Theorem grpinva 13419

Description: Deduce right inverse from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
grpinva.c	|- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )
grpinva.o	|- ( ph -> O e. B )
grpinva.i	|- ( ( ph /\ x e. B ) -> ( O .+ x ) = x )
grpinva.a	|- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
grpinva.r	|- ( ( ph /\ x e. B ) -> E. y e. B ( y .+ x ) = O )
grpinva.x	|- ( ( ph /\ ps ) -> X e. B )
grpinva.n	|- ( ( ph /\ ps ) -> N e. B )
grpinva.e	|- ( ( ph /\ ps ) -> ( N .+ X ) = O )

Assertion

Ref	Expression
grpinva	|- ( ( ph /\ ps ) -> ( X .+ N ) = O )

Distinct variable groups:   x, y, z, B   x, O, y, z   ph, x, y, z   y, N, z   x, .+ , y, z   y, X, z   ps, y

Allowed substitution hints:   ps( x, z)   N( x)   X( x)

Proof of Theorem grpinva

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

Step	Hyp	Ref	Expression
1		grpinva.c	. 2 |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )
2		grpinva.o	. 2 |- ( ph -> O e. B )
3		grpinva.i	. 2 |- ( ( ph /\ x e. B ) -> ( O .+ x ) = x )
4		grpinva.a	. 2 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
5		grpinva.r	. 2 |- ( ( ph /\ x e. B ) -> E. y e. B ( y .+ x ) = O )
6	1	3expb 1228	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x .+ y ) e. B )
7	6	caovclg 6158	. . . 4 |- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( u .+ v ) e. B )
8	7	adantlr 477	. . 3 |- ( ( ( ph /\ ps ) /\ ( u e. B /\ v e. B ) ) -> ( u .+ v ) e. B )
9		grpinva.x	. . 3 |- ( ( ph /\ ps ) -> X e. B )
10		grpinva.n	. . 3 |- ( ( ph /\ ps ) -> N e. B )
11	8, 9, 10	caovcld 6159	. 2 |- ( ( ph /\ ps ) -> ( X .+ N ) e. B )
12	4	caovassg 6164	. . . . 5 |- ( ( ph /\ ( u e. B /\ v e. B /\ w e. B ) ) -> ( ( u .+ v ) .+ w ) = ( u .+ ( v .+ w ) ) )
13	12	adantlr 477	. . . 4 |- ( ( ( ph /\ ps ) /\ ( u e. B /\ v e. B /\ w e. B ) ) -> ( ( u .+ v ) .+ w ) = ( u .+ ( v .+ w ) ) )
14	13, 9, 10, 11	caovassd 6165	. . 3 |- ( ( ph /\ ps ) -> ( ( X .+ N ) .+ ( X .+ N ) ) = ( X .+ ( N .+ ( X .+ N ) ) ) )
15		grpinva.e	. . . . . 6 |- ( ( ph /\ ps ) -> ( N .+ X ) = O )
16	15	oveq1d 6016	. . . . 5 |- ( ( ph /\ ps ) -> ( ( N .+ X ) .+ N ) = ( O .+ N ) )
17	13, 10, 9, 10	caovassd 6165	. . . . 5 |- ( ( ph /\ ps ) -> ( ( N .+ X ) .+ N ) = ( N .+ ( X .+ N ) ) )
18		oveq2 6009	. . . . . . 7 |- ( y = N -> ( O .+ y ) = ( O .+ N ) )
19		id 19	. . . . . . 7 |- ( y = N -> y = N )
20	18, 19	eqeq12d 2244	. . . . . 6 |- ( y = N -> ( ( O .+ y ) = y <-> ( O .+ N ) = N ) )
21	3	ralrimiva 2603	. . . . . . . 8 |- ( ph -> A. x e. B ( O .+ x ) = x )
22		oveq2 6009	. . . . . . . . . 10 |- ( x = y -> ( O .+ x ) = ( O .+ y ) )
23		id 19	. . . . . . . . . 10 |- ( x = y -> x = y )
24	22, 23	eqeq12d 2244	. . . . . . . . 9 |- ( x = y -> ( ( O .+ x ) = x <-> ( O .+ y ) = y ) )
25	24	cbvralvw 2769	. . . . . . . 8 |- ( A. x e. B ( O .+ x ) = x <-> A. y e. B ( O .+ y ) = y )
26	21, 25	sylib 122	. . . . . . 7 |- ( ph -> A. y e. B ( O .+ y ) = y )
27	26	adantr 276	. . . . . 6 |- ( ( ph /\ ps ) -> A. y e. B ( O .+ y ) = y )
28	20, 27, 10	rspcdva 2912	. . . . 5 |- ( ( ph /\ ps ) -> ( O .+ N ) = N )
29	16, 17, 28	3eqtr3d 2270	. . . 4 |- ( ( ph /\ ps ) -> ( N .+ ( X .+ N ) ) = N )
30	29	oveq2d 6017	. . 3 |- ( ( ph /\ ps ) -> ( X .+ ( N .+ ( X .+ N ) ) ) = ( X .+ N ) )
31	14, 30	eqtrd 2262	. 2 |- ( ( ph /\ ps ) -> ( ( X .+ N ) .+ ( X .+ N ) ) = ( X .+ N ) )
32	1, 2, 3, 4, 5, 11, 31	grpinvalem 13418	1 |- ( ( ph /\ ps ) -> ( X .+ N ) = O )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509  (class class class)co 6001

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6004

This theorem is referenced by:  grprida  13420  grprcan  13570  grprinv  13584