Theorem caovimo 6112

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 5925	. . . . . . 7 |- ( ( A F w ) = B -> ( ( A F w ) F v ) = ( B F v ) )
2	1	adantl 277	. . . . . 6 |- ( ( w e. S /\ ( A F w ) = B ) -> ( ( A F w ) F v ) = ( B F v ) )
3	2	3ad2ant2 1021	. . . . 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 982	. . . . . . . . 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 277	. . . . . . . . . . . . 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 999	. . . . . . . . . . . . 13 |- ( ( A e. S /\ w e. S /\ v e. S ) -> A e. S )
8		simp2 1000	. . . . . . . . . . . . 13 |- ( ( A e. S /\ w e. S /\ v e. S ) -> w e. S )
9		simp3 1001	. . . . . . . . . . . . 13 |- ( ( A e. S /\ w e. S /\ v e. S ) -> v e. S )
10	6, 7, 8, 9	caovassd 6078	. . . . . . . . . . . 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 277	. . . . . . . . . . . . 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 6100	. . . . . . . . . . . 12 |- ( ( A e. S /\ w e. S /\ v e. S ) -> ( A F ( w F v ) ) = ( w F ( A F v ) ) )
14	10, 13	eqtrd 2226	. . . . . . . . . . 11 |- ( ( A e. S /\ w e. S /\ v e. S ) -> ( ( A F w ) F v ) = ( w F ( A F v ) ) )
15	14	adantr 276	. . . . . . . . . 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 5926	. . . . . . . . . . . 12 |- ( ( A F v ) = B -> ( w F ( A F v ) ) = ( w F B ) )
17		oveq1 5925	. . . . . . . . . . . . . 14 |- ( x = w -> ( x F B ) = ( w F B ) )
18		id 19	. . . . . . . . . . . . . 14 |- ( x = w -> x = w )
19	17, 18	eqeq12d 2208	. . . . . . . . . . . . 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 2826	. . . . . . . . . . . 12 |- ( w e. S -> ( w F B ) = w )
22	16, 21	sylan9eqr 2248	. . . . . . . . . . 11 |- ( ( w e. S /\ ( A F v ) = B ) -> ( w F ( A F v ) ) = w )
23	22	3ad2antl2 1162	. . . . . . . . . 10 |- ( ( ( A e. S /\ w e. S /\ v e. S ) /\ ( A F v ) = B ) -> ( w F ( A F v ) ) = w )
24	15, 23	eqtrd 2226	. . . . . . . . 9 |- ( ( ( A e. S /\ w e. S /\ v e. S ) /\ ( A F v ) = B ) -> ( ( A F w ) F v ) = w )
25	4, 24	sylanbr 285	. . . . . . . 8 |- ( ( ( ( A e. S /\ w e. S ) /\ v e. S ) /\ ( A F v ) = B ) -> ( ( A F w ) F v ) = w )
26	25	anasss 399	. . . . . . 7 |- ( ( ( A e. S /\ w e. S ) /\ ( v e. S /\ ( A F v ) = B ) ) -> ( ( A F w ) F v ) = w )
27	26	3impa 1196	. . . . . 6 |- ( ( A e. S /\ w e. S /\ ( v e. S /\ ( A F v ) = B ) ) -> ( ( A F w ) F v ) = w )
28	27	3adant2r 1235	. . . . 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 277	. . . . . . . . 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 6075	. . . . . . . 8 |- ( v e. S -> ( B F v ) = ( v F B ) )
34		oveq1 5925	. . . . . . . . . 10 |- ( x = v -> ( x F B ) = ( v F B ) )
35		id 19	. . . . . . . . . 10 |- ( x = v -> x = v )
36	34, 35	eqeq12d 2208	. . . . . . . . 9 |- ( x = v -> ( ( x F B ) = x <-> ( v F B ) = v ) )
37	36, 20	vtoclga 2826	. . . . . . . 8 |- ( v e. S -> ( v F B ) = v )
38	33, 37	eqtrd 2226	. . . . . . 7 |- ( v e. S -> ( B F v ) = v )
39	38	adantr 276	. . . . . 6 |- ( ( v e. S /\ ( A F v ) = B ) -> ( B F v ) = v )
40	39	3ad2ant3 1022	. . . . 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 2234	. . . 4 |- ( ( A e. S /\ ( w e. S /\ ( A F w ) = B ) /\ ( v e. S /\ ( A F v ) = B ) ) -> w = v )
42	41	3expib 1208	. . 3 |- ( A e. S -> ( ( ( w e. S /\ ( A F w ) = B ) /\ ( v e. S /\ ( A F v ) = B ) ) -> w = v ) )
43	42	alrimivv 1886	. 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 2256	. . . 4 |- ( w = v -> ( w e. S <-> v e. S ) )
45		oveq2 5926	. . . . 5 |- ( w = v -> ( A F w ) = ( A F v ) )
46	45	eqeq1d 2202	. . . 4 |- ( w = v -> ( ( A F w ) = B <-> ( A F v ) = B ) )
47	44, 46	anbi12d 473	. . 3 |- ( w = v -> ( ( w e. S /\ ( A F w ) = B ) <-> ( v e. S /\ ( A F v ) = B ) ) )
48	47	mo4 2103	. 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 134	1 |- ( A e. S -> E* w ( w e. S /\ ( A F w ) = B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   A.wal 1362   = wceq 1364   E*wmo 2043   e. wcel 2164  (class class class)co 5918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921

This theorem is referenced by:  recmulnqg  7451