Theorem grpinvex 13538

Description: Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
grpcl.b	|- B = ( Base ` G )
grpcl.p	|- .+ = ( +g ` G )
grpinvex.p	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
grpinvex	|- ( ( G e. Grp /\ X e. B ) -> E. y e. B ( y .+ X ) = .0. )

Distinct variable groups:   y, B   y, G   y, X

Allowed substitution hints:   .+ ( y)   .0. ( y)

Proof of Theorem grpinvex

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpcl.b	. . . 4 |- B = ( Base ` G )
2		grpcl.p	. . . 4 |- .+ = ( +g ` G )
3		grpinvex.p	. . . 4 |- .0. = ( 0g ` G )
4	1, 2, 3	isgrp 13534	. . 3 |- ( G e. Grp <-> ( G e. Mnd /\ A. x e. B E. y e. B ( y .+ x ) = .0. ) )
5	4	simprbi 275	. 2 |- ( G e. Grp -> A. x e. B E. y e. B ( y .+ x ) = .0. )
6		oveq2 6008	. . . . 5 |- ( x = X -> ( y .+ x ) = ( y .+ X ) )
7	6	eqeq1d 2238	. . . 4 |- ( x = X -> ( ( y .+ x ) = .0. <-> ( y .+ X ) = .0. ) )
8	7	rexbidv 2531	. . 3 |- ( x = X -> ( E. y e. B ( y .+ x ) = .0. <-> E. y e. B ( y .+ X ) = .0. ) )
9	8	rspccva 2906	. 2 |- ( ( A. x e. B E. y e. B ( y .+ x ) = .0. /\ X e. B ) -> E. y e. B ( y .+ X ) = .0. )
10	5, 9	sylan 283	1 |- ( ( G e. Grp /\ X e. B ) -> E. y e. B ( y .+ X ) = .0. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   ` cfv 5317  (class class class)co 6000   Basecbs 13027   +g cplusg 13105   0gc0g 13284   Mndcmnd 13444   Grpcgrp 13528

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-rab 2517  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-iota 5277  df-fv 5325  df-ov 6003  df-grp 13531

This theorem is referenced by:  dfgrp2  13555  grprcan  13565  grpinveu  13566  grprinv  13579