Theorem grprida 13533

Description: Deduce right identity 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 )

Assertion

Ref	Expression
grprida	|- ( ( ph /\ x e. B ) -> ( x .+ O ) = x )

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

Proof of Theorem grprida

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

Step	Hyp	Ref	Expression
1		grpinva.r	. . . 4 |- ( ( ph /\ x e. B ) -> E. y e. B ( y .+ x ) = O )
2		oveq1 6035	. . . . . 6 |- ( y = n -> ( y .+ x ) = ( n .+ x ) )
3	2	eqeq1d 2240	. . . . 5 |- ( y = n -> ( ( y .+ x ) = O <-> ( n .+ x ) = O ) )
4	3	cbvrexvw 2773	. . . 4 |- ( E. y e. B ( y .+ x ) = O <-> E. n e. B ( n .+ x ) = O )
5	1, 4	sylib 122	. . 3 |- ( ( ph /\ x e. B ) -> E. n e. B ( n .+ x ) = O )
6		grpinva.a	. . . . . . . 8 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
7	6	caovassg 6191	. . . . . . 7 |- ( ( ph /\ ( u e. B /\ v e. B /\ w e. B ) ) -> ( ( u .+ v ) .+ w ) = ( u .+ ( v .+ w ) ) )
8	7	adantlr 477	. . . . . 6 |- ( ( ( ph /\ ( x e. B /\ ( n e. B /\ ( n .+ x ) = O ) ) ) /\ ( u e. B /\ v e. B /\ w e. B ) ) -> ( ( u .+ v ) .+ w ) = ( u .+ ( v .+ w ) ) )
9		simprl 531	. . . . . 6 |- ( ( ph /\ ( x e. B /\ ( n e. B /\ ( n .+ x ) = O ) ) ) -> x e. B )
10		simprrl 541	. . . . . 6 |- ( ( ph /\ ( x e. B /\ ( n e. B /\ ( n .+ x ) = O ) ) ) -> n e. B )
11	8, 9, 10, 9	caovassd 6192	. . . . 5 |- ( ( ph /\ ( x e. B /\ ( n e. B /\ ( n .+ x ) = O ) ) ) -> ( ( x .+ n ) .+ x ) = ( x .+ ( n .+ x ) ) )
12		grpinva.c	. . . . . . 7 |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )
13		grpinva.o	. . . . . . 7 |- ( ph -> O e. B )
14		grpinva.i	. . . . . . 7 |- ( ( ph /\ x e. B ) -> ( O .+ x ) = x )
15		simprrr 542	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ ( n e. B /\ ( n .+ x ) = O ) ) ) -> ( n .+ x ) = O )
16	12, 13, 14, 6, 1, 9, 10, 15	grpinva 13532	. . . . . 6 |- ( ( ph /\ ( x e. B /\ ( n e. B /\ ( n .+ x ) = O ) ) ) -> ( x .+ n ) = O )
17	16	oveq1d 6043	. . . . 5 |- ( ( ph /\ ( x e. B /\ ( n e. B /\ ( n .+ x ) = O ) ) ) -> ( ( x .+ n ) .+ x ) = ( O .+ x ) )
18	15	oveq2d 6044	. . . . 5 |- ( ( ph /\ ( x e. B /\ ( n e. B /\ ( n .+ x ) = O ) ) ) -> ( x .+ ( n .+ x ) ) = ( x .+ O ) )
19	11, 17, 18	3eqtr3d 2272	. . . 4 |- ( ( ph /\ ( x e. B /\ ( n e. B /\ ( n .+ x ) = O ) ) ) -> ( O .+ x ) = ( x .+ O ) )
20	19	anassrs 400	. . 3 |- ( ( ( ph /\ x e. B ) /\ ( n e. B /\ ( n .+ x ) = O ) ) -> ( O .+ x ) = ( x .+ O ) )
21	5, 20	rexlimddv 2656	. 2 |- ( ( ph /\ x e. B ) -> ( O .+ x ) = ( x .+ O ) )
22	21, 14	eqtr3d 2266	1 |- ( ( ph /\ x e. B ) -> ( x .+ O ) = x )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2202   E.wrex 2512  (class class class)co 6028

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031

This theorem is referenced by:  isgrpde  13668