Theorem grpinvex 13417

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 13413	. . 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 5965	. . . . 5 |- ( x = X -> ( y .+ x ) = ( y .+ X ) )
7	6	eqeq1d 2215	. . . 4 |- ( x = X -> ( ( y .+ x ) = .0. <-> ( y .+ X ) = .0. ) )
8	7	rexbidv 2508	. . 3 |- ( x = X -> ( E. y e. B ( y .+ x ) = .0. <-> E. y e. B ( y .+ X ) = .0. ) )
9	8	rspccva 2880	. 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 1373   e. wcel 2177   A.wral 2485   E.wrex 2486   ` cfv 5280  (class class class)co 5957   Basecbs 12907   +g cplusg 12984   0gc0g 13163   Mndcmnd 13323   Grpcgrp 13407

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-un 3174  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-iota 5241  df-fv 5288  df-ov 5960  df-grp 13410

This theorem is referenced by:  dfgrp2  13434  grprcan  13444  grpinveu  13445  grprinv  13458