Theorem caovmo 7591

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 7364	. . . . . 6 ⊢ (𝑢 = 𝐴 → (𝑢𝐹𝑤) = (𝐴𝐹𝑤))
2	1	eqeq1d 2738	. . . . 5 ⊢ (𝑢 = 𝐴 → ((𝑢𝐹𝑤) = 𝐵 ↔ (𝐴𝐹𝑤) = 𝐵))
3	2	mobidv 2547	. . . 4 ⊢ (𝑢 = 𝐴 → (∃*𝑤(𝑢𝐹𝑤) = 𝐵 ↔ ∃*𝑤(𝐴𝐹𝑤) = 𝐵))
4		oveq2 7365	. . . . . . 7 ⊢ (𝑤 = 𝑣 → (𝑢𝐹𝑤) = (𝑢𝐹𝑣))
5	4	eqeq1d 2738	. . . . . 6 ⊢ (𝑤 = 𝑣 → ((𝑢𝐹𝑤) = 𝐵 ↔ (𝑢𝐹𝑣) = 𝐵))
6	5	mo4 2564	. . . . 5 ⊢ (∃*𝑤(𝑢𝐹𝑤) = 𝐵 ↔ ∀𝑤∀𝑣(((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → 𝑤 = 𝑣))
7		simpr 485	. . . . . . . . 9 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝐹𝑣) = 𝐵)
8	7	oveq2d 7373	. . . . . . . 8 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑤𝐹(𝑢𝐹𝑣)) = (𝑤𝐹𝐵))
9		simpl 483	. . . . . . . . . 10 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝐹𝑤) = 𝐵)
10	9	oveq1d 7372	. . . . . . . . 9 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → ((𝑢𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
11		vex 3449	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
12		vex 3449	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
13		vex 3449	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
14		caovmo.ass	. . . . . . . . . . 11 ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
15	11, 12, 13, 14	caovass 7554	. . . . . . . . . 10 ⊢ ((𝑢𝐹𝑤)𝐹𝑣) = (𝑢𝐹(𝑤𝐹𝑣))
16		caovmo.com	. . . . . . . . . . 11 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
17	11, 12, 13, 16, 14	caov12 7582	. . . . . . . . . 10 ⊢ (𝑢𝐹(𝑤𝐹𝑣)) = (𝑤𝐹(𝑢𝐹𝑣))
18	15, 17	eqtri 2764	. . . . . . . . 9 ⊢ ((𝑢𝐹𝑤)𝐹𝑣) = (𝑤𝐹(𝑢𝐹𝑣))
19		caovmo.2	. . . . . . . . . . 11 ⊢ 𝐵 ∈ 𝑆
20	19	elexi 3464	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
21	20, 13, 16	caovcom 7551	. . . . . . . . 9 ⊢ (𝐵𝐹𝑣) = (𝑣𝐹𝐵)
22	10, 18, 21	3eqtr3g 2799	. . . . . . . 8 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑤𝐹(𝑢𝐹𝑣)) = (𝑣𝐹𝐵))
23	8, 22	eqtr3d 2778	. . . . . . 7 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑤𝐹𝐵) = (𝑣𝐹𝐵))
24	9, 19	eqeltrdi 2846	. . . . . . . . 9 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝐹𝑤) ∈ 𝑆)
25		caovmo.dom	. . . . . . . . . 10 ⊢ dom 𝐹 = (𝑆 × 𝑆)
26		caovmo.3	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ 𝑆
27	25, 26	ndmovrcl 7540	. . . . . . . . 9 ⊢ ((𝑢𝐹𝑤) ∈ 𝑆 → (𝑢 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))
28	24, 27	syl 17	. . . . . . . 8 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))
29		oveq1 7364	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (𝑥𝐹𝐵) = (𝑤𝐹𝐵))
30		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
31	29, 30	eqeq12d 2752	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → ((𝑥𝐹𝐵) = 𝑥 ↔ (𝑤𝐹𝐵) = 𝑤))
32		caovmo.id	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑆 → (𝑥𝐹𝐵) = 𝑥)
33	31, 32	vtoclga 3534	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑆 → (𝑤𝐹𝐵) = 𝑤)
34	28, 33	simpl2im 504	. . . . . . 7 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑤𝐹𝐵) = 𝑤)
35	7, 19	eqeltrdi 2846	. . . . . . . . 9 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢𝐹𝑣) ∈ 𝑆)
36	25, 26	ndmovrcl 7540	. . . . . . . . 9 ⊢ ((𝑢𝐹𝑣) ∈ 𝑆 → (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆))
37	35, 36	syl 17	. . . . . . . 8 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆))
38		oveq1 7364	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (𝑥𝐹𝐵) = (𝑣𝐹𝐵))
39		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → 𝑥 = 𝑣)
40	38, 39	eqeq12d 2752	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → ((𝑥𝐹𝐵) = 𝑥 ↔ (𝑣𝐹𝐵) = 𝑣))
41	40, 32	vtoclga 3534	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑆 → (𝑣𝐹𝐵) = 𝑣)
42	37, 41	simpl2im 504	. . . . . . 7 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → (𝑣𝐹𝐵) = 𝑣)
43	23, 34, 42	3eqtr3d 2784	. . . . . 6 ⊢ (((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → 𝑤 = 𝑣)
44	43	ax-gen 1797	. . . . 5 ⊢ ∀𝑣(((𝑢𝐹𝑤) = 𝐵 ∧ (𝑢𝐹𝑣) = 𝐵) → 𝑤 = 𝑣)
45	6, 44	mpgbir 1801	. . . 4 ⊢ ∃*𝑤(𝑢𝐹𝑤) = 𝐵
46	3, 45	vtoclg 3525	. . 3 ⊢ (𝐴 ∈ 𝑆 → ∃*𝑤(𝐴𝐹𝑤) = 𝐵)
47		moanimv 2619	. . 3 ⊢ (∃*𝑤(𝐴 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ↔ (𝐴 ∈ 𝑆 → ∃*𝑤(𝐴𝐹𝑤) = 𝐵))
48	46, 47	mpbir 230	. 2 ⊢ ∃*𝑤(𝐴 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵)
49		eleq1 2825	. . . . . . 7 ⊢ ((𝐴𝐹𝑤) = 𝐵 → ((𝐴𝐹𝑤) ∈ 𝑆 ↔ 𝐵 ∈ 𝑆))
50	19, 49	mpbiri 257	. . . . . 6 ⊢ ((𝐴𝐹𝑤) = 𝐵 → (𝐴𝐹𝑤) ∈ 𝑆)
51	25, 26	ndmovrcl 7540	. . . . . 6 ⊢ ((𝐴𝐹𝑤) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))
52	50, 51	syl 17	. . . . 5 ⊢ ((𝐴𝐹𝑤) = 𝐵 → (𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆))
53	52	simpld 495	. . . 4 ⊢ ((𝐴𝐹𝑤) = 𝐵 → 𝐴 ∈ 𝑆)
54	53	ancri 550	. . 3 ⊢ ((𝐴𝐹𝑤) = 𝐵 → (𝐴 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵))
55	54	moimi 2543	. 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 396  ∀wal 1539   = wceq 1541   ∈ wcel 2106  ∃*wmo 2536  ∅c0 4282   × cxp 5631  dom cdm 5633  (class class class)co 7357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-xp 5639  df-dm 5643  df-iota 6448  df-fv 6504  df-ov 7360

This theorem is referenced by:  recmulnq  10900