Theorem grpridd 6017

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
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 )

Assertion

Ref	Expression
grpridd	|- ( ( 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 grpridd

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

Step	Hyp	Ref	Expression
1		grprinvlem.n	. . . 4 |- ( ( ph /\ x e. B ) -> E. y e. B ( y .+ x ) = O )
2		oveq1 5831	. . . . . 6 |- ( y = n -> ( y .+ x ) = ( n .+ x ) )
3	2	eqeq1d 2166	. . . . 5 |- ( y = n -> ( ( y .+ x ) = O <-> ( n .+ x ) = O ) )
4	3	cbvrexv 2681	. . . 4 |- ( E. y e. B ( y .+ x ) = O <-> E. n e. B ( n .+ x ) = O )
5	1, 4	sylib 121	. . 3 |- ( ( ph /\ x e. B ) -> E. n e. B ( n .+ x ) = O )
6		grprinvlem.a	. . . . . . . 8 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
7	6	caovassg 5979	. . . . . . 7 |- ( ( ph /\ ( u e. B /\ v e. B /\ w e. B ) ) -> ( ( u .+ v ) .+ w ) = ( u .+ ( v .+ w ) ) )
8	7	adantlr 469	. . . . . 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 521	. . . . . 6 |- ( ( ph /\ ( x e. B /\ ( n e. B /\ ( n .+ x ) = O ) ) ) -> x e. B )
10		simprrl 529	. . . . . 6 |- ( ( ph /\ ( x e. B /\ ( n e. B /\ ( n .+ x ) = O ) ) ) -> n e. B )
11	8, 9, 10, 9	caovassd 5980	. . . . 5 |- ( ( ph /\ ( x e. B /\ ( n e. B /\ ( n .+ x ) = O ) ) ) -> ( ( x .+ n ) .+ x ) = ( x .+ ( n .+ x ) ) )
12		grprinvlem.c	. . . . . . 7 |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )
13		grprinvlem.o	. . . . . . 7 |- ( ph -> O e. B )
14		grprinvlem.i	. . . . . . 7 |- ( ( ph /\ x e. B ) -> ( O .+ x ) = x )
15		simprrr 530	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ ( n e. B /\ ( n .+ x ) = O ) ) ) -> ( n .+ x ) = O )
16	12, 13, 14, 6, 1, 9, 10, 15	grprinvd 6016	. . . . . 6 |- ( ( ph /\ ( x e. B /\ ( n e. B /\ ( n .+ x ) = O ) ) ) -> ( x .+ n ) = O )
17	16	oveq1d 5839	. . . . 5 |- ( ( ph /\ ( x e. B /\ ( n e. B /\ ( n .+ x ) = O ) ) ) -> ( ( x .+ n ) .+ x ) = ( O .+ x ) )
18	15	oveq2d 5840	. . . . 5 |- ( ( ph /\ ( x e. B /\ ( n e. B /\ ( n .+ x ) = O ) ) ) -> ( x .+ ( n .+ x ) ) = ( x .+ O ) )
19	11, 17, 18	3eqtr3d 2198	. . . 4 |- ( ( ph /\ ( x e. B /\ ( n e. B /\ ( n .+ x ) = O ) ) ) -> ( O .+ x ) = ( x .+ O ) )
20	19	anassrs 398	. . 3 |- ( ( ( ph /\ x e. B ) /\ ( n e. B /\ ( n .+ x ) = O ) ) -> ( O .+ x ) = ( x .+ O ) )
21	5, 20	rexlimddv 2579	. 2 |- ( ( ph /\ x e. B ) -> ( O .+ x ) = ( x .+ O ) )
22	21, 14	eqtr3d 2192	1 |- ( ( ph /\ x e. B ) -> ( x .+ O ) = x )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   = wceq 1335   e. wcel 2128   E.wrex 2436  (class class class)co 5824

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-iota 5135  df-fv 5178  df-ov 5827

This theorem is referenced by: (None)