Theorem caovmo 6871

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 e. S
caovmo.dom	|- dom F = ( S X. S )
caovmo.3	|- -. (/) e. S
caovmo.com	|- ( x F y ) = ( y F x )
caovmo.ass	|- ( ( x F y ) F z ) = ( x F ( y F z ) )
caovmo.id	|- ( x e. S -> ( x F B ) = x )

Assertion

Ref	Expression
caovmo	|- E* w ( A F w ) = 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 6657	. . . . . 6 |- ( u = A -> ( u F w ) = ( A F w ) )
2	1	eqeq1d 2624	. . . . 5 |- ( u = A -> ( ( u F w ) = B <-> ( A F w ) = B ) )
3	2	mobidv 2491	. . . 4 |- ( u = A -> ( E* w ( u F w ) = B <-> E* w ( A F w ) = B ) )
4		oveq2 6658	. . . . . . 7 |- ( w = v -> ( u F w ) = ( u F v ) )
5	4	eqeq1d 2624	. . . . . 6 |- ( w = v -> ( ( u F w ) = B <-> ( u F v ) = B ) )
6	5	mo4 2517	. . . . 5 |- ( E* w ( u F w ) = B <-> A. w A. v ( ( ( u F w ) = B /\ ( u F v ) = B ) -> w = v ) )
7		simpr 477	. . . . . . . . 9 |- ( ( ( u F w ) = B /\ ( u F v ) = B ) -> ( u F v ) = B )
8	7	oveq2d 6666	. . . . . . . 8 |- ( ( ( u F w ) = B /\ ( u F v ) = B ) -> ( w F ( u F v ) ) = ( w F B ) )
9		simpl 473	. . . . . . . . . 10 |- ( ( ( u F w ) = B /\ ( u F v ) = B ) -> ( u F w ) = B )
10	9	oveq1d 6665	. . . . . . . . 9 |- ( ( ( u F w ) = B /\ ( u F v ) = B ) -> ( ( u F w ) F v ) = ( B F v ) )
11		vex 3203	. . . . . . . . . . 11 |- u e. _V
12		vex 3203	. . . . . . . . . . 11 |- w e. _V
13		vex 3203	. . . . . . . . . . 11 |- v e. _V
14		caovmo.ass	. . . . . . . . . . 11 |- ( ( x F y ) F z ) = ( x F ( y F z ) )
15	11, 12, 13, 14	caovass 6834	. . . . . . . . . 10 |- ( ( u F w ) F v ) = ( u F ( w F v ) )
16		caovmo.com	. . . . . . . . . . 11 |- ( x F y ) = ( y F x )
17	11, 12, 13, 16, 14	caov12 6862	. . . . . . . . . 10 |- ( u F ( w F v ) ) = ( w F ( u F v ) )
18	15, 17	eqtri 2644	. . . . . . . . 9 |- ( ( u F w ) F v ) = ( w F ( u F v ) )
19		caovmo.2	. . . . . . . . . . 11 |- B e. S
20	19	elexi 3213	. . . . . . . . . 10 |- B e. _V
21	20, 13, 16	caovcom 6831	. . . . . . . . 9 |- ( B F v ) = ( v F B )
22	10, 18, 21	3eqtr3g 2679	. . . . . . . 8 |- ( ( ( u F w ) = B /\ ( u F v ) = B ) -> ( w F ( u F v ) ) = ( v F B ) )
23	8, 22	eqtr3d 2658	. . . . . . 7 |- ( ( ( u F w ) = B /\ ( u F v ) = B ) -> ( w F B ) = ( v F B ) )
24	9, 19	syl6eqel 2709	. . . . . . . . . 10 |- ( ( ( u F w ) = B /\ ( u F v ) = B ) -> ( u F w ) e. S )
25		caovmo.dom	. . . . . . . . . . 11 |- dom F = ( S X. S )
26		caovmo.3	. . . . . . . . . . 11 |- -. (/) e. S
27	25, 26	ndmovrcl 6820	. . . . . . . . . 10 |- ( ( u F w ) e. S -> ( u e. S /\ w e. S ) )
28	24, 27	syl 17	. . . . . . . . 9 |- ( ( ( u F w ) = B /\ ( u F v ) = B ) -> ( u e. S /\ w e. S ) )
29	28	simprd 479	. . . . . . . 8 |- ( ( ( u F w ) = B /\ ( u F v ) = B ) -> w e. S )
30		oveq1 6657	. . . . . . . . . 10 |- ( x = w -> ( x F B ) = ( w F B ) )
31		id 22	. . . . . . . . . 10 |- ( x = w -> x = w )
32	30, 31	eqeq12d 2637	. . . . . . . . 9 |- ( x = w -> ( ( x F B ) = x <-> ( w F B ) = w ) )
33		caovmo.id	. . . . . . . . 9 |- ( x e. S -> ( x F B ) = x )
34	32, 33	vtoclga 3272	. . . . . . . 8 |- ( w e. S -> ( w F B ) = w )
35	29, 34	syl 17	. . . . . . 7 |- ( ( ( u F w ) = B /\ ( u F v ) = B ) -> ( w F B ) = w )
36	7, 19	syl6eqel 2709	. . . . . . . . . 10 |- ( ( ( u F w ) = B /\ ( u F v ) = B ) -> ( u F v ) e. S )
37	25, 26	ndmovrcl 6820	. . . . . . . . . 10 |- ( ( u F v ) e. S -> ( u e. S /\ v e. S ) )
38	36, 37	syl 17	. . . . . . . . 9 |- ( ( ( u F w ) = B /\ ( u F v ) = B ) -> ( u e. S /\ v e. S ) )
39	38	simprd 479	. . . . . . . 8 |- ( ( ( u F w ) = B /\ ( u F v ) = B ) -> v e. S )
40		oveq1 6657	. . . . . . . . . 10 |- ( x = v -> ( x F B ) = ( v F B ) )
41		id 22	. . . . . . . . . 10 |- ( x = v -> x = v )
42	40, 41	eqeq12d 2637	. . . . . . . . 9 |- ( x = v -> ( ( x F B ) = x <-> ( v F B ) = v ) )
43	42, 33	vtoclga 3272	. . . . . . . 8 |- ( v e. S -> ( v F B ) = v )
44	39, 43	syl 17	. . . . . . 7 |- ( ( ( u F w ) = B /\ ( u F v ) = B ) -> ( v F B ) = v )
45	23, 35, 44	3eqtr3d 2664	. . . . . 6 |- ( ( ( u F w ) = B /\ ( u F v ) = B ) -> w = v )
46	45	ax-gen 1722	. . . . 5 |- A. v ( ( ( u F w ) = B /\ ( u F v ) = B ) -> w = v )
47	6, 46	mpgbir 1726	. . . 4 |- E* w ( u F w ) = B
48	3, 47	vtoclg 3266	. . 3 |- ( A e. S -> E* w ( A F w ) = B )
49		moanimv 2531	. . 3 |- ( E* w ( A e. S /\ ( A F w ) = B ) <-> ( A e. S -> E* w ( A F w ) = B ) )
50	48, 49	mpbir 221	. 2 |- E* w ( A e. S /\ ( A F w ) = B )
51		eleq1 2689	. . . . . . 7 |- ( ( A F w ) = B -> ( ( A F w ) e. S <-> B e. S ) )
52	19, 51	mpbiri 248	. . . . . 6 |- ( ( A F w ) = B -> ( A F w ) e. S )
53	25, 26	ndmovrcl 6820	. . . . . 6 |- ( ( A F w ) e. S -> ( A e. S /\ w e. S ) )
54	52, 53	syl 17	. . . . 5 |- ( ( A F w ) = B -> ( A e. S /\ w e. S ) )
55	54	simpld 475	. . . 4 |- ( ( A F w ) = B -> A e. S )
56	55	ancri 575	. . 3 |- ( ( A F w ) = B -> ( A e. S /\ ( A F w ) = B ) )
57	56	moimi 2520	. 2 |- ( E* w ( A e. S /\ ( A F w ) = B ) -> E* w ( A F w ) = B )
58	50, 57	ax-mp 5	1 |- E* w ( A F w ) = B

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   E*wmo 2471   (/)c0 3915   X. cxp 5112   dom cdm 5114  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  recmulnq  9786