Theorem grpinvex 13792

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 13788	. . 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 6083	. . . . 5 |- ( x = X -> ( y .+ x ) = ( y .+ X ) )
7	6	eqeq1d 2247	. . . 4 |- ( x = X -> ( ( y .+ x ) = .0. <-> ( y .+ X ) = .0. ) )
8	7	rexbidv 2551	. . 3 |- ( x = X -> ( E. y e. B ( y .+ x ) = .0. <-> E. y e. B ( y .+ X ) = .0. ) )
9	8	rspccva 2928	. 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 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   0gc0g 13587   Mndcmnd 13706   Grpcgrp 13782

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380  df-ov 6078  df-grp 13785

This theorem is referenced by:  dfgrp2  13809  grprcan  13819  grpinveu  13820  grprinv  13833