Theorem caovmo 7647

Description: Uniqueness of inverse element in commutative, associative operation with identity. Remark in proof of Proposition 9-2.4 of [Gleason] p. 119. (Contributed by NM, 4-Mar-1996.)

Hypotheses

Ref	Expression
caovmo.2	⊢ 𝐵 ∈ 𝑆
caovmo.dom	⊢ dom 𝐹 = (𝑆 × 𝑆)
caovmo.3	⊢ ¬ ∅ ∈ 𝑆
caovmo.com	⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
caovmo.ass	⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
caovmo.id	⊢ (𝑥 ∈ 𝑆 → (𝑥𝐹𝐵) = 𝑥)

Assertion

Ref	Expression
caovmo	⊢ ∃*𝑤(𝐴𝐹𝑤) = 𝐵

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑤,𝐴,𝑥,𝑦   𝑤,𝐵,𝑧   𝑤,𝐹   𝑤,𝑆

Proof of Theorem caovmo

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7417	. . . . . 6 ⊢ (𝑢 = 𝐴 → (𝑢𝐹𝑤) = (𝐴𝐹𝑤))
2	1	eqeq1d 2765	. . . . 5 ⊢ (𝑢 = 𝐴 → ((𝑢𝐹𝑤) = 𝐵 ↔ (𝐴𝐹𝑤) = 𝐵))
3	2	mobidv 2577	. . . 4 ⊢ (𝑢 = 𝐴 → (∃*𝑤(𝑢𝐹𝑤) = 𝐵 ↔ ∃*𝑤(𝐴𝐹𝑤) = 𝐵))
4		oveq2 7418	. . . . . . 7 ⊢ (𝑤 = 𝑣 → (𝑢𝐹𝑤) = (𝑢𝐹𝑣))
5	4	eqeq1d 2765	. . . . . 6 ⊢ (𝑤 = 𝑣 → ((𝑢𝐹𝑤) = 𝐵 ↔ (𝑢𝐹𝑣) = 𝐵))
6	5	mo4 2594	. . . . 5 ⊢ (∃*𝑤(𝑢𝐹𝑤) = 𝐵 ↔ ∀𝑤∀𝑣(((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → 𝑤 = 𝑣))
7		simpr 489	. . . . . . . . 9 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝐹𝑣) = 𝐵)
8	7	oveq2d 7426	. . . . . . . 8 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑤𝐹(𝑢𝐹𝑣)) = (𝑤𝐹𝐵))
9		simpl 487	. . . . . . . . . 10 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝐹𝑤) = 𝐵)
10	9	oveq1d 7425	. . . . . . . . 9 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → ((𝑢𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
11		vex 3459	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
12		vex 3459	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
13		vex 3459	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
14		caovmo.ass	. . . . . . . . . . 11 ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
15	11, 12, 13, 14	caovass 7610	. . . . . . . . . 10 ⊢ ((𝑢𝐹𝑤)𝐹𝑣) = (𝑢𝐹(𝑤𝐹𝑣))
16		caovmo.com	. . . . . . . . . . 11 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
17	11, 12, 13, 16, 14	caov12 7638	. . . . . . . . . 10 ⊢ (𝑢𝐹(𝑤𝐹𝑣)) = (𝑤𝐹(𝑢𝐹𝑣))
18	15, 17	eqtri 2786	. . . . . . . . 9 ⊢ ((𝑢𝐹𝑤)𝐹𝑣) = (𝑤𝐹(𝑢𝐹𝑣))
19		caovmo.2	. . . . . . . . . . 11 ⊢ 𝐵 ∈ 𝑆
20	19	elexi 3477	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
21	20, 13, 16	caovcom 7607	. . . . . . . . 9 ⊢ (𝐵𝐹𝑣) = (𝑣𝐹𝐵)
22	10, 18, 21	3eqtr3g 2821	. . . . . . . 8 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑤𝐹(𝑢𝐹𝑣)) = (𝑣𝐹𝐵))
23	8, 22	eqtr3d 2800	. . . . . . 7 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑤𝐹𝐵) = (𝑣𝐹𝐵))
24	9, 19	eqeltrdi 2871	. . . . . . . . 9 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝐹𝑤) ∈ 𝑆)
25		caovmo.dom	. . . . . . . . . 10 ⊢ dom 𝐹 = (𝑆 × 𝑆)
26		caovmo.3	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ 𝑆
27	25, 26	ndmovrcl 7596	. . . . . . . . 9 ⊢ ((𝑢𝐹𝑤) ∈ 𝑆 → (𝑢 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))
28	24, 27	syl 18	. . . . . . . 8 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))
29		oveq1 7417	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (𝑥𝐹𝐵) = (𝑤𝐹𝐵))
30		id 23	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
31	29, 30	eqeq12d 2779	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → ((𝑥𝐹𝐵) = 𝑥 ↔ (𝑤𝐹𝐵) = 𝑤))
32		caovmo.id	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑆 → (𝑥𝐹𝐵) = 𝑥)
33	31, 32	vtoclga 3541	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑆 → (𝑤𝐹𝐵) = 𝑤)
34	28, 33	simpl2im 512	. . . . . . 7 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑤𝐹𝐵) = 𝑤)
35	7, 19	eqeltrdi 2871	. . . . . . . . 9 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝐹𝑣) ∈ 𝑆)
36	25, 26	ndmovrcl 7596	. . . . . . . . 9 ⊢ ((𝑢𝐹𝑣) ∈ 𝑆 → (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆))
37	35, 36	syl 18	. . . . . . . 8 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆))
38		oveq1 7417	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (𝑥𝐹𝐵) = (𝑣𝐹𝐵))
39		id 23	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → 𝑥 = 𝑣)
40	38, 39	eqeq12d 2779	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → ((𝑥𝐹𝐵) = 𝑥 ↔ (𝑣𝐹𝐵) = 𝑣))
41	40, 32	vtoclga 3541	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑆 → (𝑣𝐹𝐵) = 𝑣)
42	37, 41	simpl2im 512	. . . . . . 7 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑣𝐹𝐵) = 𝑣)
43	23, 34, 42	3eqtr3d 2806	. . . . . 6 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → 𝑤 = 𝑣)
44	43	ax-gen 1825	. . . . 5 ⊢ ∀𝑣(((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → 𝑤 = 𝑣)
45	6, 44	mpgbir 1829	. . . 4 ⊢ ∃*𝑤(𝑢𝐹𝑤) = 𝐵
46	3, 45	vtoclg 3522	. . 3 ⊢ (𝐴 ∈ 𝑆 → ∃*𝑤(𝐴𝐹𝑤) = 𝐵)
47		moanimv 2647	. . 3 ⊢ (∃*𝑤(𝐴 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ↔ (𝐴 ∈ 𝑆 → ∃*𝑤(𝐴𝐹𝑤) = 𝐵))
48	46, 47	mpbir 234	. 2 ⊢ ∃*𝑤(𝐴 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵)
49		eleq1 2851	. . . . . . 7 ⊢ ((𝐴𝐹𝑤) = 𝐵 → ((𝐴𝐹𝑤) ∈ 𝑆 ↔ 𝐵 ∈ 𝑆))
50	19, 49	mpbiri 261	. . . . . 6 ⊢ ((𝐴𝐹𝑤) = 𝐵 → (𝐴𝐹𝑤) ∈ 𝑆)
51	25, 26	ndmovrcl 7596	. . . . . 6 ⊢ ((𝐴𝐹𝑤) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))
52	50, 51	syl 18	. . . . 5 ⊢ ((𝐴𝐹𝑤) = 𝐵 → (𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))
53	52	simpld 499	. . . 4 ⊢ ((𝐴𝐹𝑤) = 𝐵 → 𝐴 ∈ 𝑆)
54	53	ancri 558	. . 3 ⊢ ((𝐴𝐹𝑤) = 𝐵 → (𝐴 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵))
55	54	moimi 2573	. 2 ⊢ (∃*𝑤(𝐴 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) → ∃*𝑤(𝐴𝐹𝑤) = 𝐵)
56	48, 55	ax-mp 5	1 ⊢ ∃*𝑤(𝐴𝐹𝑤) = 𝐵

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400  ∀wal 1568   = wceq 1570   ∈ wcel 2143  ∃*wmo 2565  ∅c0 4286   × cxp 5659  dom cdm 5661  (class class class)co 7410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-xp 5667  df-dm 5671  df-iota 6492  df-fv 6544  df-ov 7413

This theorem is referenced by:  recmulnq  10944