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Theorem grpinvex 14812
 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
grpcl.p
grpinvex.p
Assertion
Ref Expression
grpinvex
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem grpinvex
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 grpcl.b . . . 4
2 grpcl.p . . . 4
3 grpinvex.p . . . 4
41, 2, 3isgrp 14808 . . 3
54simprbi 451 . 2
6 oveq2 6081 . . . . 5
76eqeq1d 2443 . . . 4
87rexbidv 2718 . . 3
98rspccva 3043 . 2
105, 9sylan 458 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2697  wrex 2698  cfv 5446  (class class class)co 6073  cbs 13461   cplusg 13521  c0g 13715  cmnd 14676  cgrp 14677 This theorem is referenced by:  grprcan  14830  grpinveu  14831  grprinv  14844 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-grp 14804
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