 22-Aug-2024.)
𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀 − 𝑁))) / 2)    &   𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   (𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   (𝜑 → (2 · (𝑀 · 𝑁)) = (𝐵↑2))
 
22-Aug-2024flt4lem5a 40144 Part 1 of Equation 1 of https://crypto.stanford.edu/pbc/notes/numberfield/fermatn4.html. (Contributed by SN, 22-Aug-2024.)
𝑀 = (((√‘(𝐶 + (𝐵↑2))) + (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑁 = (((√‘(𝐶 + (𝐵↑2))) − (√‘(𝐶 − (𝐵↑2)))) / 2)    &   𝑅 = (((√‘(𝑀 + 𝑁)) + (√‘(𝑀 − 𝑁))) / 2)    &   𝑆 = (((√‘(𝑀 + 𝑁)) − (√‘(𝑀 − 𝑁))) / 2)    &   (𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   (𝜑 → ((𝐴↑2) + (𝑁↑2)) = (𝑀↑2))
 
22-Aug-2024flt4lem4 40141 If the product of two coprime factors is a perfect square, the factors are perfect squares. (Contributed by SN, 22-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → (𝐴 gcd 𝐵) = 1)    &   (𝜑 → (𝐴 · 𝐵) = (𝐶↑2))    ⇒   (𝜑 → (𝐴 = ((𝐴 gcd 𝐶)↑2) ∧ 𝐵 = ((𝐵 gcd 𝐶)↑2)))
 
22-Aug-2024flt4lem3 40140 Equivalent to pythagtriplem4 16353. Show that 𝐶 + 𝐴 and 𝐶 − 𝐴 are coprime. (Contributed by SN, 22-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))    ⇒   (𝜑 → ((𝐶 + 𝐴) gcd (𝐶 − 𝐴)) = 1)
 
22-Aug-2024flt4lem2 40139 If 𝐴 is even, 𝐵 is odd. (Contributed by SN, 22-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))    ⇒   (𝜑 → ¬ 2 ∥ 𝐵)
 
22-Aug-2024fltabcoprm 40134 A counterexample to FLT with 𝐴, 𝐶 coprime also has 𝐴, 𝐵 coprime. Converse of fltaccoprm 40132. (Contributed by SN, 22-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))    ⇒   (𝜑 → (𝐴 gcd 𝐵) = 1)
 
22-Aug-2024no2inds 33806 Double induction on surreals. The many substitution instances are to cover all possible cases. (Contributed by Scott Fenton, 22-Aug-2024.)
(𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))    &   (𝑥 = 𝑧 → (𝜃 ↔ 𝜒))    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   (𝑦 = 𝐵 → (𝜏 ↔ 𝜂))    &   ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ((∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒 ∧ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓 ∧ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃) → 𝜑))    ⇒   ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝜂)
 
22-Aug-2024no2indslem 33805 Double induction on surreals with explicit notation for the relationships. (Contributed by Scott Fenton, 22-Aug-2024.)
𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}    &   𝑆 = {⟨𝑐, 𝑑⟩ ∣ (𝑐 ∈ ( No × No ) ∧ 𝑑 ∈ ( No × No ) ∧ (((1st ‘𝑐)𝑅(1st ‘𝑑) ∨ (1st ‘𝑐) = (1st ‘𝑑)) ∧ ((2nd ‘𝑐)𝑅(2nd ‘𝑑) ∨ (2nd ‘𝑐) = (2nd ‘𝑑)) ∧ 𝑐 ≠ 𝑑))}    &   (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))    &   (𝑥 = 𝑧 → (𝜃 ↔ 𝜒))    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   (𝑦 = 𝐵 → (𝜏 ↔ 𝜂))    &   ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → ((∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜒 ∧ ∀𝑧 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓 ∧ ∀𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝜃) → 𝜑))    ⇒   ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝜂)
 
22-Aug-2024xpord2ind 33482 Induction over the cross product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 22-Aug-2024.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    &   𝑅 Fr 𝐴    &   𝑅 Po 𝐴    &   𝑅 Se 𝐴    &   𝑆 Fr 𝐵    &   𝑆 Po 𝐵    &   𝑆 Se 𝐵    &   (𝑎 = 𝑐 → (𝜑 ↔ 𝜓))    &   (𝑏 = 𝑑 → (𝜓 ↔ 𝜒))    &   (𝑎 = 𝑐 → (𝜃 ↔ 𝜒))    &   (𝑎 = 𝑋 → (𝜑 ↔ 𝜏))    &   (𝑏 = 𝑌 → (𝜏 ↔ 𝜂))    &   ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜃) → 𝜑))    ⇒   ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂)
 
22-Aug-2024xpord2pred 33480 Calculate the predecessor class in frxp2 33479. (Contributed by Scott Fenton, 22-Aug-2024.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑋, 𝑌⟩) = (((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑌) ∪ {𝑌})) ∖ {⟨𝑋, 𝑌⟩}))
 
22-Aug-2024frpoins3xp3g 33476 Special case of founded partial recursion over a triple cross product. (Contributed by Scott Fenton, 22-Aug-2024.)
((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → (∀𝑤∀𝑡∀𝑢(⟨⟨𝑤, 𝑡⟩, 𝑢⟩ ∈ Pred(𝑅, ((𝐴 × 𝐵) × 𝐶), ⟨⟨𝑥, 𝑦⟩, 𝑧⟩) → 𝜃) → 𝜑))    &   (𝑥 = 𝑤 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝑡 → (𝜓 ↔ 𝜒))    &   (𝑧 = 𝑢 → (𝜒 ↔ 𝜃))    &   (𝑥 = 𝑋 → (𝜑 ↔ 𝜏))    &   (𝑦 = 𝑌 → (𝜏 ↔ 𝜂))    &   (𝑧 = 𝑍 → (𝜂 ↔ 𝜁))    ⇒   (((𝑅 Fr ((𝐴 × 𝐵) × 𝐶) ∧ 𝑅 Po ((𝐴 × 𝐵) × 𝐶) ∧ 𝑅 Se ((𝐴 × 𝐵) × 𝐶)) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶)) → 𝜁)
 
22-Aug-2024frpoins3xpg 33475 Special case of founded partial induction over a cross product. (Contributed by Scott Fenton, 22-Aug-2024.)
((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (∀𝑧∀𝑤(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒) → 𝜑))    &   (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))    &   (𝑥 = 𝑋 → (𝜑 ↔ 𝜃))    &   (𝑦 = 𝑌 → (𝜃 ↔ 𝜏))    ⇒   (((𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵)) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → 𝜏)
 
22-Aug-2024ralxp3es 33376 Restricted for-all over a triple cross product with explicit substitution. (Contributed by Scott Fenton, 22-Aug-2024.)
(∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)[(1st ‘(1st ‘𝑥)) / 𝑦][(2nd ‘(1st ‘𝑥)) / 𝑧][(2nd ‘𝑥) / 𝑤]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜑)
 
22-Aug-2024sbcoteq1a 33375 Equality theorem for substitution of a class for an ordered triple. (Contributed by Scott Fenton, 22-Aug-2024.)
(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → ([(1st ‘(1st ‘𝐴)) / 𝑥][(2nd ‘(1st ‘𝐴)) / 𝑦][(2nd ‘𝐴) / 𝑧]𝜑 ↔ 𝜑))
 
22-Aug-2024ralxp3f 33373 Restricted for all over a triple cross product. (Contributed by Scott Fenton, 22-Aug-2024.)
Ⅎ𝑦𝜑    &   Ⅎ𝑧𝜑    &   Ⅎ𝑤𝜑    &   Ⅎ𝑥𝜓    &   (𝑥 = ⟨⟨𝑦, 𝑧⟩, 𝑤⟩ → (𝜑 ↔ 𝜓))    ⇒   (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
 
22-Aug-2024f1osetex 8529 The set of bijections between two classes exists. (Contributed by AV, 30-Mar-2024.) (Revised by AV, 8-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.)
{𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐵} ∈ V
 
22-Aug-2024fsetdmprc0 8525 The set of functions with a proper class as domain is empty. (Contributed by AV, 22-Aug-2024.)
(𝐴 ∉ V → {𝑓 ∣ 𝑓 Fn 𝐴} = ∅)
 
22-Aug-2024abanssr 4207 A class abstraction with a conjunction is a subset of the class abstraction with the right conjunct only. (Contributed by AV, 7-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.)
{𝑓 ∣ (𝜑 ∧ 𝜓)} ⊆ {𝑓 ∣ 𝜓}
 
22-Aug-2024abanssl 4206 A class abstraction with a conjunction is a subset of the class abstraction with the left conjunct only. (Contributed by AV, 7-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.)
{𝑓 ∣ (𝜑 ∧ 𝜓)} ⊆ {𝑓 ∣ 𝜑}
 
21-Aug-2024flt4lem1 40138 Satisfy the antecedent used in several pythagtrip 16368 lemmas, with 𝐴, 𝐶 coprime rather than 𝐴, 𝐵. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → ¬ 2 ∥ 𝐴)    &   (𝜑 → (𝐴 gcd 𝐶) = 1)    &   (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2))    ⇒   (𝜑 → ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)))
 
21-Aug-2024flt4lem 40137 Raising a number to the fourth power is equivalent to squaring it twice. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐴 ∈ ℂ)    ⇒   (𝜑 → (𝐴↑4) = ((𝐴↑2)↑2))
 
21-Aug-2024fltdvdsabdvdsc 40130 Any factor of both 𝐴 and 𝐵 also divides 𝐶. This establishes the validity of fltabcoprmex 40131. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐶)
 
21-Aug-2024dvdsexpad 39992 Deduction associated with dvdsexpim 39988. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐴 ∈ ℤ)    &   (𝜑 → 𝐵 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝐴 ∥ 𝐵)    ⇒   (𝜑 → (𝐴↑𝑁) ∥ (𝐵↑𝑁))
 
21-Aug-2024lsubrotld 39965 Rotate the variables left in an equation with subtraction on the left, converting it into an addition. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 − 𝐵) = 𝐶)    ⇒   (𝜑 → (𝐵 + 𝐶) = 𝐴)
 
21-Aug-2024raddcom12d 39964 Swap the first two variables in an equation with addition on the right, converting it into a subtraction. Version of mvrraddd 11227 with a commuted consequent, and of mvlraddd 11225 with a commuted hypothesis. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐴 = (𝐵 + 𝐶))    ⇒   (𝜑 → 𝐵 = (𝐴 − 𝐶))
 
21-Aug-2024laddrotrd 39963 Rotate the variables right in an equation with addition on the left, converting it into a subtraction. Version of mvlladdd 11226 with a commuted consequent, and of mvrladdd 11228 with a commuted hypothesis. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 + 𝐵) = 𝐶)    ⇒   (𝜑 → (𝐶 − 𝐴) = 𝐵)
 
21-Aug-2024mvrrsubd 39962 Move a subtraction in the RHS to a right-addition in the LHS. Converse of mvlraddd 11225. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝐴 = (𝐵 − 𝐶))    ⇒   (𝜑 → (𝐴 + 𝐶) = 𝐵)
 
21-Aug-2024aks4d1p1 39774 Show inequality for existence of a non-divisor. (Contributed by metakunt, 21-Aug-2024.)
(𝜑 → 𝑁 ∈ (ℤ≥‘3))    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    ⇒   (𝜑 → 𝐴 < (2↑𝐵))
 
21-Aug-20243lexlogpow5ineq5 39759 Result for bound in AKS inequality lemma. (Contributed by metakunt, 21-Aug-2024.)
((2 logb 3)↑5) ≤ 15
 
21-Aug-20243lexlogpow2ineq2 39758 Result for bound in AKS inequality lemma. (Contributed by metakunt, 21-Aug-2024.)
(2 < ((2 logb 3)↑2) ∧ ((2 logb 3)↑2) < 3)
 
21-Aug-20243lexlogpow2ineq1 39757 Result for bound in AKS inequality lemma. (Contributed by metakunt, 21-Aug-2024.)
((3 / 2) < (2 logb 3) ∧ (2 logb 3) < (5 / 3))
 
21-Aug-20243lexlogpow5ineq4 39755 Sharper logarithm inequality chain. (Contributed by metakunt, 21-Aug-2024.)
(𝜑 → 𝑋 ∈ ℝ)    &   (𝜑 → 3 ≤ 𝑋)    ⇒   (𝜑 → 9 < ((2 logb 𝑋)↑5))
 
21-Aug-20243exp7 39752 3 to the power of 7 equals 2187. (Contributed by metakunt, 21-Aug-2024.)
(3↑7) = 2187
 
21-Aug-2024sexp3 33487 Show that the triple order is set-like. (Contributed by Scott Fenton, 21-Aug-2024.)
𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    &   (𝜑 → 𝑅 Se 𝐴)    &   (𝜑 → 𝑆 Se 𝐵)    &   (𝜑 → 𝑇 Se 𝐶)    ⇒   (𝜑 → 𝑈 Se ((𝐴 × 𝐵) × 𝐶))
 
21-Aug-2024xpord3pred 33486 Calculate the predecsessor class for the triple order. (Contributed by Scott Fenton, 21-Aug-2024.)
𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    ⇒   ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨⟨𝑋, 𝑌⟩, 𝑍⟩) = ((((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑌) ∪ {𝑌})) × (Pred(𝑇, 𝐶, 𝑍) ∪ {𝑍})) ∖ {⟨⟨𝑋, 𝑌⟩, 𝑍⟩}))
 
21-Aug-2024frxp3 33485 Give foundedness over a triple cross product. (Contributed by Scott Fenton, 21-Aug-2024.)
𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    &   (𝜑 → 𝑅 Fr 𝐴)    &   (𝜑 → 𝑆 Fr 𝐵)    &   (𝜑 → 𝑇 Fr 𝐶)    ⇒   (𝜑 → 𝑈 Fr ((𝐴 × 𝐵) × 𝐶))
 
21-Aug-2024poxp3 33484 Triple cross product partial ordering. (Contributed by Scott Fenton, 21-Aug-2024.)
𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    &   (𝜑 → 𝑅 Po 𝐴)    &   (𝜑 → 𝑆 Po 𝐵)    &   (𝜑 → 𝑇 Po 𝐶)    ⇒   (𝜑 → 𝑈 Po ((𝐴 × 𝐵) × 𝐶))
 
21-Aug-2024xpord3lem 33483 Lemma for triple ordering. Calculate the value of the relationship. (Contributed by Scott Fenton, 21-Aug-2024.)
𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    ⇒   (⟨⟨𝑎, 𝑏⟩, 𝑐⟩𝑈⟨⟨𝑑, 𝑒⟩, 𝑓⟩ ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶) ∧ (((𝑎𝑅𝑑 ∨ 𝑎 = 𝑑) ∧ (𝑏𝑆𝑒 ∨ 𝑏 = 𝑒) ∧ (𝑐𝑇𝑓 ∨ 𝑐 = 𝑓)) ∧ (𝑎 ≠ 𝑑 ∨ 𝑏 ≠ 𝑒 ∨ 𝑐 ≠ 𝑓))))
 
21-Aug-2024ralxp3 33374 Restricted for-all over a triple cross product. (Contributed by Scott Fenton, 21-Aug-2024.)
(𝑝 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   (∀𝑝 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓)
 
21-Aug-2024elxpxpss 33372 Version of elrel 5657 for triple cross products. (Contributed by Scott Fenton, 21-Aug-2024.)
((𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷) ∧ 𝐴 ∈ 𝑅) → ∃𝑥∃𝑦∃𝑧 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
 
21-Aug-2024elxpxp 33371 Membership in a triple cross product. (Contributed by Scott Fenton, 21-Aug-2024.)
(𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
 
21-Aug-2024otthne 33370 Contrapositive of the ordered triple theorem. (Contributed by Scott Fenton, 21-Aug-2024.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    ⇒   (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ≠ ⟨⟨𝐷, 𝐸⟩, 𝐹⟩ ↔ (𝐴 ≠ 𝐷 ∨ 𝐵 ≠ 𝐸 ∨ 𝐶 ≠ 𝐹))
 
21-Aug-2024ot22ndd 33369 Extract the second member of an ordered triple. Deduction version. (Contributed by Scott Fenton, 21-Aug-2024.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    ⇒   (𝑋 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ → (2nd ‘(1st ‘𝑋)) = 𝐵)
 
21-Aug-2024ot21std 33368 Extract the first member of an ordered triple. Deduction version. (Contributed by Scott Fenton, 21-Aug-2024.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    ⇒   (𝑋 = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ → (1st ‘(1st ‘𝑋)) = 𝐴)
 
21-Aug-2024ot2elxp 33367 Ordered triple membership in a triple cross product. (Contributed by Scott Fenton, 21-Aug-2024.)
(⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ ((𝐷 × 𝐸) × 𝐹) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐸 ∧ 𝐶 ∈ 𝐹))
 
21-Aug-2024ralxpes 33366 A version of ralxp 5699 with explicit substitution. (Contributed by Scott Fenton, 21-Aug-2024.)
(∀𝑥 ∈ (𝐴 × 𝐵)[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜑)
 
21-Aug-2024prmdvdssq 16256 Condition for a prime dividing a square. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by SN, 21-Aug-2024.)
((𝑃 ∈ ℙ ∧ 𝑀 ∈ ℤ) → (𝑃 ∥ 𝑀 ↔ 𝑃 ∥ (𝑀↑2)))
 
21-Aug-2024rprpwr 16101 If 𝐴 and 𝐵 are relatively prime, then so are 𝐴 and 𝐵↑𝑁. Originally a subproof of rppwr 16102. (Contributed by SN, 21-Aug-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) = 1 → (𝐴 gcd (𝐵↑𝑁)) = 1))
 
21-Aug-2024dvds2addd 15834 Deduction form of dvds2add 15832. (Contributed by SN, 21-Aug-2024.)
(𝜑 → 𝐾 ∈ ℤ)    &   (𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝐾 ∥ 𝑀)    &   (𝜑 → 𝐾 ∥ 𝑁)    ⇒   (𝜑 → 𝐾 ∥ (𝑀 + 𝑁))
 
20-Aug-2024infdesc 40135 Infinite descent. The hypotheses say that 𝑆 is lower bounded, and that if 𝜓 holds for an integer in 𝑆, it holds for a smaller integer in 𝑆. By infinite descent, eventually we cannot go any smaller, therefore 𝜓 holds for no integer in 𝑆. (Contributed by SN, 20-Aug-2024.)
(𝑦 = 𝑥 → (𝜓 ↔ 𝜒))    &   (𝑦 = 𝑧 → (𝜓 ↔ 𝜃))    &   (𝜑 → 𝑆 ⊆ (ℤ≥‘𝑀))    &   ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝜒)) → ∃𝑧 ∈ 𝑆 (𝜃 ∧ 𝑧 < 𝑥))    ⇒   (𝜑 → {𝑦 ∈ 𝑆 ∣ 𝜓} = ∅)
 
20-Aug-2024fltbccoprm 40133 A counterexample to FLT with 𝐴, 𝐵 coprime also has 𝐵, 𝐶 coprime. Proven from fltaccoprm 40132 using commutativity of addition. (Contributed by SN, 20-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    &   (𝜑 → (𝐴 gcd 𝐵) = 1)    ⇒   (𝜑 → (𝐵 gcd 𝐶) = 1)
 
20-Aug-2024fltaccoprm 40132 A counterexample to FLT with 𝐴, 𝐵 coprime also has 𝐴, 𝐶 coprime. (Contributed by SN, 20-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    &   (𝜑 → (𝐴 gcd 𝐵) = 1)    ⇒   (𝜑 → (𝐴 gcd 𝐶) = 1)
 
20-Aug-2024fltabcoprmex 40131 A counterexample to FLT implies a counterexample to FLT with 𝐴, 𝐵 (assigned to 𝐴 / (𝐴 gcd 𝐵) and 𝐵 / (𝐴 gcd 𝐵)) coprime (by divgcdcoprm0 16203). (Contributed by SN, 20-Aug-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   (𝜑 → (((𝐴 / (𝐴 gcd 𝐵))↑𝑁) + ((𝐵 / (𝐴 gcd 𝐵))↑𝑁)) = ((𝐶 / (𝐴 gcd 𝐵))↑𝑁))
 
20-Aug-2024flt0 40129 A counterexample for FLT does not exist for 𝑁 = 0. (Contributed by SN, 20-Aug-2024.)
(𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   (𝜑 → 𝑁 ∈ ℕ)
 
20-Aug-2024fltdiv 40128 A counterexample to FLT stays valid when scaled. The hypotheses are more general than they need to be for convenience. (Contributed by SN, 20-Aug-2024.)
(𝜑 → 𝑆 ∈ ℂ)    &   (𝜑 → 𝑆 ≠ 0)    &   (𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   (𝜑 → (((𝐴 / 𝑆)↑𝑁) + ((𝐵 / 𝑆)↑𝑁)) = ((𝐶 / 𝑆)↑𝑁))
 
20-Aug-2024fltmul 40127 A counterexample to FLT stays valid when scaled. The hypotheses are more general than they need to be for convenience. (There does not seem to be a standard term for Fermat or Pythagorean triples extended to any 𝑁 ∈ ℕ0, hence the label is more about the context in which this theorem is used). (Contributed by SN, 20-Aug-2024.)
(𝜑 → 𝑆 ∈ ℂ)    &   (𝜑 → 𝐴 ∈ ℂ)    &   (𝜑 → 𝐵 ∈ ℂ)    &   (𝜑 → 𝐶 ∈ ℂ)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁))    ⇒   (𝜑 → (((𝑆 · 𝐴)↑𝑁) + ((𝑆 · 𝐵)↑𝑁)) = ((𝑆 · 𝐶)↑𝑁))
 
20-Aug-2024dvdsexpnn 40000 dvdssqlem 16104 generalized to positive integer exponents. (Contributed by SN, 20-Aug-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴 ∥ 𝐵 ↔ (𝐴↑𝑁) ∥ (𝐵↑𝑁)))
 
20-Aug-20243rspcedvdw 39856 Triple application of rspcedvdw 39854. (Contributed by SN, 20-Aug-2024.)
(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    &   (𝑦 = 𝐵 → (𝜒 ↔ 𝜃))    &   (𝑧 = 𝐶 → (𝜃 ↔ 𝜏))    &   (𝜑 → 𝐴 ∈ 𝑋)    &   (𝜑 → 𝐵 ∈ 𝑌)    &   (𝜑 → 𝐶 ∈ 𝑍)    &   (𝜑 → 𝜏)    ⇒   (𝜑 → ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑌 ∃𝑧 ∈ 𝑍 𝜓)
 
20-Aug-2024rspcedvdw 39854 Version of rspcedvd 3533 where the implicit substitution hypothesis does not have an antecedent, which also avoids a disjoint variable condition on 𝜑, 𝑥. (Contributed by SN, 20-Aug-2024.)
(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))    &   (𝜑 → 𝐴 ∈ 𝐵)    &   (𝜑 → 𝜒)    ⇒   (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
 
20-Aug-2024addscomd 33824 Surreal addition commutes. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)
(𝜑 → 𝐴 ∈ No )    &   (𝜑 → 𝐵 ∈ No )    ⇒   (𝜑 → (𝐴 +s 𝐵) = (𝐵 +s 𝐴))
 
20-Aug-2024addscom 33823 Surreal addition commutes. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (𝐵 +s 𝐴))
 
20-Aug-2024addsid1d 33822 Surreal addition to zero is identity. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)
(𝜑 → 𝐴 ∈ No )    ⇒   (𝜑 → (𝐴 +s 0s ) = 𝐴)
 
20-Aug-2024addsid1 33821 Surreal addition to zero is identity. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)
(𝐴 ∈ No → (𝐴 +s 0s ) = 𝐴)
 
20-Aug-2024addsval 33820 The value of surreal addition. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 20-Aug-2024.)
((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑦 = (𝑟 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘𝐵)𝑧 = (𝐴 +s 𝑟)})))
 
20-Aug-2024addsfn 33819 Surreal addition is a function over pairs of surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
+s Fn ( No × No )
 
20-Aug-2024negs0s 33818 Negative surreal zero is surreal zero. (Contributed by Scott Fenton, 20-Aug-2024.)
( -us ‘ 0s ) = 0s
 
20-Aug-2024negsval 33817 The value of the surreal negation function. (Contributed by Scott Fenton, 20-Aug-2024.)
(𝐴 ∈ No → ( -us ‘𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
 
20-Aug-2024negsfn 33816 Surreal negation is a function over surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
-us Fn No
 
20-Aug-2024df-subs 33815 Define surreal subtraction. (Contributed by Scott Fenton, 20-Aug-2024.)
-s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +s ( -us ‘𝑦)))
 
20-Aug-2024df-negs 33814 Define surreal negation. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 20-Aug-2024.)
-us = norec ((𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥)))))
 
20-Aug-2024df-adds 33813 Define surreal addition. This is the first of the field operations on the surreals. Definition from [Conway] p. 5. Definition from [Gonshor] p. 13. (Contributed by Scott Fenton, 20-Aug-2024.)
+s = norec2 ((𝑥 ∈ V, 𝑎 ∈ V ↦ (({𝑦 ∣ ∃𝑙 ∈ ( L ‘(1st ‘𝑥))𝑦 = (𝑙𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑙 ∈ ( L ‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑙)}) |s ({𝑦 ∣ ∃𝑟 ∈ ( R ‘(1st ‘𝑥))𝑦 = (𝑟𝑎(2nd ‘𝑥))} ∪ {𝑧 ∣ ∃𝑟 ∈ ( R ‘(2nd ‘𝑥))𝑧 = ((1st ‘𝑥)𝑎𝑟)}))))
 
20-Aug-2024norec2ov 33808 The value of the double-recursion surreal function. (Contributed by Scott Fenton, 20-Aug-2024.)
𝐹 = norec2 (𝐺)    ⇒   ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝐹𝐵) = (⟨𝐴, 𝐵⟩𝐺(𝐹 ↾ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))))
 
20-Aug-2024norec2fn 33807 The double-recursion operator on surreals yields a function on pairs of surreals. (Contributed by Scott Fenton, 20-Aug-2024.)
𝐹 = norec2 (𝐺)    ⇒   𝐹 Fn ( No × No )
 
20-Aug-2024df-norec2 33800 Define surreal recursion on two variables. This function is key to the development of most of surreal numbers. (Contributed by Scott Fenton, 20-Aug-2024.)
norec2 (𝐹) = frecs({⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ ( No × No ) ∧ 𝑏 ∈ ( No × No ) ∧ (((1st ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd ‘𝑎){⟨𝑐, 𝑑⟩ ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No × No ), 𝐹)
 
20-Aug-2024right0s 33770 The right set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.)
( R ‘ 0s ) = ∅
 
20-Aug-2024left0s 33769 The left set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024.)
( L ‘ 0s ) = ∅
 
20-Aug-2024ssltsepc 33681 Two elements of separated sets obey less than. (Contributed by Scott Fenton, 20-Aug-2024.)
((𝐴 <<s 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 <s 𝑌)
 
19-Aug-2024aks4d1p1p5 39773 Show inequality for existence of a non-divisor. (Contributed by metakunt, 19-Aug-2024.)
(𝜑 → 𝑁 ∈ ℕ)    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   (𝜑 → 4 ≤ 𝑁)    &   𝐶 = (2 logb (((2 logb 𝑁)↑5) + 1))    &   𝐷 = ((2 logb 𝑁)↑2)    &   𝐸 = ((2 logb 𝑁)↑4)    ⇒   (𝜑 → 𝐴 < (2↑𝐵))
 
19-Aug-2024aks4d1p1p7 39772 Bound of intermediary of inequality step. (Contributed by metakunt, 19-Aug-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 4 ≤ 𝐴)    ⇒   (𝜑 → ((2 · ((1 / ((((2 logb 𝐴)↑5) + 1) · (log‘2))) · (((5 · ((2 logb 𝐴)↑4)) · (1 / (𝐴 · (log‘2)))) + 0))) + ((2 / ((log‘2)↑2)) · (((log‘𝐴)↑(2 − 1)) / 𝐴))) ≤ ((4 / ((log‘2)↑4)) · (((log‘𝐴)↑3) / 𝐴)))
 
19-Aug-2024aks4d1p1p6 39771 Inequality lift to differentiable functions for a term in AKS inequality lemma. (Contributed by metakunt, 19-Aug-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 3 ≤ 𝐴)    &   (𝜑 → 𝐴 ≤ 𝐵)    ⇒   (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 · (2 logb (((2 logb 𝑥)↑5) + 1))) + ((2 logb 𝑥)↑2)))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 · ((1 / ((((2 logb 𝑥)↑5) + 1) · (log‘2))) · (((5 · ((2 logb 𝑥)↑4)) · (1 / (𝑥 · (log‘2)))) + 0))) + ((2 / ((log‘2)↑2)) · (((log‘𝑥)↑(2 − 1)) / 𝑥)))))
 
19-Aug-2024dvle2 39770 Collapsed dvle 24876. (Contributed by metakunt, 19-Aug-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐸) ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝐺) ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐸)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐹))    &   (𝜑 → (ℝ D (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐺)) = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝐻))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝐹 ≤ 𝐻)    &   (𝑥 = 𝐴 → 𝐸 = 𝑃)    &   (𝑥 = 𝐴 → 𝐺 = 𝑄)    &   (𝑥 = 𝐵 → 𝐸 = 𝑅)    &   (𝑥 = 𝐵 → 𝐺 = 𝑆)    &   (𝜑 → 𝑃 ≤ 𝑄)    &   (𝜑 → 𝐴 ≤ 𝐵)    ⇒   (𝜑 → 𝑅 ≤ 𝑆)
 
19-Aug-2024aks4d1p1p4 39769 Technical step for inequality. The hard work is in to prove the final hypothesis. (Contributed by metakunt, 19-Aug-2024.)
(𝜑 → 𝑁 ∈ ℕ)    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   (𝜑 → 3 ≤ 𝑁)    &   𝐶 = (2 logb (((2 logb 𝑁)↑5) + 1))    &   𝐷 = ((2 logb 𝑁)↑2)    &   𝐸 = ((2 logb 𝑁)↑4)    &   (𝜑 → ((2 · 𝐶) + 𝐷) ≤ 𝐸)    ⇒   (𝜑 → 𝐴 < (2↑𝐵))
 
19-Aug-2024aks4d1p1p2 39768 Rewrite 𝐴 in more suitable form. (Contributed by metakunt, 19-Aug-2024.)
(𝜑 → 𝑁 ∈ ℕ)    &   𝐴 = ((𝑁↑(⌊‘(2 logb 𝐵))) · ∏𝑘 ∈ (1...(⌊‘((2 logb 𝑁)↑2)))((𝑁↑𝑘) − 1))    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   (𝜑 → 3 ≤ 𝑁)    ⇒   (𝜑 → 𝐴 < (𝑁↑𝑐(((2 logb (((2 logb 𝑁)↑5) + 1)) + (((2 logb 𝑁)↑2) / 2)) + (((2 logb 𝑁)↑4) / 2))))
 
19-Aug-2024aks4d1p1p3 39767 Bound of a ceiling of the binary logarithm to the fifth power. (Contributed by metakunt, 19-Aug-2024.)
(𝜑 → 𝑁 ∈ ℕ)    &   𝐵 = (⌈‘((2 logb 𝑁)↑5))    &   (𝜑 → 3 ≤ 𝑁)    ⇒   (𝜑 → (𝑁↑𝑐(⌊‘(2 logb 𝐵))) < (𝑁↑𝑐(2 logb (((2 logb 𝑁)↑5) + 1))))
 
19-Aug-2024noxpordpred 33804 Next we calculate the predecessor class of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}    &   𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → Pred(𝑆, ( No × No ), ⟨𝐴, 𝐵⟩) = ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {⟨𝐴, 𝐵⟩}))
 
19-Aug-2024noxpordse 33803 Next we establish the set-like nature of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}    &   𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   𝑆 Se ( No × No )
 
19-Aug-2024noxpordfr 33802 Next we establish the foundedness of the relationship. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}    &   𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   𝑆 Fr ( No × No )
 
19-Aug-2024noxpordpo 33801 To get through most of the textbook defintions in surreal numbers we will need recursion on two variables. This set of theorems sets up the preconditions for double recursion. This theorem establishes the partial ordering. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑎, 𝑏⟩ ∣ 𝑎 ∈ (( L ‘𝑏) ∪ ( R ‘𝑏))}    &   𝑆 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ( No × No ) ∧ 𝑦 ∈ ( No × No ) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑅(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   𝑆 Po ( No × No )
 
19-Aug-2024norecov 33798 Calculate the value of the surreal recursion operation. (Contributed by Scott Fenton, 19-Aug-2024.)
𝐹 = norec (𝐺)    ⇒   (𝐴 ∈ No → (𝐹‘𝐴) = (𝐴𝐺(𝐹 ↾ (( L ‘𝐴) ∪ ( R ‘𝐴)))))
 
19-Aug-2024norecfn 33797 Surreal recursion over one variable is a function over the surreals. (Contributed by Scott Fenton, 19-Aug-2024.)
𝐹 = norec (𝐺)    ⇒   𝐹 Fn No
 
19-Aug-2024noinds 33796 Induction principle for a single surreal. If a property passes from a surreal's left and right sets to the surreal itself, then it holds for all surreals. (Contributed by Scott Fenton, 19-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜒))    &   (𝑥 ∈ No → (∀𝑦 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))𝜓 → 𝜑))    ⇒   (𝐴 ∈ No → 𝜒)
 
19-Aug-2024lrrecpred 33795 Finally, we calculate the value of the predecessor class over 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}    ⇒   (𝐴 ∈ No → Pred(𝑅, No , 𝐴) = (( L ‘𝐴) ∪ ( R ‘𝐴)))
 
19-Aug-2024lrrecfr 33794 Now we show that 𝑅 is founded over No . (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}    ⇒   𝑅 Fr No
 
19-Aug-2024lrrecse 33793 Next, we show that 𝑅 is set-like over No . (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}    ⇒   𝑅 Se No
 
19-Aug-2024lrrecpo 33792 Now, we establish that 𝑅 is a partial ordering on No . (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}    ⇒   𝑅 Po No
 
19-Aug-2024lrrecval2 33791 Next, we establish an alternate expression for 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}    ⇒   ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝑅𝐵 ↔ ( bday ‘𝐴) ∈ ( bday ‘𝐵)))
 
19-Aug-2024lrrecval 33790 The next step in the development of the surreals is to establish induction and recursion across left and right sets. To that end, we are going to develop a relationship 𝑅 that is founded, partial, and set-like across the surreals. This first theorem just establishes the value of 𝑅. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}    ⇒   ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴𝑅𝐵 ↔ 𝐴 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))))
 
19-Aug-2024df-norec 33789 Define the recursion generator for surreal functions of one variable. This generator creates a recursive function of surreals from their value on their left and right sets. (Contributed by Scott Fenton, 19-Aug-2024.)
norec (𝐹) = frecs({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))}, No , 𝐹)
 
19-Aug-2024lrcut 33777 A surreal is equal to the cut of its left and right sets. (Contributed by Scott Fenton, 19-Aug-2024.)
(𝑋 ∈ No → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
 
19-Aug-2024newbday 33776 A surreal is an element of a new set iff its birthday is equal to that ordinal. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)
((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( N ‘𝐴) ↔ ( bday ‘𝑋) = 𝐴))
 
19-Aug-2024oldbday 33775 A surreal is part of the set older than ordinal 𝐴 iff its birthday is less than 𝐴. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)
((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( O ‘𝐴) ↔ ( bday ‘𝑋) ∈ 𝐴))
 
19-Aug-2024madebday 33774 A surreal is part of the set made by ordinal 𝐴 iff its birthday is less than or equal to 𝐴. Remark in [Conway] p. 29. (Contributed by Scott Fenton, 19-Aug-2024.)
((𝐴 ∈ On ∧ 𝑋 ∈ No ) → (𝑋 ∈ ( M ‘𝐴) ↔ ( bday ‘𝑋) ⊆ 𝐴))
 
19-Aug-2024madebdaylemlrcut 33773 Lemma for madebday 33774. If the inductive hypothesis of madebday 33774 is satisfied up to the birthday of 𝑋, then the conclusion of lrcut 33777 holds. (Contributed by Scott Fenton, 19-Aug-2024.)
((∀𝑏 ∈ ( bday ‘𝑋)∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
 
19-Aug-2024madebdaylemold 33772 Lemma for madebday 33774. If the inductive hypothesis of madebday 33774 is satisfied, the converse of oldbdayim 33765 holds. (Contributed by Scott Fenton, 19-Aug-2024.)
((𝐴 ∈ On ∧ ∀𝑏 ∈ 𝐴 ∀𝑦 ∈ No (( bday ‘𝑦) ⊆ 𝑏 → 𝑦 ∈ ( M ‘𝑏)) ∧ 𝑋 ∈ No ) → (( bday ‘𝑋) ∈ 𝐴 → 𝑋 ∈ ( O ‘𝐴)))
 
19-Aug-2024sexp2 33481 Condition for the relationship in frxp2 33479 to be set-like. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    &   (𝜑 → 𝑅 Se 𝐴)    &   (𝜑 → 𝑆 Se 𝐵)    ⇒   (𝜑 → 𝑇 Se (𝐴 × 𝐵))
 
19-Aug-2024frxp2 33479 Another way of giving a founded order to a cross product of two classes. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    &   (𝜑 → 𝑅 Fr 𝐴)    &   (𝜑 → 𝑆 Fr 𝐵)    ⇒   (𝜑 → 𝑇 Fr (𝐴 × 𝐵))
 
19-Aug-2024poxp2 33478 Another way of partially ordering a cross product of two classes. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    &   (𝜑 → 𝑅 Po 𝐴)    &   (𝜑 → 𝑆 Po 𝐵)    ⇒   (𝜑 → 𝑇 Po (𝐴 × 𝐵))
 
19-Aug-2024xpord2lem 33477 Lemma for cross product ordering. Calculate the value of the cross product relationship. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵) ∧ ((𝑎𝑅𝑐 ∨ 𝑎 = 𝑐) ∧ (𝑏𝑆𝑑 ∨ 𝑏 = 𝑑) ∧ (𝑎 ≠ 𝑐 ∨ 𝑏 ≠ 𝑑))))
 
19-Aug-2024dfse3 33365 Alternate definition of set-like relationships. (Contributed by Scott Fenton, 19-Aug-2024.)
(𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 Pred(𝑅, 𝐴, 𝑥) ∈ V)
 
19-Aug-2024xpab 33364 Cross product of two class abstractions. (Contributed by Scott Fenton, 19-Aug-2024.)
({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
 
19-Aug-2024frpoins3g 6189 Well-Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 19-Aug-2024.)
(𝑥 ∈ 𝐴 → (∀𝑦 ∈ Pred (𝑅, 𝐴, 𝑥)𝜓 → 𝜑))    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   (𝑥 = 𝐵 → (𝜑 ↔ 𝜒))    ⇒   (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝐵 ∈ 𝐴) → 𝜒)
 
18-Aug-2024prdsco 16945 Structure product composition operation. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝐻 = (Hom ‘𝑃)    &    ∙ = (comp‘𝑃)    ⇒   (𝜑 → ∙ = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))))
 
18-Aug-2024prdshom 16944 Structure product hom-sets. (Contributed by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝐻 = (Hom ‘𝑃)    ⇒   (𝜑 → 𝐻 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))))
 
18-Aug-2024prdstset 16943 Structure product topology. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝑂 = (TopSet‘𝑃)    ⇒   (𝜑 → 𝑂 = (∏t‘(TopOpen ∘ 𝑅)))
 
18-Aug-2024prdsds 16941 Structure product distance function. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝐷 = (dist‘𝑃)    ⇒   (𝜑 → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )))
 
18-Aug-2024prdsle 16939 Structure product weak ordering. (Contributed by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &    ≤ = (le‘𝑃)    ⇒   (𝜑 → ≤ = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})
 
18-Aug-2024prdsip 16938 Inner product in a structure product. (Contributed by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &    , = (·𝑖‘𝑃)    ⇒   (𝜑 → , = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))))
 
18-Aug-2024prdsvsca 16937 Scalar multiplication in a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &   𝐾 = (Base‘𝑆)    &    · = ( ·𝑠 ‘𝑃)    ⇒   (𝜑 → · = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))
 
18-Aug-2024prdsmulr 16936 Multiplication in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &    · = (.r‘𝑃)    ⇒   (𝜑 → · = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))))
 
18-Aug-2024prdsplusg 16935 Addition in a structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    &    + = (+g‘𝑃)    ⇒   (𝜑 → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))))
 
18-Aug-2024prdsbas 16934 Base set of a structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → dom 𝑅 = 𝐼)    ⇒   (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
 
18-Aug-2024prdssca 16933 Scalar ring of a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   (𝜑 → 𝑆 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    ⇒   (𝜑 → 𝑆 = (Scalar‘𝑃))
 
18-Aug-2024prdsval 16932 Value of the structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
𝑃 = (𝑆Xs𝑅)    &   𝐾 = (Base‘𝑆)    &   (𝜑 → dom 𝑅 = 𝐼)    &   (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))    &   (𝜑 → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))))    &   (𝜑 → × = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))))    &   (𝜑 → · = (𝑓 ∈ 𝐾, 𝑔 ∈ 𝐵 ↦ (𝑥 ∈ 𝐼 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))    &   (𝜑 → , = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑆 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))))    &   (𝜑 → 𝑂 = (∏t‘(TopOpen ∘ 𝑅)))    &   (𝜑 → ≤ = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})    &   (𝜑 → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )))    &   (𝜑 → 𝐻 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ X𝑥 ∈ 𝐼 ((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))))    &   (𝜑 → ∙ = (𝑎 ∈ (𝐵 × 𝐵), 𝑐 ∈ 𝐵 ↦ (𝑑 ∈ ((2nd ‘𝑎)𝐻𝑐), 𝑒 ∈ (𝐻‘𝑎) ↦ (𝑥 ∈ 𝐼 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))))    &   (𝜑 → 𝑆 ∈ 𝑊)    &   (𝜑 → 𝑅 ∈ 𝑍)    ⇒   (𝜑 → 𝑃 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), × ⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}) ∪ ({⟨(TopSet‘ndx), 𝑂⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), 𝐷⟩} ∪ {⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), ∙ ⟩})))
 
18-Aug-2024reldmprds 16925 The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
Rel dom Xs
 
18-Aug-2024df-prds 16924 Define a structure product. This can be a product of groups, rings, modules, or ordered topological fields; any unused components will have garbage in them but this is usually not relevant for the purpose of inheriting the structures present in the factors. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ ⦋X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) / 𝑣⦌⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ ((2nd ‘𝑎)ℎ𝑐), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})))
 
18-Aug-2024dif1enlem 8827 Lemma for rexdif1en 8828 and dif1en 8829. (Contributed by BTernaryTau, 18-Aug-2024.)
((𝐹 ∈ 𝑉 ∧ 𝑀 ∈ ω ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝐹‘𝑀)}) ≈ 𝑀)
 
16-Aug-2024fnssintima 33363 Condition for subset of an intersection of an image. (Contributed by Scott Fenton, 16-Aug-2024.)
((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ⊆ ∩ (𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐵 𝐶 ⊆ (𝐹‘𝑥)))
 
16-Aug-2024fnunop 6481 Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by AV, 16-Aug-2024.)
(𝜑 → 𝑋 ∈ 𝑉)    &   (𝜑 → 𝑌 ∈ 𝑊)    &   (𝜑 → 𝐹 Fn 𝐷)    &   𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})    &   𝐸 = (𝐷 ∪ {𝑋})    &   (𝜑 → ¬ 𝑋 ∈ 𝐷)    ⇒   (𝜑 → 𝐺 Fn 𝐸)
 
16-Aug-2024preqsnd 4759 Equivalence for a pair equal to a singleton, deduction form. (Contributed by Thierry Arnoux, 27-Dec-2016.) (Revised by AV, 13-Jun-2022.) (Revised by AV, 16-Aug-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → ({𝐴, 𝐵} = {𝐶} ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐶)))
 
16-Aug-2024elab3 3588 Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.) (Revised by AV, 16-Aug-2024.)
(𝜓 → 𝐴 ∈ 𝑉)    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
 
16-Aug-2024spcedv 3506 Existential specialization, using implicit substitution, deduction version. (Contributed by RP, 12-Aug-2020.) (Revised by AV, 16-Aug-2024.)
(𝜑 → 𝑋 ∈ 𝑉)    &   (𝜑 → 𝜒)    &   (𝑥 = 𝑋 → (𝜓 ↔ 𝜒))    ⇒   (𝜑 → ∃𝑥𝜓)
 
15-Aug-2024resdifdir 6089 Distributive law for restriction over difference. (Contributed by BTernaryTau, 15-Aug-2024.)
((𝐴 ∖ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∖ (𝐵 ↾ 𝐶))
 
15-Aug-2024resdifdi 6088 Distributive law for restriction over difference. (Contributed by BTernaryTau, 15-Aug-2024.)
(𝐴 ↾ (𝐵 ∖ 𝐶)) = ((𝐴 ↾ 𝐵) ∖ (𝐴 ↾ 𝐶))
 
15-Aug-2024ifssun 4446 A conditional class is included in the union of its two alternatives. (Contributed by BJ, 15-Aug-2024.)
if(𝜑, 𝐴, 𝐵) ⊆ (𝐴 ∪ 𝐵)
 
14-Aug-2024frlm0vald 39926 All coordinates of the zero vector are zero. (Contributed by SN, 14-Aug-2024.)
𝐹 = (𝑅 freeLMod 𝐼)    &    0 = (0g‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐼 ∈ 𝑊)    &   (𝜑 → 𝐽 ∈ 𝐼)    ⇒   (𝜑 → ((0g‘𝐹)‘𝐽) = 0 )
 
14-Aug-2024drnginvmuld 39919 Inverse of a nonzero product. (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   𝐼 = (invr‘𝑅)    &   (𝜑 → 𝑅 ∈ DivRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑋 ≠ 0 )    &   (𝜑 → 𝑌 ≠ 0 )    ⇒   (𝜑 → (𝐼‘(𝑋 · 𝑌)) = ((𝐼‘𝑌) · (𝐼‘𝑋)))
 
14-Aug-2024drngmulcan2ad 39918 Cancellation of a nonzero factor on the right for multiplication. (mulcan2ad 11451 analog). (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ DivRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝑍 ≠ 0 )    &   (𝜑 → (𝑋 · 𝑍) = (𝑌 · 𝑍))    ⇒   (𝜑 → 𝑋 = 𝑌)
 
14-Aug-2024drngmulcanad 39917 Cancellation of a nonzero factor on the left for multiplication. (mulcanad 11450 analog). (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ DivRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    &   (𝜑 → 𝑍 ≠ 0 )    &   (𝜑 → (𝑍 · 𝑋) = (𝑍 · 𝑌))    ⇒   (𝜑 → 𝑋 = 𝑌)
 
14-Aug-2024drnginvrrd 39916 Property of the multiplicative inverse in a division ring. (recidd 11586 analog). (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝑅)    &   𝐼 = (invr‘𝑅)    &   (𝜑 → 𝑅 ∈ DivRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑋 ≠ 0 )    ⇒   (𝜑 → (𝑋 · (𝐼‘𝑋)) = 1 )
 
14-Aug-2024drnginvrld 39915 Property of the multiplicative inverse in a division ring. (recid2d 11587 analog). (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝑅)    &   𝐼 = (invr‘𝑅)    &   (𝜑 → 𝑅 ∈ DivRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑋 ≠ 0 )    ⇒   (𝜑 → ((𝐼‘𝑋) · 𝑋) = 1 )
 
14-Aug-2024drnginvrn0d 39914 A multiplicative inverse in a division ring is nonzero. (recne0d 11585 analog). (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝐼 = (invr‘𝑅)    &   (𝜑 → 𝑅 ∈ DivRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑋 ≠ 0 )    ⇒   (𝜑 → (𝐼‘𝑋) ≠ 0 )
 
14-Aug-2024drnginvrcld 39913 Closure of the multiplicative inverse in a division ring. (reccld 11584 analog). (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝐼 = (invr‘𝑅)    &   (𝜑 → 𝑅 ∈ DivRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑋 ≠ 0 )    ⇒   (𝜑 → (𝐼‘𝑋) ∈ 𝐵)
 
14-Aug-2024ringridmd 39908 The unit element of a ring is a right multiplicative identity. (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 · 1 ) = 𝑋)
 
14-Aug-2024ringlidmd 39907 The unit element of a ring is a left multiplicative identity. (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &    1 = (1r‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → ( 1 · 𝑋) = 𝑋)
 
14-Aug-2024ringassd 39906 Associative law for multiplication in a ring. (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍)))
 
14-Aug-2024brif12 39865 Move a relation inside and outside the conditional operator. (Contributed by SN, 14-Aug-2024.)
(if(𝜑, 𝐴, 𝐵)𝑅if(𝜑, 𝐶, 𝐷) ↔ if-(𝜑, 𝐴𝑅𝐶, 𝐵𝑅𝐷))
 
14-Aug-2024brif2 39864 Move a relation inside and outside the conditional operator. (Contributed by SN, 14-Aug-2024.)
(𝐶𝑅if(𝜑, 𝐴, 𝐵) ↔ if-(𝜑, 𝐶𝑅𝐴, 𝐶𝑅𝐵))
 
14-Aug-2024brif1 39863 Move a relation inside and outside the conditional operator. (Contributed by SN, 14-Aug-2024.)
(if(𝜑, 𝐴, 𝐵)𝑅𝐶 ↔ if-(𝜑, 𝐴𝑅𝐶, 𝐵𝑅𝐶))
 
14-Aug-2024symgplusg 18747 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.) (Revised by AV, 29-Mar-2024.) (Proof shortened by AV, 14-Aug-2024.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (𝐴 ↑m 𝐴)    &    + = (+g‘𝐺)    ⇒    + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))
 
14-Aug-2024hashf1lem1 14003 Lemma for hashf1 14006. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 14-Aug-2024.)
(𝜑 → 𝐴 ∈ Fin)    &   (𝜑 → 𝐵 ∈ Fin)    &   (𝜑 → ¬ 𝑧 ∈ 𝐴)    &   (𝜑 → ((♯‘𝐴) + 1) ≤ (♯‘𝐵))    &   (𝜑 → 𝐹:𝐴–1-1→𝐵)    ⇒   (𝜑 → {𝑓 ∣ ((𝑓 ↾ 𝐴) = 𝐹 ∧ 𝑓:(𝐴 ∪ {𝑧})–1-1→𝐵)} ≈ (𝐵 ∖ ran 𝐹))
 
14-Aug-2024f1setex 8527 The set of injections between two classes exists if the codomain exists. (Contributed by AV, 14-Aug-2024.)
(𝐵 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴–1-1→𝐵} ∈ V)
 
14-Aug-2024fsetex 8526 The set of functions between two classes exists if the codomain exists. Generalization of mapex 8503 to arbitrary domains. (Contributed by AV, 14-Aug-2024.)
(𝐵 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)
 
14-Aug-2024indifdir 4189 Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.) (Revised by BTernaryTau, 14-Aug-2024.)
((𝐴 ∖ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∖ (𝐵 ∩ 𝐶))
 
14-Aug-2024indifdi 4188 Distribute intersection over difference. (Contributed by BTernaryTau, 14-Aug-2024.)
(𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ (𝐴 ∩ 𝐶))
 
12-Aug-2024dvrelogpow2b 39766 Derivative of the power of the binary logarithm. (Contributed by metakunt, 12-Aug-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    &   (𝜑 → 𝐴 ≤ 𝐵)    &   𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ ((2 logb 𝑥)↑𝑁))    &   𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (𝐶 · (((log‘𝑥)↑(𝑁 − 1)) / 𝑥)))    &   𝐶 = (𝑁 / ((log‘2)↑𝑁))    &   (𝜑 → 𝑁 ∈ ℕ)    ⇒   (𝜑 → (ℝ D 𝐹) = 𝐺)
 
12-Aug-20240nonelalab 39765 Technical lemma for open interval. (Contributed by metakunt, 12-Aug-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    &   (𝜑 → 𝐴 ≤ 𝐵)    &   (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))    ⇒   (𝜑 → 0 ≠ 𝐶)
 
12-Aug-2024ssfi 8840 A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138. For a shorter proof using ax-pow 5247, see ssfiALT 8841. (Contributed by NM, 24-Jun-1998.) Avoid ax-pow 5247. (Revised by BTernaryTau, 12-Aug-2024.)
((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)
 
11-Aug-2024dvrelog2b 39764 Derivative of the binary logarithm. (Contributed by metakunt, 11-Aug-2024.)
(𝜑 → 𝐴 ∈ ℝ*)    &   (𝜑 → 𝐵 ∈ ℝ*)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 𝐴 ≤ 𝐵)    &   𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (2 logb 𝑥))    &   𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / (𝑥 · (log‘2))))    ⇒   (𝜑 → (ℝ D 𝐹) = 𝐺)
 
11-Aug-2024dvrelog3 39763 The derivative of the logarithm on an open interval. (Contributed by metakunt, 11-Aug-2024.)
(𝜑 → 𝐴 ∈ ℝ*)    &   (𝜑 → 𝐵 ∈ ℝ*)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → 𝐴 ≤ 𝐵)    &   𝐹 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (log‘𝑥))    &   𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))    ⇒   (𝜑 → (ℝ D 𝐹) = 𝐺)
 
11-Aug-2024dvrelog2 39762 The derivative of the logarithm, ftc2 24913 version. (Contributed by metakunt, 11-Aug-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 0 < 𝐴)    &   (𝜑 → 𝐴 ≤ 𝐵)    &   𝐹 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (log‘𝑥))    &   𝐺 = (𝑥 ∈ (𝐴(,)𝐵) ↦ (1 / 𝑥))    ⇒   (𝜑 → (ℝ D 𝐹) = 𝐺)
 
10-Aug-2024oldirr 33766 No surreal is a member of its birthday's old set. (Contributed by Scott Fenton, 10-Aug-2024.)
¬ 𝑋 ∈ ( O ‘( bday ‘𝑋))
 
10-Aug-2024oldbdayim 33765 If 𝑋 is in the old set for 𝐴, then the birthday of 𝑋 is less than 𝐴. (Contributed by Scott Fenton, 10-Aug-2024.)
(𝑋 ∈ ( O ‘𝐴) → ( bday ‘𝑋) ∈ 𝐴)
 
10-Aug-2024madebdayim 33764 If a surreal is a member of a made set, its birthday is less than or equal to the level. (Contributed by Scott Fenton, 10-Aug-2024.)
(𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴)
 
10-Aug-2024scutbdaybnd 33703 An upper bound on the birthday of a surreal cut. (Contributed by Scott Fenton, 10-Aug-2024.)
((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) → ( bday ‘(𝐴 |s 𝐵)) ⊆ 𝑂)
 
10-Aug-2024etasslt 33701 A restatement of noeta 33640 using set less than. (Contributed by Scott Fenton, 10-Aug-2024.)
((𝐴 <<s 𝐵 ∧ 𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂) → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ 𝑂))
 
10-Aug-2024noetalem2 33639 Lemma for noeta 33640. The full statement of the theorem with hypotheses in place. (Contributed by Scott Fenton, 10-Aug-2024.)
𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ((((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ∃𝑐 ∈ No (∀𝑎 ∈ 𝐴 𝑎 <s 𝑐 ∧ ∀𝑏 ∈ 𝐵 𝑐 <s 𝑏 ∧ ( bday ‘𝑐) ⊆ 𝑂))
 
10-Aug-2024mapssfset 8521 The value of the set exponentiation (𝐵 ↑m 𝐴) is a subset of the class of functions from 𝐴 to 𝐵. (Contributed by AV, 10-Aug-2024.)
(𝐵 ↑m 𝐴) ⊆ {𝑓 ∣ 𝑓:𝐴⟶𝐵}
 
10-Aug-2024fmptapd 6975 Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.) (Revised by AV, 10-Aug-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)    &   ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵)    ⇒   (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶))
 
9-Aug-2024noeta 33640 The full-eta axiom for the surreal numbers. This is the single most important property of the surreals. It says that, given two sets of surreals such that one comes completely before the other, there is a surreal lying strictly between the two. Furthermore, if the birthdays of members of 𝐴 and 𝐵 are strictly bounded above by 𝑂, then 𝑂 non-strictly bounds the separator. Axiom FE of [Alling] p. 185. (Contributed by Scott Fenton, 9-Aug-2024.)
((((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ∃𝑧 ∈ No (∀𝑥 ∈ 𝐴 𝑥 <s 𝑧 ∧ ∀𝑦 ∈ 𝐵 𝑧 <s 𝑦 ∧ ( bday ‘𝑧) ⊆ 𝑂))
 
9-Aug-2024noetalem1 33638 Lemma for noeta 33640. Either 𝑆 or 𝑇 satisfies the final condition. (Contributed by Scott Fenton, 9-Aug-2024.)
𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))    &   𝑊 = (𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}))    ⇒   ((((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) ∧ (𝑂 ∈ On ∧ ( bday “ (𝐴 ∪ 𝐵)) ⊆ 𝑂)) → ((𝑆 ∈ No ∧ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑆 ∧ ∀𝑏 ∈ 𝐵 𝑆 <s 𝑏 ∧ ( bday ‘𝑆) ⊆ 𝑂)) ∨ (𝑇 ∈ No ∧ (∀𝑎 ∈ 𝐴 𝑎 <s 𝑇 ∧ ∀𝑏 ∈ 𝐵 𝑇 <s 𝑏 ∧ ( bday ‘𝑇) ⊆ 𝑂))))
 
9-Aug-2024noetainflem4 33637 Lemma for noeta 33640. If 𝐴 precedes 𝐵, then 𝑊 is greater than 𝐴. (Contributed by Scott Fenton, 9-Aug-2024.)
𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   𝑊 = (𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}))    ⇒   (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ∀𝑎 ∈ 𝐴 𝑎 <s 𝑊)
 
9-Aug-2024noetainflem3 33636 Lemma for noeta 33640. 𝑊 bounds 𝐵 below . (Contributed by Scott Fenton, 9-Aug-2024.)
𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   𝑊 = (𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}))    ⇒   (((𝐴 ∈ V ∧ 𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑌 ∈ 𝐵) → 𝑊 <s 𝑌)
 
9-Aug-2024noetainflem2 33635 Lemma for noeta 33640. The restriction of 𝑊 to the domain of 𝑇 is 𝑇. (Contributed by Scott Fenton, 9-Aug-2024.)
𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   𝑊 = (𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}))    ⇒   ((𝐵 ⊆ No ∧ 𝐵 ∈ V) → (𝑊 ↾ dom 𝑇) = 𝑇)
 
9-Aug-2024noetainflem1 33634 Lemma for noeta 33640. Establish that this particular construction gives a surreal. (Contributed by Scott Fenton, 9-Aug-2024.)
𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   𝑊 = (𝑇 ∪ ((suc ∪ ( bday “ 𝐴) ∖ dom 𝑇) × {2o}))    ⇒   ((𝐴 ∈ V ∧ 𝐵 ⊆ No ∧ 𝐵 ∈ V) → 𝑊 ∈ No )
 
9-Aug-2024noetasuplem2 33631 Lemma for noeta 33640. The restriction of 𝑍 to dom 𝑆 is 𝑆. (Contributed by Scott Fenton, 9-Aug-2024.)
𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1o}))    ⇒   ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑍 ↾ dom 𝑆) = 𝑆)
 
9-Aug-2024nosupinfsep 33629 Given two sets of surreals, a surreal 𝑊 separates them iff its restriction to the maximum of dom 𝑆 and dom 𝑇 separates them. Corollary 4.4 of [Lipparini] p. 7. (Contributed by Scott Fenton, 9-Aug-2024.)
𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    &   𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ 𝑊 ∈ No ) → ((∀𝑎 ∈ 𝐴 𝑎 <s 𝑊 ∧ ∀𝑏 ∈ 𝐵 𝑊 <s 𝑏) ↔ (∀𝑎 ∈ 𝐴 𝑎 <s (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) ∧ ∀𝑏 ∈ 𝐵 (𝑊 ↾ (dom 𝑆 ∪ dom 𝑇)) <s 𝑏)))
 
9-Aug-2024noinfbnd2 33628 Bounding law from below for the surreal infimum. Analagous to proposition 4.3 of [Lipparini] p. 6. (Contributed by Scott Fenton, 9-Aug-2024.)
𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) → (∀𝑏 ∈ 𝐵 𝑍 <s 𝑏 ↔ ¬ 𝑇 <s (𝑍 ↾ dom 𝑇)))
 
9-Aug-2024noinfbnd2lem1 33627 Bounding law from below when a set of surreals has a minimum. (Contributed by Scott Fenton, 9-Aug-2024.)
(((𝑈 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑈) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ No ) ∧ ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) → ¬ (𝑈 ∪ {⟨dom 𝑈, 1o⟩}) <s (𝑍 ↾ suc dom 𝑈))
 
9-Aug-2024noinfbnd1 33626 Bounding law from above for the surreal infimum. Analagous to proposition 4.2 of [Lipparini] p. 6. (Contributed by Scott Fenton, 9-Aug-2024.)
𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉 ∧ 𝑈 ∈ 𝐵) → 𝑇 <s (𝑈 ↾ dom 𝑇))
 
9-Aug-2024noinfbnd1lem6 33625 Lemma for noinfbnd1 33626. Establish a hard lower bound when there is no minimum. (Contributed by Scott Fenton, 9-Aug-2024.)
𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → 𝑇 <s (𝑈 ↾ dom 𝑇))
 
9-Aug-2024noinfbnd1lem5 33624 Lemma for noinfbnd1 33626. If 𝑈 is a prolongment of 𝑇 and in 𝐵, then (𝑈‘dom 𝑇) is not 2o. (Contributed by Scott Fenton, 9-Aug-2024.)
𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ 2o)
 
9-Aug-2024noinfbnd1lem4 33623 Lemma for noinfbnd1 33626. If 𝑈 is a prolongment of 𝑇 and in 𝐵, then (𝑈‘dom 𝑇) is not undefined. (Contributed by Scott Fenton, 9-Aug-2024.)
𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ ∅)
 
9-Aug-2024noinfbnd1lem3 33622 Lemma for noinfbnd1 33626. If 𝑈 is a prolongment of 𝑇 and in 𝐵, then (𝑈‘dom 𝑇) is not 1o. (Contributed by Scott Fenton, 9-Aug-2024.)
𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇)) → (𝑈‘dom 𝑇) ≠ 1o)
 
9-Aug-2024noinfbnd1lem2 33621 Lemma for noinfbnd1 33626. When there is no minimum, if any member of 𝐵 is a prolongment of 𝑇, then so are all elements below it. (Contributed by Scott Fenton, 9-Aug-2024.)
𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ ((𝑈 ∈ 𝐵 ∧ (𝑈 ↾ dom 𝑇) = 𝑇) ∧ (𝑊 ∈ 𝐵 ∧ ¬ 𝑈 <s 𝑊))) → (𝑊 ↾ dom 𝑇) = 𝑇)
 
9-Aug-2024noinfbnd1lem1 33620 Lemma for noinfbnd1 33626. Establish a soft lower bound. (Contributed by Scott Fenton, 9-Aug-2024.)
𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ 𝑈 ∈ 𝐵) → ¬ (𝑈 ↾ dom 𝑇) <s 𝑇)
 
9-Aug-2024noinfcbv 33614 Change bound variables for surreal infimum. (Contributed by Scott Fenton, 9-Aug-2024.)
𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   𝑇 = if(∃𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎, ((℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎) ∪ {⟨dom (℩𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ¬ 𝑏 <s 𝑎), 1o⟩}), (𝑐 ∈ {𝑏 ∣ ∃𝑑 ∈ 𝐵 (𝑏 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑏) = (𝑒 ↾ suc 𝑏)))} ↦ (℩𝑎∃𝑑 ∈ 𝐵 (𝑐 ∈ dom 𝑑 ∧ ∀𝑒 ∈ 𝐵 (¬ 𝑑 <s 𝑒 → (𝑑 ↾ suc 𝑐) = (𝑒 ↾ suc 𝑐)) ∧ (𝑑‘𝑐) = 𝑎))))
 
9-Aug-2024nosupcbv 33599 Lemma to change bound variables in a surreal supremum. (Contributed by Scott Fenton, 9-Aug-2024.)
𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   𝑆 = if(∃𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏, ((℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏) ∪ {⟨dom (℩𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎 <s 𝑏), 2o⟩}), (𝑐 ∈ {𝑑 ∣ ∃𝑒 ∈ 𝐴 (𝑑 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑑) = (𝑓 ↾ suc 𝑑)))} ↦ (℩𝑎∃𝑒 ∈ 𝐴 (𝑐 ∈ dom 𝑒 ∧ ∀𝑓 ∈ 𝐴 (¬ 𝑓 <s 𝑒 → (𝑒 ↾ suc 𝑐) = (𝑓 ↾ suc 𝑐)) ∧ (𝑒‘𝑐) = 𝑎))))
 
9-Aug-2024nogt01o 33593 Given 𝐴 greater than 𝐵, equal to 𝐵 up to 𝑋, and 𝐵(𝑋) undefined, then 𝐴(𝑋) = 1o. (Contributed by Scott Fenton, 9-Aug-2024.)
(((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐵‘𝑋) = ∅) → (𝐴‘𝑋) = 1o)
 
9-Aug-2024nogesgn1o 33570 Given 𝐴 greater than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 1o, then 𝐵(𝑋) = 1o. (Contributed by Scott Fenton, 9-Aug-2024.)
(((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 1o) ∧ ¬ 𝐴 <s 𝐵) → (𝐵‘𝑋) = 1o)
 
9-Aug-2024snres0 33362 Condition for restriction of a singleton to be empty. (Contributed by Scott Fenton, 9-Aug-2024.)
𝐵 ∈ V    ⇒   (({⟨𝐴, 𝐵⟩} ↾ 𝐶) = ∅ ↔ ¬ 𝐴 ∈ 𝐶)
 
9-Aug-2024oprabexd 7737 Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by AV, 9-Aug-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ∃*𝑧𝜓)    &   (𝜑 → 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)})    ⇒   (𝜑 → 𝐹 ∈ V)
 
9-Aug-2024opabex3rd 7728 Existence of an ordered pair abstraction if the second components are elements of a set. (Contributed by AV, 17-Sep-2023.) (Revised by AV, 9-Aug-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑦 ∈ 𝐴) → {𝑥 ∣ 𝜓} ∈ V)    ⇒   (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)} ∈ V)
 
9-Aug-2024opabex3d 7727 Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.) (Revised by AV, 9-Aug-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝜓} ∈ V)    ⇒   (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} ∈ V)
 
9-Aug-2024onun2 6306 The union of two ordinals is an ordinal. (Contributed by Scott Fenton, 9-Aug-2024.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)
 
9-Aug-2024resdmss 6087 Subset relationship for the domain of a restriction. (Contributed by Scott Fenton, 9-Aug-2024.)
dom (𝐴 ↾ 𝐵) ⊆ 𝐵
 
8-Aug-2024prjspnvs 40119 A nonzero multiple of a vector is equivalent to the vector. This converts the equivalence relation used in prjspvs 40109 (see prjspnerlem 40116). (Contributed by SN, 8-Aug-2024.)
∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   𝑆 = (Base‘𝐾)    &    · = ( ·𝑠 ‘𝑊)    &    0 = (0g‘𝐾)    &   (𝜑 → 𝐾 ∈ DivRing)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐶 ∈ 𝑆)    &   (𝜑 → 𝐶 ≠ 0 )    ⇒   (𝜑 → (𝐶 · 𝑋) ∼ 𝑋)
 
8-Aug-2024oldlim 33763 The value of the old set at a limit ordinal. (Contributed by Scott Fenton, 8-Aug-2024.)
((Lim 𝐴 ∧ 𝐴 ∈ 𝑉) → ( O ‘𝐴) = ∪ ( O “ 𝐴))
 
8-Aug-2024oldsuc 33762 The value of the old set at a successor. (Contributed by Scott Fenton, 8-Aug-2024.)
(𝐴 ∈ On → ( O ‘suc 𝐴) = (( O ‘𝐴) ∪ ( N ‘𝐴)))
 
8-Aug-2024madeoldsuc 33761 The made set is the old set of its successor. (Contributed by Scott Fenton, 8-Aug-2024.)
(𝐴 ∈ On → ( M ‘𝐴) = ( O ‘suc 𝐴))
 
8-Aug-2024new0 33752 The only surreal new on day ∅ is 0s. (Contributed by Scott Fenton, 8-Aug-2024.)
( N ‘∅) = { 0s }
 
8-Aug-2024bday1s 33719 The birthday of surreal one is ordinal one. (Contributed by Scott Fenton, 8-Aug-2024.)
( bday ‘ 1s ) = 1o
 
8-Aug-2024bday0b 33718 The only surreal with birthday ∅ is 0s. (Contributed by Scott Fenton, 8-Aug-2024.)
(𝑋 ∈ No → (( bday ‘𝑋) = ∅ ↔ 𝑋 = 0s ))
 
8-Aug-2024eqscut2 33694 Condition for equality to a surreal cut. (Contributed by Scott Fenton, 8-Aug-2024.)
((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No ) → ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ∀𝑦 ∈ No ((𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅) → ( bday ‘𝑋) ⊆ ( bday ‘𝑦)))))
 
8-Aug-2024eqscut 33693 Condition for equality to a surreal cut. (Contributed by Scott Fenton, 8-Aug-2024.)
((𝐿 <<s 𝑅 ∧ 𝑋 ∈ No ) → ((𝐿 |s 𝑅) = 𝑋 ↔ (𝐿 <<s {𝑋} ∧ {𝑋} <<s 𝑅 ∧ ( bday ‘𝑋) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐿 <<s {𝑦} ∧ {𝑦} <<s 𝑅)}))))
 
8-Aug-2024noinfres 33619 The restriction of surreal infimum when there is no minimum. (Contributed by Scott Fenton, 8-Aug-2024.)
𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑇 ↾ suc 𝐺) = (𝑈 ↾ suc 𝐺))
 
8-Aug-2024noinffv 33618 The value of surreal infimum when there is no minimum. (Contributed by Scott Fenton, 8-Aug-2024.)
𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ((¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑈 ∈ 𝐵 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑈 <s 𝑣 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑇‘𝐺) = (𝑈‘𝐺))
 
8-Aug-2024noinfbday 33617 Birthday bounding law for surreal infimum. (Contributed by Scott Fenton, 8-Aug-2024.)
𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   (((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) ∧ (𝑂 ∈ On ∧ ( bday “ 𝐵) ⊆ 𝑂)) → ( bday ‘𝑇) ⊆ 𝑂)
 
8-Aug-2024noinfdm 33616 Next, we calculate the domain of 𝑇. This is mostly to change bound variables. (Contributed by Scott Fenton, 8-Aug-2024.)
𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   (¬ ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥 → dom 𝑇 = {𝑧 ∣ ∃𝑝 ∈ 𝐵 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐵 (¬ 𝑝 <s 𝑞 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))})
 
8-Aug-2024noinfno 33615 The next several theorems deal with a surreal "infimum". This surreal will ultimately be shown to bound 𝐵 above and bound the restriction of any surreal below. We begin by showing that the given expression actually defines a surreal number. (Contributed by Scott Fenton, 8-Aug-2024.)
𝑇 = if(∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥, ((℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥) ∪ {⟨dom (℩𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ 𝑦 <s 𝑥), 1o⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐵 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐵 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐵 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥))))    ⇒   ((𝐵 ⊆ No ∧ 𝐵 ∈ 𝑉) → 𝑇 ∈ No )
 
8-Aug-2024noinfprefixmo 33598 In any class of surreals, there is at most one value of the prefix property. (Contributed by Scott Fenton, 8-Aug-2024.)
(𝐴 ⊆ No → ∃*𝑥∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑢 <s 𝑣 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥))
 
8-Aug-2024nominmo 33596 A class of surreals has at most one minimum. (Contributed by Scott Fenton, 8-Aug-2024.)
(𝑆 ⊆ No → ∃*𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ¬ 𝑦 <s 𝑥)
 
8-Aug-2024reurab 33361 Restricted existential uniqueness of a restricted abstraction. (Contributed by Scott Fenton, 8-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (∃!𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓}𝜒 ↔ ∃!𝑥 ∈ 𝐴 (𝜑 ∧ 𝜒))
 
8-Aug-2024riotarab 33360 Restricted iota of a restricted abstraction. (Contributed by Scott Fenton, 8-Aug-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   (℩𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜓}𝜒) = (℩𝑥 ∈ 𝐴 (𝜑 ∧ 𝜒))
 
8-Aug-2024onelssex 33348 Ordinal less than is equivalent to having an ordinal between them. (Contributed by Scott Fenton, 8-Aug-2024.)
((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐶 ↔ ∃𝑏 ∈ 𝐶 𝐴 ⊆ 𝑏))
 
8-Aug-2024fosetex 8528 The set of surjections between two classes exists (without any precondition). (Contributed by AV, 8-Aug-2024.)
{𝑓 ∣ 𝑓:𝐴–onto→𝐵} ∈ V
 
8-Aug-2024mapfset 8520 If 𝐵 is a set, the value of the set exponentiation (𝐵 ↑m 𝐴) is the class of all functions from 𝐴 to 𝐵. Generalisation of mapvalg 8507 (which does not require ax-rep 5168) to arbitrary domains. Note that the class {𝑓 ∣ 𝑓:𝐴⟶𝐵} can only contain set-functions, as opposed to arbitrary class-functions. When 𝐴 is a proper class, there can be no set-functions on it, so the above class is empty (see also fsetdmprc0 8525), hence a set. In this case, both sides of the equality in this theorem are the empty set. (Contributed by AV, 8-Aug-2024.)
(𝐵 ∈ 𝑉 → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = (𝐵 ↑m 𝐴))
 
8-Aug-2024fdmexb 7676 The domain of a function is a set iff the function is a set. (Contributed by AV, 8-Aug-2024.)
(𝐹:𝐴⟶𝐵 → (𝐴 ∈ V ↔ 𝐹 ∈ V))
 
8-Aug-2024fndmexb 7675 The domain of a function is a set iff the function is a set. (Contributed by AV, 8-Aug-2024.)
(𝐹 Fn 𝐴 → (𝐴 ∈ V ↔ 𝐹 ∈ V))
 
8-Aug-2024fnsnfv 6779 Singleton of function value. (Contributed by NM, 22-May-1998.) (Proof shortened by Scott Fenton, 8-Aug-2024.)
((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))
 
8-Aug-2024ssrab2 3983 Subclass relation for a restricted class. (Contributed by NM, 19-Mar-1997.) (Proof shortened by BJ and SN, 8-Aug-2024.)
{𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴
 
7-Aug-2024madecut 33759 Given a section that is a subset of an old set, the cut is a member of the made set. (Contributed by Scott Fenton, 7-Aug-2024.)
(((𝐴 ∈ On ∧ 𝐿 <<s 𝑅) ∧ (𝐿 ⊆ ( O ‘𝐴) ∧ 𝑅 ⊆ ( O ‘𝐴))) → (𝐿 |s 𝑅) ∈ ( M ‘𝐴))
 
7-Aug-2024made0 33751 The only surreal made on day ∅ is 0s. (Contributed by Scott Fenton, 7-Aug-2024.)
( M ‘∅) = { 0s }
 
7-Aug-2024ssltright 33749 A surreal is less than its right options. Theorem 0(i) of [Conway] p. 16. (Contributed by Scott Fenton, 7-Aug-2024.)
(𝐴 ∈ No → {𝐴} <<s ( R ‘𝐴))
 
7-Aug-2024ssltleft 33748 A surreal is greater than its left options. Theorem 0(ii) of [Conway] p. 16. (Contributed by Scott Fenton, 7-Aug-2024.)
(𝐴 ∈ No → ( L ‘𝐴) <<s {𝐴})
 
7-Aug-2024elold 33747 Membership in an old set. (Contributed by Scott Fenton, 7-Aug-2024.)
(𝐴 ∈ On → (𝑋 ∈ ( O ‘𝐴) ↔ ∃𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘𝑏)))
 
7-Aug-2024elmade2 33746 Membership in the made function in terms of the old function. (Contributed by Scott Fenton, 7-Aug-2024.)
(𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ( O ‘𝐴)∃𝑟 ∈ 𝒫 ( O ‘𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
 
7-Aug-2024old0 33737 No surreal is older than ∅. (Contributed by Scott Fenton, 7-Aug-2024.)
( O ‘∅) = ∅
 
7-Aug-20240slt1s 33717 Surreal zero is less than surreal one. Theorem from [Conway] p. 7. (Contributed by Scott Fenton, 7-Aug-2024.)
0s <s 1s
 
7-Aug-2024bday0s 33716 Calculate the birthday of surreal zero. (Contributed by Scott Fenton, 7-Aug-2024.)
( bday ‘ 0s ) = ∅
 
7-Aug-20241sno 33715 Surreal one is a surreal. (Contributed by Scott Fenton, 7-Aug-2024.)
1s ∈ No
 
7-Aug-20240sno 33714 Surreal zero is a surreal. (Contributed by Scott Fenton, 7-Aug-2024.)
0s ∈ No
 
7-Aug-2024df-1s 33713 Define surreal one. This is the simplest number greater than surreal zero. Definition from [Conway] p. 18. (Contributed by Scott Fenton, 7-Aug-2024.)
1s = ({ 0s } |s ∅)
 
7-Aug-2024df-0s 33712 Define surreal zero. This is the simplest cut of surreal number sets. Definition from [Conway] p. 17. (Contributed by Scott Fenton, 7-Aug-2024.)
0s = (∅ |s ∅)
 
7-Aug-2024scutbdaybnd2lim 33705 An upper bound on the birthday of a surreal cut when it is a limit birthday. (Contributed by Scott Fenton, 7-Aug-2024.)
((𝐴 <<s 𝐵 ∧ Lim ( bday ‘(𝐴 |s 𝐵))) → ( bday ‘(𝐴 |s 𝐵)) ⊆ ∪ ( bday “ (𝐴 ∪ 𝐵)))
 
7-Aug-2024slerflex 33660 Surreal less than or equal is reflexive. Theorem 0(iii) of [Conway] p. 16. (Contributed by Scott Fenton, 7-Aug-2024.)
(𝐴 ∈ No → 𝐴 ≤s 𝐴)
 
7-Aug-2024tdeglem4 24929 There is only one multi-index with total degree 0. (Contributed by Stefan O'Rear, 29-Mar-2015.) Remove a sethood antecedent. (Revised by SN, 7-Aug-2024.)
𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ))    ⇒   (𝑋 ∈ 𝐴 → ((𝐻‘𝑋) = 0 ↔ 𝑋 = (𝐼 × {0})))
 
7-Aug-2024tdeglem3 24927 Additivity of the total degree helper function. (Contributed by Stefan O'Rear, 26-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) Remove a sethood antecedent. (Revised by SN, 7-Aug-2024.)
𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ))    ⇒   ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐻‘(𝑋 ∘f + 𝑌)) = ((𝐻‘𝑋) + (𝐻‘𝑌)))
 
7-Aug-2024tdeglem1 24925 Functionality of the total degree helper function. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) Remove sethood antecedent. (Revised by SN, 7-Aug-2024.)
𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin}    &   𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ))    ⇒   𝐻:𝐴⟶ℕ0
 
7-Aug-2024psrbagev2 21009 Closure of a sum using a bag of multipliers. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Proof shortened by AV, 18-Jul-2019.) (Revised by AV, 11-Apr-2024.) Remove a sethood hypothesis. (Revised by SN, 7-Aug-2024.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   𝐶 = (Base‘𝑇)    &    · = (.g‘𝑇)    &   (𝜑 → 𝑇 ∈ CMnd)    &   (𝜑 → 𝐵 ∈ 𝐷)    &   (𝜑 → 𝐺:𝐼⟶𝐶)    ⇒   (𝜑 → (𝑇 Σg (𝐵 ∘f · 𝐺)) ∈ 𝐶)
 
7-Aug-2024psrbagev1 21007 A bag of multipliers provides the conditions for a valid sum. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) (Revised by AV, 11-Apr-2024.) Remove a sethood hypothesis. (Revised by SN, 7-Aug-2024.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   𝐶 = (Base‘𝑇)    &    · = (.g‘𝑇)    &    0 = (0g‘𝑇)    &   (𝜑 → 𝑇 ∈ CMnd)    &   (𝜑 → 𝐵 ∈ 𝐷)    &   (𝜑 → 𝐺:𝐼⟶𝐶)    ⇒   (𝜑 → ((𝐵 ∘f · 𝐺):𝐼⟶𝐶 ∧ (𝐵 ∘f · 𝐺) finSupp 0 ))
 
7-Aug-2024psrass1lem 20874 A group sum commutation used by psrass1 20902. (Contributed by Mario Carneiro, 5-Jan-2015.) Remove a sethood hypothesis. (Revised by SN, 7-Aug-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹}    &   (𝜑 → 𝐹 ∈ 𝐷)    &   𝐵 = (Base‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   ((𝜑 ∧ (𝑗 ∈ 𝑆 ∧ 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)})) → 𝑋 ∈ 𝐵)    &   (𝑘 = (𝑛 ∘f − 𝑗) → 𝑋 = 𝑌)    ⇒   (𝜑 → (𝐺 Σg (𝑛 ∈ 𝑆 ↦ (𝐺 Σg (𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ 𝑛} ↦ 𝑌)))) = (𝐺 Σg (𝑗 ∈ 𝑆 ↦ (𝐺 Σg (𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)} ↦ 𝑋)))))
 
7-Aug-2024psrbagaddcl 20859 The sum of two finite bags is a finite bag. (Contributed by Mario Carneiro, 9-Jan-2015.) Shorten proof and remove a sethood antecedent. (Revised by SN, 7-Aug-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ((𝐹 ∈ 𝐷 ∧ 𝐺 ∈ 𝐷) → (𝐹 ∘f + 𝐺) ∈ 𝐷)
 
7-Aug-2024psrbagfsupp 20851 Finite bags have finite support. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) Remove a sethood antecedent. (Revised by SN, 7-Aug-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   (𝐹 ∈ 𝐷 → 𝐹 finSupp 0)
 
7-Aug-2024hashfacen 14001 The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.) (Proof shortened by AV, 7-Aug-2024.)
((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ≈ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷})
 
7-Aug-2024unfi 8839 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 16-Nov-2002.) Avoid ax-pow 5247. (Revised by BTernaryTau, 7-Aug-2024.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)
 
7-Aug-2024mapfoss 8522 The value of the set exponentiation (𝐵 ↑m 𝐴) is a superset of the set of all functions from 𝐴 onto 𝐵. (Contributed by AV, 7-Aug-2024.)
{𝑓 ∣ 𝑓:𝐴–onto→𝐵} ⊆ (𝐵 ↑m 𝐴)
 
7-Aug-2024rexbi 3157 Distribute restricted quantification over a biconditional. (Contributed by Scott Fenton, 7-Aug-2024.)
(∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓))
 
6-Aug-2024lltropt 33750 The left options of a surreal are strictly less than the right options of the same surreal. (Contributed by Scott Fenton, 6-Aug-2024.)
(𝐴 ∈ No → ( L ‘𝐴) <<s ( R ‘𝐴))
 
6-Aug-2024elmade 33745 Membership in the made function. (Contributed by Scott Fenton, 6-Aug-2024.)
(𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) ↔ ∃𝑙 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑟 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑋)))
 
6-Aug-2024rightf 33744 The functionality of the right options function. (Contributed by Scott Fenton, 6-Aug-2024.)
R : No ⟶𝒫 No
 
6-Aug-2024leftf 33743 The functionality of the left options function. (Contributed by Scott Fenton, 6-Aug-2024.)
L : No ⟶𝒫 No
 
6-Aug-2024newf 33736 The new function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.)
N :On⟶𝒫 No
 
6-Aug-2024oldf 33735 The older function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.)
O :On⟶𝒫 No
 
6-Aug-2024madef 33734 The made function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.)
M :On⟶𝒫 No
 
6-Aug-2024oldval 33732 The value of the old options function. (Contributed by Scott Fenton, 6-Aug-2024.)
(𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴))
 
6-Aug-2024df-right 33729 Define the right options of a surreal. This is the set of surreals that are simpler and greater than the given surreal. (Contributed by Scott Fenton, 6-Aug-2024.)
R = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑥 <s 𝑦})
 
6-Aug-2024df-left 33728 Define the left options of a surreal. This is the set of surreals that are simpler and less than the given surreal. (Contributed by Scott Fenton, 6-Aug-2024.)
L = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ 𝑦 <s 𝑥})
 
6-Aug-2024gsumbagdiag 20873 Two-dimensional commutation of a group sum over a "triangular" region. fsum0diag 15322 analogue for finite bags. (Contributed by Mario Carneiro, 5-Jan-2015.) Remove a sethood hypothesis. (Revised by SN, 6-Aug-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹}    &   (𝜑 → 𝐹 ∈ 𝐷)    &   𝐵 = (Base‘𝐺)    &   (𝜑 → 𝐺 ∈ CMnd)    &   ((𝜑 ∧ (𝑗 ∈ 𝑆 ∧ 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)})) → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝐺 Σg (𝑗 ∈ 𝑆, 𝑘 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑗)} ↦ 𝑋)) = (𝐺 Σg (𝑘 ∈ 𝑆, 𝑗 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑘)} ↦ 𝑋)))
 
6-Aug-2024gsumbagdiaglem 20872 Lemma for gsumbagdiag 20873. (Contributed by Mario Carneiro, 5-Jan-2015.) Remove a sethood hypothesis. (Revised by SN, 6-Aug-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹}    &   (𝜑 → 𝐹 ∈ 𝐷)    ⇒   ((𝜑 ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑋)})) → (𝑌 ∈ 𝑆 ∧ 𝑋 ∈ {𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ (𝐹 ∘f − 𝑌)}))
 
6-Aug-2024psrbagconf1o 20867 Bag complementation is a bijection on the set of bags dominated by a given bag 𝐹. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 6-Aug-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹}    ⇒   (𝐹 ∈ 𝐷 → (𝑥 ∈ 𝑆 ↦ (𝐹 ∘f − 𝑥)):𝑆–1-1-onto→𝑆)
 
6-Aug-2024psrbagconcl 20865 The complement of a bag is a bag. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 6-Aug-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   𝑆 = {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹}    ⇒   ((𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆) → (𝐹 ∘f − 𝑋) ∈ 𝑆)
 
5-Aug-2024psrbaglefi 20863 There are finitely many bags dominated by a given bag. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 25-Jan-2015.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   (𝐹 ∈ 𝐷 → {𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹} ∈ Fin)
 
5-Aug-2024psrbagcon 20861 The analogue of the statement "0 ≤ 𝐺 ≤ 𝐹 implies 0 ≤ 𝐹 − 𝐺 ≤ 𝐹 " for finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → ((𝐹 ∘f − 𝐺) ∈ 𝐷 ∧ (𝐹 ∘f − 𝐺) ∘r ≤ 𝐹))
 
5-Aug-2024psrbaglecl 20857 The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 𝐺 ∈ 𝐷)
 
5-Aug-2024psrbaglesupp 20855 The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ))
 
5-Aug-2024frnnn0fsuppg 12132 Version of frnnn0fsupp 12130 avoiding ax-rep 5168 by assuming 𝐹 is a set rather than its domain 𝐼. (Contributed by SN, 5-Aug-2024.)
((𝐹 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 finSupp 0 ↔ (◡𝐹 “ ℕ) ∈ Fin))
 
5-Aug-2024frnnn0suppg 12131 Version of frnnn0supp 12129 avoiding ax-rep 5168 by assuming 𝐹 is a set rather than its domain 𝐼. (Contributed by SN, 5-Aug-2024.)
((𝐹 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
 
5-Aug-2024suppss 7925 Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.) (Proof shortened by SN, 5-Aug-2024.)
(𝜑 → 𝐹:𝐴⟶𝐵)    &   ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍)    ⇒   (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
 
5-Aug-2024elsuppfng 7901 An element of the support of a function with a given domain. This version of elsuppfn 7902 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-rep 5168. (Contributed by SN, 5-Aug-2024.)
((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑆 ∈ (𝐹 supp 𝑍) ↔ (𝑆 ∈ 𝑋 ∧ (𝐹‘𝑆) ≠ 𝑍)))
 
5-Aug-2024suppvalfng 7899 The value of the operation constructing the support of a function with a given domain. This version of suppvalfn 7900 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-rep 5168. (Contributed by SN, 5-Aug-2024.)
((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍})
 
5-Aug-2024ofrfvalg 7465 Value of a relation applied to two functions. Originally part of ofrfval 7467, this version assumes the functions are sets rather than their domains, avoiding ax-rep 5168. (Contributed by SN, 5-Aug-2024.)
(𝜑 → 𝐹 Fn 𝐴)    &   (𝜑 → 𝐺 Fn 𝐵)    &   (𝜑 → 𝐹 ∈ 𝑉)    &   (𝜑 → 𝐺 ∈ 𝑊)    &   (𝐴 ∩ 𝐵) = 𝑆    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)    &   ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷)    ⇒   (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷))
 
5-Aug-2024elrabi 3589 Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.) Remove disjoint variable condition on 𝐴, 𝑥 and avoid ax-10 2141, ax-11 2158, ax-12 2175. (Revised by SN, 5-Aug-2024.)
(𝐴 ∈ {𝑥 ∈ 𝑉 ∣ 𝜑} → 𝐴 ∈ 𝑉)
 
4-Aug-2024nsgqusf1o 31287 The canonical projection homomorphism 𝐸 defines a bijective correspondence between the set 𝑆 of subgroups of 𝐺 containing a normal subgroup 𝑁 and the subgroups of the quotient group 𝐺 / 𝑁. This theorem is sometimes called the correspondence theorem, or the fourth isomorphism theorem. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}    &   𝑇 = (SubGrp‘𝑄)    &    ≤ = (le‘(toInc‘𝑆))    &    ≲ = (le‘(toInc‘𝑇))    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    ⊕ = (LSSum‘𝐺)    &   𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))    &   𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})    &   (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   (𝜑 → 𝐸:𝑆–1-1-onto→𝑇)
 
4-Aug-2024nsgqusf1olem3 31286 Lemma for nsgqusf1o 31287. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}    &   𝑇 = (SubGrp‘𝑄)    &    ≤ = (le‘(toInc‘𝑆))    &    ≲ = (le‘(toInc‘𝑇))    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    ⊕ = (LSSum‘𝐺)    &   𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))    &   𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})    &   (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   (𝜑 → ran 𝐹 = 𝑆)
 
4-Aug-2024nsgqusf1olem2 31285 Lemma for nsgqusf1o 31287. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}    &   𝑇 = (SubGrp‘𝑄)    &    ≤ = (le‘(toInc‘𝑆))    &    ≲ = (le‘(toInc‘𝑇))    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    ⊕ = (LSSum‘𝐺)    &   𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))    &   𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})    &   (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   (𝜑 → ran 𝐸 = 𝑇)
 
4-Aug-2024nsgqusf1olem1 31284 Lemma for nsgqusf1o 31287. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}    &   𝑇 = (SubGrp‘𝑄)    &    ≤ = (le‘(toInc‘𝑆))    &    ≲ = (le‘(toInc‘𝑇))    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    ⊕ = (LSSum‘𝐺)    &   𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))    &   𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})    &   (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇)
 
4-Aug-2024en3d 8654 Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by AV, 4-Aug-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵))    &   (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴))    &   (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)))    ⇒   (𝜑 → 𝐴 ≈ 𝐵)
 
4-Aug-2024en2d 8653 Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by AV, 4-Aug-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝑋))    &   (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝑌))    &   (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)))    ⇒   (𝜑 → 𝐴 ≈ 𝐵)
 
3-Aug-2024qliftval 8477 The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)
𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   (𝜑 → 𝑅 Er 𝑋)    &   (𝜑 → 𝑋 ∈ 𝑉)    &   (𝑥 = 𝐶 → 𝐴 = 𝐵)    &   (𝜑 → Fun 𝐹)    ⇒   ((𝜑 ∧ 𝐶 ∈ 𝑋) → (𝐹‘[𝐶]𝑅) = 𝐵)
 
3-Aug-2024qliftf 8476 The domain and range of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)
𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   (𝜑 → 𝑅 Er 𝑋)    &   (𝜑 → 𝑋 ∈ 𝑉)    ⇒   (𝜑 → (Fun 𝐹 ↔ 𝐹:(𝑋 / 𝑅)⟶𝑌))
 
3-Aug-2024qliftfuns 8475 The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)
𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   (𝜑 → 𝑅 Er 𝑋)    &   (𝜑 → 𝑋 ∈ 𝑉)    ⇒   (𝜑 → (Fun 𝐹 ↔ ∀𝑦∀𝑧(𝑦𝑅𝑧 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)))
 
3-Aug-2024qliftfund 8474 The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)
𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   (𝜑 → 𝑅 Er 𝑋)    &   (𝜑 → 𝑋 ∈ 𝑉)    &   (𝑥 = 𝑦 → 𝐴 = 𝐵)    &   ((𝜑 ∧ 𝑥𝑅𝑦) → 𝐴 = 𝐵)    ⇒   (𝜑 → Fun 𝐹)
 
3-Aug-2024qliftfun 8473 The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)
𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   (𝜑 → 𝑅 Er 𝑋)    &   (𝜑 → 𝑋 ∈ 𝑉)    &   (𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   (𝜑 → (Fun 𝐹 ↔ ∀𝑥∀𝑦(𝑥𝑅𝑦 → 𝐴 = 𝐵)))
 
3-Aug-2024qliftel1 8472 Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)
𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   (𝜑 → 𝑅 Er 𝑋)    &   (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅𝐹𝐴)
 
3-Aug-2024qliftel 8471 Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)
𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   (𝜑 → 𝑅 Er 𝑋)    &   (𝜑 → 𝑋 ∈ 𝑉)    ⇒   (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶𝑅𝑥 ∧ 𝐷 = 𝐴)))
 
3-Aug-2024qliftrel 8470 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)
𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   (𝜑 → 𝑅 Er 𝑋)    &   (𝜑 → 𝑋 ∈ 𝑉)    ⇒   (𝜑 → 𝐹 ⊆ ((𝑋 / 𝑅) × 𝑌))
 
3-Aug-2024qliftlem 8469 Lemma for theorems about a function lift. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)
𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)    &   ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)    &   (𝜑 → 𝑅 Er 𝑋)    &   (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
 
3-Aug-2024fvprc 6698 A function's value at a proper class is the empty set. See fvprcALT 6699 for a proof that uses ax-pow 5247 instead of ax-sep 5181 and ax-pr 5311. (Contributed by NM, 20-May-1998.) Avoid ax-pow 5247. (Revised by BTernaryTau, 3-Aug-2024.)
(¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
 
1-Aug-2024dfopif 4770 Rewrite df-op 4538 using if. When both arguments are sets, it reduces to the standard Kuratowski definition; otherwise, it is defined to be the empty set. Avoid directly depending on this detail so that theorems will not depend on the Kuratowski construction. (Contributed by Mario Carneiro, 26-Apr-2015.) Avoid ax-10 2141, ax-11 2158, ax-12 2175. (Revised by SN, 1-Aug-2024.) (Avoid depending on this detail.)
⟨𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
 
31-Jul-2024pssnn 8835 A proper subset of a natural number is equinumerous to some smaller number. Lemma 6F of [Enderton] p. 137. (Contributed by NM, 22-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.) Avoid ax-pow 5247. (Revised by BTernaryTau, 31-Jul-2024.)
((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)
 
31-Jul-2024enrefnn 8713 Equinumerosity is reflexive for finite ordinals, proved without using the Axiom of Power Sets (unlike enrefg 8649). (Contributed by BTernaryTau, 31-Jul-2024.)
(𝐴 ∈ ω → 𝐴 ≈ 𝐴)
 
31-Jul-2024en2snOLD 8708 Obsolete version of en2sn 8707 as of 25-Sep-2024. (Contributed by NM, 9-Nov-2003.) Avoid ax-pow 5247. (Revised by BTernaryTau, 31-Jul-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵})
 
31-Jul-2024en0OLD 8681 Obsolete version of en0 8680 as of 23-Sep-2024. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5247. (Revised by BTernaryTau, 31-Jul-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ≈ ∅ ↔ 𝐴 = ∅)
 
30-Jul-2024mhphflem 39946 Lemma for mhphf 39947. Add several multiples of 𝐿 together, in a case where the total amount of multiplies is 𝑁. (Contributed by SN, 30-Jul-2024.)
𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   𝐻 = {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}    &   𝐵 = (Base‘𝐺)    &    · = (.g‘𝐺)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝐺 ∈ Mnd)    &   (𝜑 → 𝐿 ∈ 𝐵)    &   (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ((𝜑 ∧ 𝑎 ∈ 𝐻) → (𝐺 Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) · 𝐿))) = (𝑁 · 𝐿))
 
30-Jul-2024pwsgprod 39933 Finite products in a power structure are taken componentwise. Compare pwsgsum 19339. (Contributed by SN, 30-Jul-2024.)
𝑌 = (𝑅 ↑s 𝐼)    &   𝐵 = (Base‘𝑅)    &    1 = (1r‘𝑌)    &   𝑀 = (mulGrp‘𝑌)    &   𝑇 = (mulGrp‘𝑅)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝐽 ∈ 𝑊)    &   (𝜑 → 𝑅 ∈ CRing)    &   ((𝜑 ∧ (𝑥 ∈ 𝐼 ∧ 𝑦 ∈ 𝐽)) → 𝑈 ∈ 𝐵)    &   (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈)) finSupp 1 )    ⇒   (𝜑 → (𝑀 Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈))) = (𝑥 ∈ 𝐼 ↦ (𝑇 Σg (𝑦 ∈ 𝐽 ↦ 𝑈))))
 
30-Jul-2024pwsexpg 39932 Value of a group exponentiation in a structure power. Compare pwsmulg 18508. (Contributed by SN, 30-Jul-2024.)
𝑌 = (𝑅 ↑s 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝑀 = (mulGrp‘𝑌)    &   𝑇 = (mulGrp‘𝑅)    &    ∙ = (.g‘𝑀)    &    · = (.g‘𝑇)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝐴 ∈ 𝐼)    ⇒   (𝜑 → ((𝑁 ∙ 𝑋)‘𝐴) = (𝑁 · (𝑋‘𝐴)))
 
30-Jul-2024pwspjmhmmgpd 39931 The projection given by pwspjmhm 18228 is also a monoid homomorphism between the respective multiplicative groups. (Contributed by SN, 30-Jul-2024.)
𝑌 = (𝑅 ↑s 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝑀 = (mulGrp‘𝑌)    &   𝑇 = (mulGrp‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ 𝐼)    ⇒   (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) ∈ (𝑀 MndHom 𝑇))
 
30-Jul-2024bj-stabpeirce 34428 This minimal implicational calculus tautology is used in the following argument: When 𝜑, 𝜓, 𝜒, 𝜃, 𝜏 are replaced respectively by (𝜑 → ⊥), ⊥, 𝜑, ⊥, ⊥, the antecedent becomes ¬ ¬ (¬ ¬ 𝜑 → 𝜑), that is, the double negation of the stability of 𝜑. If that statement were provable in minimal calculus, then, since ⊥ plays no particular role in minimal calculus, also the statement with 𝜓 in place of ⊥ would be provable. The corresponding consequent is (((𝜓 → 𝜑) → 𝜓) → 𝜓), that is, the non-intuitionistic Peirce law. Therefore, the double negation of the stability of any formula is not provable in minimal calculus. However, it is provable both in intuitionistic calculus (see iset.mm/bj-nnst) and in classical refutability calculus (see bj-peircestab 34427). (Contributed by BJ, 30-Nov-2023.) (Revised by BJ, 30-Jul-2024.) (Proof modification is discouraged.)
(((((𝜑 → 𝜓) → 𝜒) → 𝜃) → 𝜏) → (((𝜓 → 𝜒) → 𝜃) → 𝜏))
 
30-Jul-2024bj-nnclavci 34423 Inference associated with bj-nnclavc 34422. Its associated inference is an instance of syl 17. Notice the non-intuitionistic proof from peirce 205 and syl 17. (Contributed by BJ, 30-Jul-2024.)
(𝜑 → 𝜓)    ⇒   (((𝜑 → 𝜓) → 𝜑) → 𝜓)
 
30-Jul-2024bj-nnclavc 34422 Commuted form of bj-nnclav 34420. Notice the non-intuitionistic proof from bj-peircei 34440 and imim1i 63. (Contributed by BJ, 30-Jul-2024.) A proof which is shorter when compressed uses embantd 59. (Proof modification is discouraged.)
((𝜑 → 𝜓) → (((𝜑 → 𝜓) → 𝜑) → 𝜓))
 
30-Jul-2024bj-nnclavi 34421 Inference associated with bj-nnclav 34420. Its associated inference is an instance of syl 17. Notice the non-intuitionistic proof from bj-peircei 34440 and bj-poni 34419. (Contributed by BJ, 30-Jul-2024.)
((𝜑 → 𝜓) → 𝜑)    ⇒   ((𝜑 → 𝜓) → 𝜓)
 
30-Jul-2024bj-poni 34419 Inference associated with "pon", pm2.27 42. Its associated inference is ax-mp 5. (Contributed by BJ, 30-Jul-2024.)
𝜑    ⇒   ((𝜑 → 𝜓) → 𝜓)
 
30-Jul-2024psrbagf 20849 A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 30-Jul-2024.)
𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}    ⇒   (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0)
 
30-Jul-2024frnsuppeqg 7907 Version of frnsuppeq 7906 avoiding ax-rep 5168 by assuming 𝐹 is a set rather than its domain 𝐼. (Contributed by SN, 30-Jul-2024.)
((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍}))))
 
29-Jul-2024evlsbagval 39937 Polynomial evaluation builder for a bag of variables. EDITORIAL: This theorem should stay in my mathbox until there's another use, since 0 and 1 using 𝑈 instead of 𝑆 is convenient for its sole use case mhphf 39947, but may not be convenient for other uses. (Contributed by SN, 29-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑃 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝑊 = (Base‘𝑃)    &   𝐾 = (Base‘𝑆)    &   𝑀 = (mulGrp‘𝑆)    &    ↑ = (.g‘𝑀)    &    0 = (0g‘𝑈)    &    1 = (1r‘𝑈)    &   𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   𝐹 = (𝑠 ∈ 𝐷 ↦ if(𝑠 = 𝐵, 1 , 0 ))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    &   (𝜑 → 𝐵 ∈ 𝐷)    ⇒   (𝜑 → (𝐹 ∈ 𝑊 ∧ ((𝑄‘𝐹)‘𝐴) = (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣))))))
 
29-Jul-2024pwselbasr 39930 The reverse direction of pwselbasb 16965: a function between the index and base set of a structure is an element of the structure power. (Contributed by SN, 29-Jul-2024.)
𝑌 = (𝑅 ↑s 𝐼)    &   𝐵 = (Base‘𝑅)    &   𝑉 = (Base‘𝑌)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   (𝜑 → 𝐼 ∈ 𝑍)    &   (𝜑 → 𝑋:𝐼⟶𝐵)    ⇒   (𝜑 → 𝑋 ∈ 𝑉)
 
29-Jul-2024ringcld 39905 Closure of the multiplication operation of a ring. (Contributed by SN, 29-Jul-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 · 𝑌) ∈ 𝐵)
 
29-Jul-2024isfsuppd 39882 Deduction form of isfsupp 8978. (Contributed by SN, 29-Jul-2024.)
(𝜑 → 𝑅 ∈ 𝑉)    &   (𝜑 → 𝑍 ∈ 𝑊)    &   (𝜑 → Fun 𝑅)    &   (𝜑 → (𝑅 supp 𝑍) ∈ Fin)    ⇒   (𝜑 → 𝑅 finSupp 𝑍)
 
29-Jul-2024elmapdd 39881 Deduction associated with elmapd 8511. (Contributed by SN, 29-Jul-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐶:𝐵⟶𝐴)    ⇒   (𝜑 → 𝐶 ∈ (𝐴 ↑m 𝐵))
 
29-Jul-2024grpcld 18350 Closure of the operation of a group. (Contributed by SN, 29-Jul-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &   (𝜑 → 𝐺 ∈ Grp)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)
 
28-Jul-2024mhphf 39947 A homogeneous polynomial defines a homogeneous function. Equivalently, an algebraic form is a homogeneous function. (An algebraic form is the function corresponding to a homogeneous polynomial, which in this case is the (𝑄‘𝑋) which corresponds to 𝑋). (Contributed by SN, 28-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝐻 = (𝐼 mHomP 𝑈)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝐾 = (Base‘𝑆)    &    · = (.r‘𝑆)    &    ↑ = (.g‘(mulGrp‘𝑆))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝐿 ∈ 𝑅)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    &   (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   (𝜑 → ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
 
28-Jul-2024mhpind 39945 The homogeneous polynomials of degree 𝑁 are generated by the terms of degree 𝑁 and addition. (Contributed by SN, 28-Jul-2024.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g‘𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &    + = (+g‘𝑃)    &   𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   𝑆 = {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ Grp)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    &   (𝜑 → (𝐷 × { 0 }) ∈ 𝐺)    &   ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → (𝑠 ∈ 𝐷 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∈ 𝐺)    &   ((𝜑 ∧ (𝑥 ∈ ((𝐻‘𝑁) ∩ 𝐺) ∧ 𝑦 ∈ ((𝐻‘𝑁) ∩ 𝐺))) → (𝑥 + 𝑦) ∈ 𝐺)    ⇒   (𝜑 → 𝑋 ∈ 𝐺)
 
28-Jul-2024ordtypelem9 9131 Lemma for ordtype 9137. Either the function OrdIso is an isomorphism onto all of 𝐴, or OrdIso is not a set, which by oif 9135 implies that either ran 𝑂 ⊆ 𝐴 is a proper class or dom 𝑂 = On. (Contributed by Mario Carneiro, 25-Jun-2015.) (Revised by AV, 28-Jul-2024.)
𝐹 = recs(𝐺)    &   𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤}    &   𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣))    &   𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}    &   𝑂 = OrdIso(𝑅, 𝐴)    &   (𝜑 → 𝑅 We 𝐴)    &   (𝜑 → 𝑅 Se 𝐴)    &   (𝜑 → 𝑂 ∈ 𝑉)    ⇒   (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
 
28-Jul-2024mapfien 9013 A bijection of the base sets induces a bijection on the set of finitely supported functions. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.)
𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊}    &   𝑊 = (𝐺‘𝑍)    &   (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)    &   (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)    &   (𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑋)    &   (𝜑 → 𝐷 ∈ 𝑌)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   (𝜑 → (𝑓 ∈ 𝑆 ↦ (𝐺 ∘ (𝑓 ∘ 𝐹))):𝑆–1-1-onto→𝑇)
 
28-Jul-2024mapfienlem3 9012 Lemma 3 for mapfien 9013. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.)
𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊}    &   𝑊 = (𝐺‘𝑍)    &   (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)    &   (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)    &   (𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑋)    &   (𝜑 → 𝐷 ∈ 𝑌)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) ∈ 𝑆)
 
28-Jul-2024mapfienlem2 9011 Lemma 2 for mapfien 9013. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.)
𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊}    &   𝑊 = (𝐺‘𝑍)    &   (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)    &   (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)    &   (𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑋)    &   (𝜑 → 𝐷 ∈ 𝑌)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((◡𝐺 ∘ 𝑔) ∘ ◡𝐹) finSupp 𝑍)
 
28-Jul-2024mapfienlem1 9010 Lemma 1 for mapfien 9013. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.)
𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊}    &   𝑊 = (𝐺‘𝑍)    &   (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)    &   (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)    &   (𝜑 → 𝐴 ∈ 𝑈)    &   (𝜑 → 𝐵 ∈ 𝑉)    &   (𝜑 → 𝐶 ∈ 𝑋)    &   (𝜑 → 𝐷 ∈ 𝑌)    &   (𝜑 → 𝑍 ∈ 𝐵)    ⇒   ((𝜑 ∧ 𝑓 ∈ 𝑆) → (𝐺 ∘ (𝑓 ∘ 𝐹)) finSupp 𝑊)
 
27-Jul-2024evlsmulval 39940 Polynomial evaluation builder for multiplication. (Contributed by SN, 27-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑃 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → 𝐼 ∈ 𝑍)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    &   (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉))    &   (𝜑 → (𝑁 ∈ 𝐵 ∧ ((𝑄‘𝑁)‘𝐴) = 𝑊))    &    ∙ = (.r‘𝑃)    &    · = (.r‘𝑆)    ⇒   (𝜑 → ((𝑀 ∙ 𝑁) ∈ 𝐵 ∧ ((𝑄‘(𝑀 ∙ 𝑁))‘𝐴) = (𝑉 · 𝑊)))
 
27-Jul-2024evlsaddval 39939 Polynomial evaluation builder for addition. (Contributed by SN, 27-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑃 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → 𝐼 ∈ 𝑍)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    &   (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉))    &   (𝜑 → (𝑁 ∈ 𝐵 ∧ ((𝑄‘𝑁)‘𝐴) = 𝑊))    &    ✚ = (+g‘𝑃)    &    + = (+g‘𝑆)    ⇒   (𝜑 → ((𝑀 ✚ 𝑁) ∈ 𝐵 ∧ ((𝑄‘(𝑀 ✚ 𝑁))‘𝐴) = (𝑉 + 𝑊)))
 
27-Jul-2024evlsexpval 39938 Polynomial evaluation builder for exponentiation. (Contributed by SN, 27-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑃 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → 𝐼 ∈ 𝑍)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    &   (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉))    &    ∙ = (.g‘(mulGrp‘𝑃))    &    ↑ = (.g‘(mulGrp‘𝑆))    &   (𝜑 → 𝑁 ∈ ℕ0)    ⇒   (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝐵 ∧ ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = (𝑁 ↑ 𝑉)))
 
27-Jul-2024evlsvarval 39936 Polynomial evaluation builder for a variable. (Contributed by SN, 27-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑃 = (𝐼 mPoly 𝑈)    &   𝑉 = (𝐼 mVar 𝑈)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑃)    &   (𝜑 → 𝐼 ∈ 𝑊)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝑋 ∈ 𝐼)    &   (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   (𝜑 → ((𝑉‘𝑋) ∈ 𝐵 ∧ ((𝑄‘(𝑉‘𝑋))‘𝐴) = (𝐴‘𝑋)))
 
27-Jul-2024evlsscaval 39935 Polynomial evaluation builder for a scalar. Compare evl1scad 21223. Note that scalar multiplication by 𝑋 is the same as vector multiplication by (𝐴‘𝑋) by asclmul1 20817. (Contributed by SN, 27-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑃 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (algSc‘𝑃)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   (𝜑 → 𝑋 ∈ 𝑅)    &   (𝜑 → 𝐿 ∈ (𝐾 ↑m 𝐼))    ⇒   (𝜑 → ((𝐴‘𝑋) ∈ 𝐵 ∧ ((𝑄‘(𝐴‘𝑋))‘𝐿) = 𝑋))
 
27-Jul-2024ismhmd 39903 Deduction version of ismhm 18192. (Contributed by SN, 27-Jul-2024.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)    &    + = (+g‘𝑆)    &    ⨣ = (+g‘𝑇)    &    0 = (0g‘𝑆)    &   𝑍 = (0g‘𝑇)    &   (𝜑 → 𝑆 ∈ Mnd)    &   (𝜑 → 𝑇 ∈ Mnd)    &   (𝜑 → 𝐹:𝐵⟶𝐶)    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))    &   (𝜑 → (𝐹‘ 0 ) = 𝑍)    ⇒   (𝜑 → 𝐹 ∈ (𝑆 MndHom 𝑇))
 
27-Jul-2024nsgmgc 31283 There is a monotone Galois connection between the lattice of subgroups of a group 𝐺 containing a normal subgroup 𝑁 and the lattice of subgroups of the quotient group 𝐺 / 𝑁. This is sometimes called the lattice theorem. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}    &   𝑇 = (SubGrp‘𝑄)    &   𝐽 = (𝑉MGalConn𝑊)    &   𝑉 = (toInc‘𝑆)    &   𝑊 = (toInc‘𝑇)    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    ⊕ = (LSSum‘𝐺)    &   𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))    &   𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})    &   (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   (𝜑 → 𝐸𝐽𝐹)
 
27-Jul-2024nsgmgclem 31282 Lemma for nsgmgc 31283. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    ⊕ = (LSSum‘𝐺)    &   (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    &   (𝜑 → 𝐹 ∈ (SubGrp‘𝑄))    ⇒   (𝜑 → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝐹} ∈ (SubGrp‘𝐺))
 
27-Jul-2024nsgqus0 31281 A normal subgroup 𝑁 is a member of all subgroups 𝐹 of the quotient group by 𝑁. In fact, it is the identity element of the quotient group. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    ⇒   ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ 𝐹)
 
27-Jul-2024qusima 31280 The image of a subgroup by the natural map from elements to their cosets. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    ⊕ = (LSSum‘𝐺)    &   𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))    &   𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁))    &   (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    &   (𝜑 → 𝐻 ∈ 𝑆)    &   (𝜑 → 𝑆 ⊆ (SubGrp‘𝐺))    ⇒   (𝜑 → (𝐸‘𝐻) = (𝐹 “ 𝐻))
 
27-Jul-2024quslsm 31279 Express the image by the quotient map in terms of direct sum. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &    ⊕ = (LSSum‘𝐺)    &   (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → [𝑋](𝐺 ~QG 𝑆) = ({𝑋} ⊕ 𝑆))
 
27-Jul-2024grplsmid 31278 The direct sum of an element 𝑋 of a subgroup 𝐴 is the subgroup itself. (Contributed by Thierry Arnoux, 27-Jul-2024.)
⊕ = (LSSum‘𝐺)    ⇒   ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝐴) → ({𝑋} ⊕ 𝐴) = 𝐴)
 
27-Jul-2024grplsm0l 31277 Sumset with the identity singleton is the original set. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &    ⊕ = (LSSum‘𝐺)    &    0 = (0g‘𝐺)    ⇒   ((𝐺 ∈ Grp ∧ 𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ({ 0 } ⊕ 𝐴) = 𝐴)
 
26-Jul-2024evlsval3 39934 Give a formula for the polynomial evaluation homomorphism. (Contributed by SN, 26-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑃 = (𝐼 mPoly 𝑈)    &   𝐵 = (Base‘𝑃)    &   𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   𝐾 = (Base‘𝑆)    &   𝑈 = (𝑆 ↾s 𝑅)    &   𝑇 = (𝑆 ↑s (𝐾 ↑m 𝐼))    &   𝑀 = (mulGrp‘𝑇)    &    ↑ = (.g‘𝑀)    &    · = (.r‘𝑇)    &   𝐸 = (𝑝 ∈ 𝐵 ↦ (𝑇 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑀 Σg (𝑏 ∘f ↑ 𝐺))))))    &   𝐹 = (𝑥 ∈ 𝑅 ↦ ((𝐾 ↑m 𝐼) × {𝑥}))    &   𝐺 = (𝑥 ∈ 𝐼 ↦ (𝑎 ∈ (𝐾 ↑m 𝐼) ↦ (𝑎‘𝑥)))    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑆 ∈ CRing)    &   (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    ⇒   (𝜑 → 𝑄 = 𝐸)
 
26-Jul-2024fvmptd4 39875 Deduction version of fvmpt 6807 (where the substitution hypothesis does not have the antecedent 𝜑). (Contributed by SN, 26-Jul-2024.)
(𝑥 = 𝐴 → 𝐵 = 𝐶)    &   (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵))    &   (𝜑 → 𝐴 ∈ 𝐷)    &   (𝜑 → 𝐶 ∈ 𝑉)    ⇒   (𝜑 → (𝐹‘𝐴) = 𝐶)
 
26-Jul-2024mgcf1o 30972 Given a Galois connection, exhibit an order isomorphism. (Contributed by Thierry Arnoux, 26-Jul-2024.)
𝐻 = (𝑉MGalConn𝑊)    &   𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    ≤ = (le‘𝑉)    &    ≲ = (le‘𝑊)    &   (𝜑 → 𝑉 ∈ Poset)    &   (𝜑 → 𝑊 ∈ Poset)    &   (𝜑 → 𝐹𝐻𝐺)    ⇒   (𝜑 → (𝐹 ↾ ran 𝐺) Isom ≤ , ≲ (ran 𝐺, ran 𝐹))
 
26-Jul-2024mgcf1olem2 30971 Property of a Galois connection, lemma for mgcf1o 30972. (Contributed by Thierry Arnoux, 26-Jul-2024.)
𝐻 = (𝑉MGalConn𝑊)    &   𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    ≤ = (le‘𝑉)    &    ≲ = (le‘𝑊)    &   (𝜑 → 𝑉 ∈ Poset)    &   (𝜑 → 𝑊 ∈ Poset)    &   (𝜑 → 𝐹𝐻𝐺)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝐺‘(𝐹‘(𝐺‘𝑌))) = (𝐺‘𝑌))
 
26-Jul-2024mgcf1olem1 30970 Property of a Galois connection, lemma for mgcf1o 30972. (Contributed by Thierry Arnoux, 26-Jul-2024.)
𝐻 = (𝑉MGalConn𝑊)    &   𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    ≤ = (le‘𝑉)    &    ≲ = (le‘𝑊)    &   (𝜑 → 𝑉 ∈ Poset)    &   (𝜑 → 𝑊 ∈ Poset)    &   (𝜑 → 𝐹𝐻𝐺)    &   (𝜑 → 𝑋 ∈ 𝐴)    ⇒   (𝜑 → (𝐹‘(𝐺‘(𝐹‘𝑋))) = (𝐹‘𝑋))
 
26-Jul-2024mhppwdeg 21062 Degree of a homogeneous polynomial raised to a power. General version of deg1pw 24990. (Contributed by SN, 26-Jul-2024.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝑇 = (mulGrp‘𝑃)    &    ↑ = (.g‘𝑇)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑀 ∈ ℕ0)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ (𝐻‘𝑀))    ⇒   (𝜑 → (𝑁 ↑ 𝑋) ∈ (𝐻‘(𝑀 · 𝑁)))
 
26-Jul-2024ifexd 4477 Existence of the conditional operator (deduction form). (Contributed by SN, 26-Jul-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ V)
 
24-Jul-2024flt4lem6 40150 Remove shared factors in a solution to 𝐴↑4 + 𝐵↑4 = 𝐶↑2. (Contributed by SN, 24-Jul-2024.)
(𝜑 → 𝐴 ∈ ℕ)    &   (𝜑 → 𝐵 ∈ ℕ)    &   (𝜑 → 𝐶 ∈ ℕ)    &   (𝜑 → ((𝐴↑4) + (𝐵↑4)) = (𝐶↑2))    ⇒   (𝜑 → (((𝐴 / (𝐴 gcd 𝐵)) ∈ ℕ ∧ (𝐵 / (𝐴 gcd 𝐵)) ∈ ℕ ∧ (𝐶 / ((𝐴 gcd 𝐵)↑2)) ∈ ℕ) ∧ (((𝐴 / (𝐴 gcd 𝐵))↑4) + ((𝐵 / (𝐴 gcd 𝐵))↑4)) = ((𝐶 / ((𝐴 gcd 𝐵)↑2))↑2)))
 
24-Jul-2024ply1fermltl 31356 Fermat's little theorem for polynomials. If 𝑃 is prime, Then (𝑋 + 𝐴)↑𝑃 = ((𝑋↑𝑃) + 𝐴) modulo 𝑃. (Contributed by Thierry Arnoux, 24-Jul-2024.)
𝑍 = (ℤ/nℤ‘𝑃)    &   𝑊 = (Poly1‘𝑍)    &   𝑋 = (var1‘𝑍)    &    + = (+g‘𝑊)    &   𝑁 = (mulGrp‘𝑊)    &    ↑ = (.g‘𝑁)    &   𝐶 = (algSc‘𝑊)    &   𝐴 = (𝐶‘((ℤRHom‘𝑍)‘𝐸))    &   (𝜑 → 𝑃 ∈ ℙ)    &   (𝜑 → 𝐸 ∈ ℤ)    ⇒   (𝜑 → (𝑃 ↑ (𝑋 + 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))
 
24-Jul-2024ply1chr 31355 The characteristic of a polynomial ring is the characteristic of the underlying ring. (Contributed by Thierry Arnoux, 24-Jul-2024.)
𝑃 = (Poly1‘𝑅)    ⇒   (𝑅 ∈ CRing → (chr‘𝑃) = (chr‘𝑅))
 
24-Jul-2024ply1scleq 31354 Equality of a constant polynomial is the same as equality of the constant term. (Contributed by Thierry Arnoux, 24-Jul-2024.)
𝑃 = (Poly1‘𝑅)    &   𝐵 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝐸 ∈ 𝐵)    &   (𝜑 → 𝐹 ∈ 𝐵)    ⇒   (𝜑 → ((𝐴‘𝐸) = (𝐴‘𝐹) ↔ 𝐸 = 𝐹))
 
24-Jul-2024asclmulg 31352 Apply group multiplication to the algebra scalars. (Contributed by Thierry Arnoux, 24-Jul-2024.)
𝐴 = (algSc‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    ↑ = (.g‘𝑊)    &    ∗ = (.g‘𝐹)    ⇒   ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → (𝐴‘(𝑁 ∗ 𝑋)) = (𝑁 ↑ (𝐴‘𝑋)))
 
24-Jul-2024lsmsnorb2 31266 The sumset of a single element with a group is the element's orbit by the group action. See gaorb 18673. (Contributed by Thierry Arnoux, 24-Jul-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g‘𝐺)    &    ⊕ = (LSSum‘𝐺)    &    ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔 ∈ 𝐴 (𝑥 + 𝑔) = 𝑦)}    &   (𝜑 → 𝐺 ∈ Mnd)    &   (𝜑 → 𝐴 ⊆ 𝐵)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → ({𝑋} ⊕ 𝐴) = [𝑋] ∼ )
 
24-Jul-2024znfermltl 31248 Fermat's little theorem in ℤ/nℤ. (Contributed by Thierry Arnoux, 24-Jul-2024.)
𝑍 = (ℤ/nℤ‘𝑃)    &   𝐵 = (Base‘𝑍)    &    ↑ = (.g‘(mulGrp‘𝑍))    ⇒   ((𝑃 ∈ ℙ ∧ 𝐴 ∈ 𝐵) → (𝑃 ↑ 𝐴) = 𝐴)
 
23-Jul-2024offun 7471 The function operation produces a function. (Contributed by SN, 23-Jul-2024.)
(𝜑 → 𝐹 Fn 𝐴)    &   (𝜑 → 𝐺 Fn 𝐵)    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → Fun (𝐹 ∘f 𝑅𝐺))
 
23-Jul-2024mpofun 7323 The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.) (Proof shortened by SN, 23-Jul-2024.)
𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   Fun 𝐹
 
22-Jul-2024aks4d1p1p1 39761 Exponential law for finite products, special case. (Contributed by metakunt, 22-Jul-2024.)
(𝜑 → 𝐴 ∈ ℝ+)    &   (𝜑 → 𝑁 ∈ ℕ)    ⇒   (𝜑 → ∏𝑘 ∈ (1...𝑁)(𝐴↑𝑐𝑘) = (𝐴↑𝑐Σ𝑘 ∈ (1...𝑁)𝑘))
 
22-Jul-2024intlewftc 39760 Inequality inference by invoking fundamental theorem of calculus. (Contributed by metakunt, 22-Jul-2024.)
(𝜑 → 𝐴 ∈ ℝ)    &   (𝜑 → 𝐵 ∈ ℝ)    &   (𝜑 → 𝐴 ≤ 𝐵)    &   (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ))    &   (𝜑 → 𝐷 = (ℝ D 𝐹))    &   (𝜑 → 𝐸 = (ℝ D 𝐺))    &   (𝜑 → 𝐷 ∈ ((𝐴(,)𝐵)–cn→ℝ))    &   (𝜑 → 𝐸 ∈ ((𝐴(,)𝐵)–cn→ℝ))    &   (𝜑 → 𝐷 ∈ 𝐿1)    &   (𝜑 → 𝐸 ∈ 𝐿1)    &   (𝜑 → 𝐷 = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑃))    &   (𝜑 → 𝐸 = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑄))    &   ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑃 ≤ 𝑄)    &   (𝜑 → (𝐹‘𝐴) ≤ (𝐺‘𝐴))    ⇒   (𝜑 → (𝐹‘𝐵) ≤ (𝐺‘𝐵))
 
22-Jul-2024mhpmulcl 21061 A product of homogeneous polynomials is a homogeneous polynomial whose degree is the sum of the degrees of the factors. Compare mdegmulle2 24949 (which shows less-than-or-equal instead of equal). (Contributed by SN, 22-Jul-2024.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑌 = (𝐼 mPoly 𝑅)    &    · = (.r‘𝑌)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑀 ∈ ℕ0)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑃 ∈ (𝐻‘𝑀))    &   (𝜑 → 𝑄 ∈ (𝐻‘𝑁))    ⇒   (𝜑 → (𝑃 · 𝑄) ∈ (𝐻‘(𝑀 + 𝑁)))
 
22-Jul-2024ismhp3 21055 A polynomial is homogeneous iff the degree of every nonzero term is the same. (Contributed by SN, 22-Jul-2024.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &    0 = (0g‘𝑅)    &   𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑅 ∈ 𝑊)    &   (𝜑 → 𝑁 ∈ ℕ0)    &   (𝜑 → 𝑋 ∈ 𝐵)    ⇒   (𝜑 → (𝑋 ∈ (𝐻‘𝑁) ↔ ∀𝑑 ∈ 𝐷 ((𝑋‘𝑑) ≠ 0 → ((ℂfld ↾s ℕ0) Σg 𝑑) = 𝑁)))
 
21-Jul-2024wl-ifp-ncond2 35330 If one case of an if- condition is false, the other automatically follows. (Contributed by Wolf Lammen, 21-Jul-2024.)
(¬ 𝜒 → (if-(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∧ 𝜓)))
 
21-Jul-2024wl-ifp-ncond1 35329 If one case of an if- condition is false, the other automatically follows. (Contributed by Wolf Lammen, 21-Jul-2024.)
(¬ 𝜓 → (if-(𝜑, 𝜓, 𝜒) ↔ (¬ 𝜑 ∧ 𝜒)))
 
21-Jul-2024mgcmnt2d 30967 Galois connection implies monotonicity of the right adjoint. (Contributed by Thierry Arnoux, 21-Jul-2024.)
𝐻 = (𝑉MGalConn𝑊)    &   (𝜑 → 𝑉 ∈ Proset )    &   (𝜑 → 𝑊 ∈ Proset )    &   (𝜑 → 𝐹𝐻𝐺)    ⇒   (𝜑 → 𝐺 ∈ (𝑊Monot𝑉))
 
21-Jul-2024mgcmnt1d 30966 Galois connection implies monotonicity of the left adjoint. (Contributed by Thierry Arnoux, 21-Jul-2024.)
𝐻 = (𝑉MGalConn𝑊)    &   (𝜑 → 𝑉 ∈ Proset )    &   (𝜑 → 𝑊 ∈ Proset )    &   (𝜑 → 𝐹𝐻𝐺)    ⇒   (𝜑 → 𝐹 ∈ (𝑉Monot𝑊))
 
21-Jul-2024wemapso 9156 Construct lexicographic order on a function space based on a well-ordering of the indices and a total ordering of the values. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 21-Jul-2024.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    ⇒   ((𝑅 We 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐵 ↑m 𝐴))
 
21-Jul-2024wemapsolem 9155 Lemma for wemapso 9156. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 21-Jul-2024.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   𝑈 ⊆ (𝐵 ↑m 𝐴)    &   (𝜑 → 𝑅 Or 𝐴)    &   (𝜑 → 𝑆 Or 𝐵)    &   ((𝜑 ∧ ((𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈) ∧ 𝑎 ≠ 𝑏)) → ∃𝑐 ∈ dom (𝑎 ∖ 𝑏)∀𝑑 ∈ dom (𝑎 ∖ 𝑏) ¬ 𝑑𝑅𝑐)    ⇒   (𝜑 → 𝑇 Or 𝑈)
 
21-Jul-2024wemappo 9154 Construct lexicographic order on a function space based on a well-ordering of the indices and a total ordering of the values.

Without totality on the values or least differing indices, the best we can prove here is a partial order. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by AV, 21-Jul-2024.)

𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    ⇒   ((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) → 𝑇 Po (𝐵 ↑m 𝐴))
 
21-Jul-2024wemaplem3 9153 Lemma for wemapso 9156. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.) (Revised by AV, 21-Jul-2024.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   (𝜑 → 𝑃 ∈ (𝐵 ↑m 𝐴))    &   (𝜑 → 𝑋 ∈ (𝐵 ↑m 𝐴))    &   (𝜑 → 𝑄 ∈ (𝐵 ↑m 𝐴))    &   (𝜑 → 𝑅 Or 𝐴)    &   (𝜑 → 𝑆 Po 𝐵)    &   (𝜑 → 𝑃𝑇𝑋)    &   (𝜑 → 𝑋𝑇𝑄)    ⇒   (𝜑 → 𝑃𝑇𝑄)
 
21-Jul-2024wemaplem2 9152 Lemma for wemapso 9156. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.) (Revised by AV, 21-Jul-2024.)
𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   (𝜑 → 𝑃 ∈ (𝐵 ↑m 𝐴))    &   (𝜑 → 𝑋 ∈ (𝐵 ↑m 𝐴))    &   (𝜑 → 𝑄 ∈ (𝐵 ↑m 𝐴))    &   (𝜑 → 𝑅 Or 𝐴)    &   (𝜑 → 𝑆 Po 𝐵)    &   (𝜑 → 𝑎 ∈ 𝐴)    &   (𝜑 → (𝑃‘𝑎)𝑆(𝑋‘𝑎))    &   (𝜑 → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑎 → (𝑃‘𝑐) = (𝑋‘𝑐)))    &   (𝜑 → 𝑏 ∈ 𝐴)    &   (𝜑 → (𝑋‘𝑏)𝑆(𝑄‘𝑏))    &   (𝜑 → ∀𝑐 ∈ 𝐴 (𝑐𝑅𝑏 → (𝑋‘𝑐) = (𝑄‘𝑐)))    ⇒   (𝜑 → 𝑃𝑇𝑄)
 
20-Jul-2024fpwwe 10243 Given any function 𝐹 from the powerset of 𝐴 to 𝐴, canth2 8788 gives that the function is not injective, but we can say rather more than that. There is a unique well-ordered subset ⟨𝑋, (𝑊‘𝑋)⟩ which "agrees" with 𝐹 in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 9627. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))}    &   (𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹‘𝑥) ∈ 𝐴)    &   𝑋 = ∪ dom 𝑊    ⇒   (𝜑 → ((𝑌𝑊𝑅 ∧ (𝐹‘𝑌) ∈ 𝑌) ↔ (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋))))
 
20-Jul-2024fpwwelem 10242 Lemma for fpwwe 10243. (Contributed by Mario Carneiro, 15-May-2015.) (Revised by AV, 20-Jul-2024.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))}    &   (𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝐹‘(◡𝑅 “ {𝑦})) = 𝑦))))
 
20-Jul-2024fpwwe2 10240 Given any function 𝐹 from well-orderings of subsets of 𝐴 to 𝐴, there is a unique well-ordered subset ⟨𝑋, (𝑊‘𝑋)⟩ which "agrees" with 𝐹 in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 9627. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   𝑋 = ∪ dom 𝑊    ⇒   (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋))))
 
20-Jul-2024fpwwe2lem12 10239 Lemma for fpwwe2 10240. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   𝑋 = ∪ dom 𝑊    ⇒   (𝜑 → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)
 
20-Jul-2024fpwwe2lem11 10238 Lemma for fpwwe2 10240. (Contributed by Mario Carneiro, 18-May-2015.) (Proof shortened by Peter Mazsa, 23-Sep-2022.) (Revised by AV, 20-Jul-2024.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   𝑋 = ∪ dom 𝑊    ⇒   (𝜑 → 𝑋 ∈ dom 𝑊)
 
20-Jul-2024fpwwe2lem10 10237 Lemma for fpwwe2 10240. (Contributed by Mario Carneiro, 15-May-2015.) (Revised by AV, 20-Jul-2024.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   𝑋 = ∪ dom 𝑊    ⇒   (𝜑 → 𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))
 
20-Jul-2024fpwwe2lem9 10236 Lemma for fpwwe2 10240. Given two well-orders ⟨𝑋, 𝑅⟩ and ⟨𝑌, 𝑆⟩ of parts of 𝐴, one is an initial segment of the other. (Contributed by Mario Carneiro, 15-May-2015.) (Revised by AV, 20-Jul-2024.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑 → 𝑋𝑊𝑅)    &   (𝜑 → 𝑌𝑊𝑆)    ⇒   (𝜑 → ((𝑋 ⊆ 𝑌 ∧ 𝑅 = (𝑆 ∩ (𝑌 × 𝑋))) ∨ (𝑌 ⊆ 𝑋 ∧ 𝑆 = (𝑅 ∩ (𝑋 × 𝑌)))))
 
20-Jul-2024fpwwe2lem8 10235 Lemma for fpwwe2 10240. Given two well-orders ⟨𝑋, 𝑅⟩ and ⟨𝑌, 𝑆⟩ of parts of 𝐴, one is an initial segment of the other. (The 𝑂 ⊆ 𝑃 hypothesis is in order to break the symmetry of 𝑋 and 𝑌.) (Contributed by Mario Carneiro, 15-May-2015.) (Proof shortened by Peter Mazsa, 23-Sep-2022.) (Revised by AV, 20-Jul-2024.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑 → 𝑋𝑊𝑅)    &   (𝜑 → 𝑌𝑊𝑆)    &   𝑀 = OrdIso(𝑅, 𝑋)    &   𝑁 = OrdIso(𝑆, 𝑌)    &   (𝜑 → dom 𝑀 ⊆ dom 𝑁)    ⇒   (𝜑 → (𝑋 ⊆ 𝑌 ∧ 𝑅 = (𝑆 ∩ (𝑌 × 𝑋))))
 
20-Jul-2024fpwwe2lem7 10234 Lemma for fpwwe2 10240. Show by induction that the two isometries 𝑀 and 𝑁 agree on their common domain. (Contributed by Mario Carneiro, 15-May-2015.) (Proof shortened by Peter Mazsa, 23-Sep-2022.) (Revised by AV, 20-Jul-2024.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑 → 𝑋𝑊𝑅)    &   (𝜑 → 𝑌𝑊𝑆)    &   𝑀 = OrdIso(𝑅, 𝑋)    &   𝑁 = OrdIso(𝑆, 𝑌)    &   (𝜑 → dom 𝑀 ⊆ dom 𝑁)    ⇒   (𝜑 → 𝑀 = (𝑁 ↾ dom 𝑀))
 
20-Jul-2024fpwwe2lem6 10233 Lemma for fpwwe2 10240. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑 → 𝑋𝑊𝑅)    &   (𝜑 → 𝑌𝑊𝑆)    &   𝑀 = OrdIso(𝑅, 𝑋)    &   𝑁 = OrdIso(𝑆, 𝑌)    &   (𝜑 → 𝐵 ∈ dom 𝑀)    &   (𝜑 → 𝐵 ∈ dom 𝑁)    &   (𝜑 → (𝑀 ↾ 𝐵) = (𝑁 ↾ 𝐵))    ⇒   ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐶𝑆(𝑁‘𝐵) ∧ (𝐷𝑅(𝑀‘𝐵) → (𝐶𝑅𝐷 ↔ 𝐶𝑆𝐷))))
 
20-Jul-2024fpwwe2lem5 10232 Lemma for fpwwe2 10240. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑 → 𝑋𝑊𝑅)    &   (𝜑 → 𝑌𝑊𝑆)    &   𝑀 = OrdIso(𝑅, 𝑋)    &   𝑁 = OrdIso(𝑆, 𝑌)    &   (𝜑 → 𝐵 ∈ dom 𝑀)    &   (𝜑 → 𝐵 ∈ dom 𝑁)    &   (𝜑 → (𝑀 ↾ 𝐵) = (𝑁 ↾ 𝐵))    ⇒   ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐶 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌 ∧ (◡𝑀‘𝐶) = (◡𝑁‘𝐶)))
 
20-Jul-2024fpwwe2lem4 10231 Lemma for fpwwe2 10240. (Contributed by Mario Carneiro, 15-May-2015.) (Revised by AV, 20-Jul-2024.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑 → 𝐴 ∈ 𝑉)    &   ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    ⇒   ((𝜑 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴)
 
20-Jul-2024fpwwe2lem3 10230 Lemma for fpwwe2 10240. (Contributed by Mario Carneiro, 19-May-2015.) (Revised by AV, 20-Jul-2024.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝑋𝑊𝑅)    ⇒   ((𝜑 ∧ 𝐵 ∈ 𝑋) → ((◡𝑅 “ {𝐵})𝐹(𝑅 ∩ ((◡𝑅 “ {𝐵}) × (◡𝑅 “ {𝐵})))) = 𝐵)
 
20-Jul-2024fpwwe2lem2 10229 Lemma for fpwwe2 10240. (Contributed by Mario Carneiro, 19-May-2015.) (Revised by AV, 20-Jul-2024.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
 
18-Jul-2024metakunt34 39832 𝐷 is a permutation. (Contributed by metakunt, 18-Jul-2024.)
(𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝐼 ∈ ℕ)    &   (𝜑 → 𝐼 ≤ 𝑀)    &   𝐷 = (𝑤 ∈ (1...𝑀) ↦ if(𝑤 = 𝐼, 𝑤, if(𝑤 < 𝐼, ((𝑤 + (𝑀 − 𝐼)) + if(𝐼 ≤ (𝑤 + (𝑀 − 𝐼)), 1, 0)), ((𝑤 − 𝐼) + if(𝐼 ≤ (𝑤 − 𝐼), 1, 0)))))    ⇒   (𝜑 → 𝐷:(1...𝑀)–1-1-onto→(1...𝑀))
 
18-Jul-2024metakunt33 39831 Construction of one solution of the increment equation system. (Contributed by metakunt, 18-Jul-2024.)
(𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝐼 ∈ ℕ)    &   (𝜑 → 𝐼 ≤ 𝑀)    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼)))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   𝐷 = (𝑤 ∈ (1...𝑀) ↦ if(𝑤 = 𝐼, 𝑤, if(𝑤 < 𝐼, ((𝑤 + (𝑀 − 𝐼)) + if(𝐼 ≤ (𝑤 + (𝑀 − 𝐼)), 1, 0)), ((𝑤 − 𝐼) + if(𝐼 ≤ (𝑤 − 𝐼), 1, 0)))))    ⇒   (𝜑 → (𝐶 ∘ (𝐵 ∘ 𝐴)) = 𝐷)
 
18-Jul-2024metakunt32 39830 Construction of one solution of the increment equation system. (Contributed by metakunt, 18-Jul-2024.)
(𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝐼 ∈ ℕ)    &   (𝜑 → 𝐼 ≤ 𝑀)    &   (𝜑 → 𝑋 ∈ (1...𝑀))    &   𝐷 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑥, if(𝑥 < 𝐼, ((𝑥 + (𝑀 − 𝐼)) + if(𝐼 ≤ (𝑥 + (𝑀 − 𝐼)), 1, 0)), ((𝑥 − 𝐼) + if(𝐼 ≤ (𝑥 − 𝐼), 1, 0)))))    &   𝐺 = if(𝐼 ≤ (𝑋 + (𝑀 − 𝐼)), 1, 0)    &   𝐻 = if(𝐼 ≤ (𝑋 − 𝐼), 1, 0)    &   𝑅 = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻)))    ⇒   (𝜑 → (𝐷‘𝑋) = 𝑅)
 
18-Jul-2024metakunt31 39829 Construction of one solution of the increment equation system. (Contributed by metakunt, 18-Jul-2024.)
(𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝐼 ∈ ℕ)    &   (𝜑 → 𝐼 ≤ 𝑀)    &   (𝜑 → 𝑋 ∈ (1...𝑀))    &   𝐴 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝐼, 𝑀, if(𝑥 < 𝐼, 𝑥, (𝑥 − 1))))    &   𝐵 = (𝑧 ∈ (1...𝑀) ↦ if(𝑧 = 𝑀, 𝑀, if(𝑧 < 𝐼, (𝑧 + (𝑀 − 𝐼)), (𝑧 + (1 − 𝐼)))))    &   𝐶 = (𝑦 ∈ (1...𝑀) ↦ if(𝑦 = 𝑀, 𝐼, if(𝑦 < 𝐼, 𝑦, (𝑦 + 1))))    &   𝐺 = if(𝐼 ≤ (𝑋 + (𝑀 − 𝐼)), 1, 0)    &   𝐻 = if(𝐼 ≤ (𝑋 − 𝐼), 1, 0)    &   𝑅 = if(𝑋 = 𝐼, 𝑋, if(𝑋 < 𝐼, ((𝑋 + (𝑀 − 𝐼)) + 𝐺), ((𝑋 − 𝐼) + 𝐻)))    ⇒   (𝜑 → (𝐶‘(𝐵‘(𝐴‘𝑋))) = 𝑅)
 
18-Jul-2024nummin 32748 Every nonempty class of numerable sets has a minimal element. (Contributed by BTernaryTau, 18-Jul-2024.)
((𝐴 ⊆ dom card ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 Pred( ≺ , 𝐴, 𝑥) = ∅)
 
18-Jul-2024cardpred 32747 The cardinality function preserves predecessors. (Contributed by BTernaryTau, 18-Jul-2024.)
((𝐴 ⊆ dom card ∧ 𝐵 ∈ dom card) → Pred( E , (card “ 𝐴), (card‘𝐵)) = (card “ Pred( ≺ , 𝐴, 𝐵)))
 
16-Jul-2024fnrelpredd 32746 A function that preserves a relation also preserves predecessors. (Contributed by BTernaryTau, 16-Jul-2024.)
(𝜑 → 𝐹 Fn 𝐴)    &   (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)))    &   (𝜑 → 𝐶 ⊆ 𝐴)    &   (𝜑 → 𝐷 ∈ 𝐴)    ⇒   (𝜑 → Pred(𝑆, (𝐹 “ 𝐶), (𝐹‘𝐷)) = (𝐹 “ Pred(𝑅, 𝐶, 𝐷)))
 
16-Jul-2024unexd 7528 The union of two sets is a set. (Contributed by SN, 16-Jul-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    &   (𝜑 → 𝐵 ∈ 𝑊)    ⇒   (𝜑 → (𝐴 ∪ 𝐵) ∈ V)
 
16-Jul-2024difexd 5211 Existence of a difference. (Contributed by SN, 16-Jul-2024.)
(𝜑 → 𝐴 ∈ 𝑉)    ⇒   (𝜑 → (𝐴 ∖ 𝐵) ∈ V)
 
16-Jul-2024sylbida 595 A syllogism deduction. (Contributed by SN, 16-Jul-2024.)
(𝜑 → (𝜓 ↔ 𝜒))    &   ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ((𝜑 ∧ 𝜓) → 𝜃)
 
15-Jul-2024fsuppssindlem2 39943 Lemma for fsuppssind 39944. Write a function as a union. (Contributed by SN, 15-Jul-2024.)
(𝜑 → 𝐵 ∈ 𝑊)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝑆 ⊆ 𝐼)    ⇒   (𝜑 → (𝐹 ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝑓‘𝑥), 0 )) ∈ 𝐻} ↔ (𝐹:𝑆⟶𝐵 ∧ (𝐹 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻)))
 
15-Jul-2024fsuppssindlem1 39942 Lemma for fsuppssind 39944. Functions are zero outside of their support. (Contributed by SN, 15-Jul-2024.)
(𝜑 → 0 ∈ 𝑊)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝐹:𝐼⟶𝐵)    &   (𝜑 → (𝐹 supp 0 ) ⊆ 𝑆)    ⇒   (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 )))
 
13-Jul-2024rtrclind 14611 Principle of transitive induction. The first three hypotheses give various existences, the next four give necessary substitutions and the last two are the basis and the induction step. (Contributed by Drahflow, 12-Nov-2015.) (Revised by AV, 13-Jul-2024.)
(𝜂 → Rel 𝑅)    &   (𝜂 → 𝑆 ∈ 𝑉)    &   (𝜂 → 𝑋 ∈ 𝑊)    &   (𝑖 = 𝑆 → (𝜑 ↔ 𝜒))    &   (𝑖 = 𝑥 → (𝜑 ↔ 𝜓))    &   (𝑖 = 𝑗 → (𝜑 ↔ 𝜃))    &   (𝑥 = 𝑋 → (𝜓 ↔ 𝜏))    &   (𝜂 → 𝜒)    &   (𝜂 → (𝑗𝑅𝑥 → (𝜃 → 𝜓)))    ⇒   (𝜂 → (𝑆(t*‘𝑅)𝑋 → 𝜏))
 
13-Jul-2024relexpind 14610 Principle of transitive induction, finite version. The first three hypotheses give various existences, the next four give necessary substitutions and the last two are the basis and the induction hypothesis. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 13-Jul-2024.)
(𝜂 → Rel 𝑅)    &   (𝜂 → 𝑆 ∈ 𝑉)    &   (𝜂 → 𝑋 ∈ 𝑊)    &   (𝑖 = 𝑆 → (𝜑 ↔ 𝜒))    &   (𝑖 = 𝑥 → (𝜑 ↔ 𝜓))    &   (𝑖 = 𝑗 → (𝜑 ↔ 𝜃))    &   (𝑥 = 𝑋 → (𝜓 ↔ 𝜏))    &   (𝜂 → 𝜒)    &   (𝜂 → (𝑗𝑅𝑥 → (𝜃 → 𝜓)))    ⇒   (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑋 → 𝜏)))

Older news:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(4-Feb-2014) Mario Carneiro writes:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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