Theorem grprinvd 5716

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

Assertion

Ref	Expression
grprinvd	|- ( ( 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 grprinvd

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

Step	Hyp	Ref	Expression
1		grprinvlem.c	. 2 |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )
2		grprinvlem.o	. 2 |- ( ph -> O e. B )
3		grprinvlem.i	. 2 |- ( ( ph /\ x e. B ) -> ( O .+ x ) = x )
4		grprinvlem.a	. 2 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
5		grprinvlem.n	. 2 |- ( ( ph /\ x e. B ) -> E. y e. B ( y .+ x ) = O )
6	1	3expb 1139	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x .+ y ) e. B )
7	6	caovclg 5673	. . . 4 |- ( ( ph /\ ( u e. B /\ v e. B ) ) -> ( u .+ v ) e. B )
8	7	adantlr 460	. . 3 |- ( ( ( ph /\ ps ) /\ ( u e. B /\ v e. B ) ) -> ( u .+ v ) e. B )
9		grprinvd.x	. . 3 |- ( ( ph /\ ps ) -> X e. B )
10		grprinvd.n	. . 3 |- ( ( ph /\ ps ) -> N e. B )
11	8, 9, 10	caovcld 5674	. 2 |- ( ( ph /\ ps ) -> ( X .+ N ) e. B )
12	4	caovassg 5679	. . . . 5 |- ( ( ph /\ ( u e. B /\ v e. B /\ w e. B ) ) -> ( ( u .+ v ) .+ w ) = ( u .+ ( v .+ w ) ) )
13	12	adantlr 460	. . . 4 |- ( ( ( ph /\ ps ) /\ ( u e. B /\ v e. B /\ w e. B ) ) -> ( ( u .+ v ) .+ w ) = ( u .+ ( v .+ w ) ) )
14	13, 9, 10, 11	caovassd 5680	. . 3 |- ( ( ph /\ ps ) -> ( ( X .+ N ) .+ ( X .+ N ) ) = ( X .+ ( N .+ ( X .+ N ) ) ) )
15		grprinvd.e	. . . . . 6 |- ( ( ph /\ ps ) -> ( N .+ X ) = O )
16	15	oveq1d 5547	. . . . 5 |- ( ( ph /\ ps ) -> ( ( N .+ X ) .+ N ) = ( O .+ N ) )
17	13, 10, 9, 10	caovassd 5680	. . . . 5 |- ( ( ph /\ ps ) -> ( ( N .+ X ) .+ N ) = ( N .+ ( X .+ N ) ) )
18	3	ralrimiva 2434	. . . . . . . 8 |- ( ph -> A. x e. B ( O .+ x ) = x )
19		oveq2 5540	. . . . . . . . . 10 |- ( x = y -> ( O .+ x ) = ( O .+ y ) )
20		id 19	. . . . . . . . . 10 |- ( x = y -> x = y )
21	19, 20	eqeq12d 2095	. . . . . . . . 9 |- ( x = y -> ( ( O .+ x ) = x <-> ( O .+ y ) = y ) )
22	21	cbvralv 2577	. . . . . . . 8 |- ( A. x e. B ( O .+ x ) = x <-> A. y e. B ( O .+ y ) = y )
23	18, 22	sylib 120	. . . . . . 7 |- ( ph -> A. y e. B ( O .+ y ) = y )
24	23	adantr 270	. . . . . 6 |- ( ( ph /\ ps ) -> A. y e. B ( O .+ y ) = y )
25		oveq2 5540	. . . . . . . 8 |- ( y = N -> ( O .+ y ) = ( O .+ N ) )
26		id 19	. . . . . . . 8 |- ( y = N -> y = N )
27	25, 26	eqeq12d 2095	. . . . . . 7 |- ( y = N -> ( ( O .+ y ) = y <-> ( O .+ N ) = N ) )
28	27	rspcv 2697	. . . . . 6 |- ( N e. B -> ( A. y e. B ( O .+ y ) = y -> ( O .+ N ) = N ) )
29	10, 24, 28	sylc 61	. . . . 5 |- ( ( ph /\ ps ) -> ( O .+ N ) = N )
30	16, 17, 29	3eqtr3d 2121	. . . 4 |- ( ( ph /\ ps ) -> ( N .+ ( X .+ N ) ) = N )
31	30	oveq2d 5548	. . 3 |- ( ( ph /\ ps ) -> ( X .+ ( N .+ ( X .+ N ) ) ) = ( X .+ N ) )
32	14, 31	eqtrd 2113	. 2 |- ( ( ph /\ ps ) -> ( ( X .+ N ) .+ ( X .+ N ) ) = ( X .+ N ) )
33	1, 2, 3, 4, 5, 11, 32	grprinvlem 5715	1 |- ( ( ph /\ ps ) -> ( X .+ N ) = O )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   = wceq 1284   e. wcel 1433   A.wral 2348   E.wrex 2349  (class class class)co 5532

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by:  grpridd  5717