Theorem caovimo 5972

Description: Uniqueness of inverse element in commutative, associative operation with identity. The identity element is B. (Contributed by Jim Kingdon, 18-Sep-2019.)

Hypotheses

Ref	Expression
caovimo.idel	|- B e. S
caovimo.com	|- ( ( x e. S /\ y e. S ) -> ( x F y ) = ( y F x ) )
caovimo.ass	|- ( ( x e. S /\ y e. S /\ z e. S ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
caovimo.id	|- ( x e. S -> ( x F B ) = x )

Assertion

Ref	Expression
caovimo	|- ( A e. S -> E* w ( w e. S /\ ( A F w ) = B ) )

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

Allowed substitution hint:   B( z)

Proof of Theorem caovimo

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 5789	. . . . . . 7 |- ( ( A F w ) = B -> ( ( A F w ) F v ) = ( B F v ) )
2	1	adantl 275	. . . . . 6 |- ( ( w e. S /\ ( A F w ) = B ) -> ( ( A F w ) F v ) = ( B F v ) )
3	2	3ad2ant2 1004	. . . . 5 |- ( ( A e. S /\ ( w e. S /\ ( A F w ) = B ) /\ ( v e. S /\ ( A F v ) = B ) ) -> ( ( A F w ) F v ) = ( B F v ) )
4		df-3an 965	. . . . . . . . 9 |- ( ( A e. S /\ w e. S /\ v e. S ) <-> ( ( A e. S /\ w e. S ) /\ v e. S ) )
5		caovimo.ass	. . . . . . . . . . . . . 14 |- ( ( x e. S /\ y e. S /\ z e. S ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
6	5	adantl 275	. . . . . . . . . . . . 13 |- ( ( ( A e. S /\ w e. S /\ v e. S ) /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
7		simp1 982	. . . . . . . . . . . . 13 |- ( ( A e. S /\ w e. S /\ v e. S ) -> A e. S )
8		simp2 983	. . . . . . . . . . . . 13 |- ( ( A e. S /\ w e. S /\ v e. S ) -> w e. S )
9		simp3 984	. . . . . . . . . . . . 13 |- ( ( A e. S /\ w e. S /\ v e. S ) -> v e. S )
10	6, 7, 8, 9	caovassd 5938	. . . . . . . . . . . 12 |- ( ( A e. S /\ w e. S /\ v e. S ) -> ( ( A F w ) F v ) = ( A F ( w F v ) ) )
11		caovimo.com	. . . . . . . . . . . . . 14 |- ( ( x e. S /\ y e. S ) -> ( x F y ) = ( y F x ) )
12	11	adantl 275	. . . . . . . . . . . . 13 |- ( ( ( A e. S /\ w e. S /\ v e. S ) /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
13	7, 8, 9, 12, 6	caov12d 5960	. . . . . . . . . . . 12 |- ( ( A e. S /\ w e. S /\ v e. S ) -> ( A F ( w F v ) ) = ( w F ( A F v ) ) )
14	10, 13	eqtrd 2173	. . . . . . . . . . 11 |- ( ( A e. S /\ w e. S /\ v e. S ) -> ( ( A F w ) F v ) = ( w F ( A F v ) ) )
15	14	adantr 274	. . . . . . . . . 10 |- ( ( ( A e. S /\ w e. S /\ v e. S ) /\ ( A F v ) = B ) -> ( ( A F w ) F v ) = ( w F ( A F v ) ) )
16		oveq2 5790	. . . . . . . . . . . 12 |- ( ( A F v ) = B -> ( w F ( A F v ) ) = ( w F B ) )
17		oveq1 5789	. . . . . . . . . . . . . 14 |- ( x = w -> ( x F B ) = ( w F B ) )
18		id 19	. . . . . . . . . . . . . 14 |- ( x = w -> x = w )
19	17, 18	eqeq12d 2155	. . . . . . . . . . . . 13 |- ( x = w -> ( ( x F B ) = x <-> ( w F B ) = w ) )
20		caovimo.id	. . . . . . . . . . . . 13 |- ( x e. S -> ( x F B ) = x )
21	19, 20	vtoclga 2755	. . . . . . . . . . . 12 |- ( w e. S -> ( w F B ) = w )
22	16, 21	sylan9eqr 2195	. . . . . . . . . . 11 |- ( ( w e. S /\ ( A F v ) = B ) -> ( w F ( A F v ) ) = w )
23	22	3ad2antl2 1145	. . . . . . . . . 10 |- ( ( ( A e. S /\ w e. S /\ v e. S ) /\ ( A F v ) = B ) -> ( w F ( A F v ) ) = w )
24	15, 23	eqtrd 2173	. . . . . . . . 9 |- ( ( ( A e. S /\ w e. S /\ v e. S ) /\ ( A F v ) = B ) -> ( ( A F w ) F v ) = w )
25	4, 24	sylanbr 283	. . . . . . . 8 |- ( ( ( ( A e. S /\ w e. S ) /\ v e. S ) /\ ( A F v ) = B ) -> ( ( A F w ) F v ) = w )
26	25	anasss 397	. . . . . . 7 |- ( ( ( A e. S /\ w e. S ) /\ ( v e. S /\ ( A F v ) = B ) ) -> ( ( A F w ) F v ) = w )
27	26	3impa 1177	. . . . . 6 |- ( ( A e. S /\ w e. S /\ ( v e. S /\ ( A F v ) = B ) ) -> ( ( A F w ) F v ) = w )
28	27	3adant2r 1212	. . . . 5 |- ( ( A e. S /\ ( w e. S /\ ( A F w ) = B ) /\ ( v e. S /\ ( A F v ) = B ) ) -> ( ( A F w ) F v ) = w )
29	11	adantl 275	. . . . . . . . 9 |- ( ( v e. S /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
30		caovimo.idel	. . . . . . . . . 10 |- B e. S
31	30	a1i 9	. . . . . . . . 9 |- ( v e. S -> B e. S )
32		id 19	. . . . . . . . 9 |- ( v e. S -> v e. S )
33	29, 31, 32	caovcomd 5935	. . . . . . . 8 |- ( v e. S -> ( B F v ) = ( v F B ) )
34		oveq1 5789	. . . . . . . . . 10 |- ( x = v -> ( x F B ) = ( v F B ) )
35		id 19	. . . . . . . . . 10 |- ( x = v -> x = v )
36	34, 35	eqeq12d 2155	. . . . . . . . 9 |- ( x = v -> ( ( x F B ) = x <-> ( v F B ) = v ) )
37	36, 20	vtoclga 2755	. . . . . . . 8 |- ( v e. S -> ( v F B ) = v )
38	33, 37	eqtrd 2173	. . . . . . 7 |- ( v e. S -> ( B F v ) = v )
39	38	adantr 274	. . . . . 6 |- ( ( v e. S /\ ( A F v ) = B ) -> ( B F v ) = v )
40	39	3ad2ant3 1005	. . . . 5 |- ( ( A e. S /\ ( w e. S /\ ( A F w ) = B ) /\ ( v e. S /\ ( A F v ) = B ) ) -> ( B F v ) = v )
41	3, 28, 40	3eqtr3d 2181	. . . 4 |- ( ( A e. S /\ ( w e. S /\ ( A F w ) = B ) /\ ( v e. S /\ ( A F v ) = B ) ) -> w = v )
42	41	3expib 1185	. . 3 |- ( A e. S -> ( ( ( w e. S /\ ( A F w ) = B ) /\ ( v e. S /\ ( A F v ) = B ) ) -> w = v ) )
43	42	alrimivv 1848	. 2 |- ( A e. S -> A. w A. v ( ( ( w e. S /\ ( A F w ) = B ) /\ ( v e. S /\ ( A F v ) = B ) ) -> w = v ) )
44		eleq1 2203	. . . 4 |- ( w = v -> ( w e. S <-> v e. S ) )
45		oveq2 5790	. . . . 5 |- ( w = v -> ( A F w ) = ( A F v ) )
46	45	eqeq1d 2149	. . . 4 |- ( w = v -> ( ( A F w ) = B <-> ( A F v ) = B ) )
47	44, 46	anbi12d 465	. . 3 |- ( w = v -> ( ( w e. S /\ ( A F w ) = B ) <-> ( v e. S /\ ( A F v ) = B ) ) )
48	47	mo4 2061	. 2 |- ( E* w ( w e. S /\ ( A F w ) = B ) <-> A. w A. v ( ( ( w e. S /\ ( A F w ) = B ) /\ ( v e. S /\ ( A F v ) = B ) ) -> w = v ) )
49	43, 48	sylibr 133	1 |- ( A e. S -> E* w ( w e. S /\ ( A F w ) = B ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   A.wal 1330   = wceq 1332   e. wcel 1481   E*wmo 2001  (class class class)co 5782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139  df-ov 5785

This theorem is referenced by:  recmulnqg  7223