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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  |-  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.
StepHypRef Expression
1 grpcl.b . . . 4  |-  B  =  ( Base `  G
)
2 grpcl.p . . . 4  |-  .+  =  ( +g  `  G )
3 grpinvex.p . . . 4  |-  .0.  =  ( 0g `  G )
41, 2, 3isgrp 14808 . . 3  |-  ( G  e.  Grp  <->  ( G  e.  Mnd  /\  A. x  e.  B  E. y  e.  B  ( y  .+  x )  =  .0.  ) )
54simprbi 451 . 2  |-  ( G  e.  Grp  ->  A. x  e.  B  E. y  e.  B  ( y  .+  x )  =  .0.  )
6 oveq2 6081 . . . . 5  |-  ( x  =  X  ->  (
y  .+  x )  =  ( y  .+  X ) )
76eqeq1d 2443 . . . 4  |-  ( x  =  X  ->  (
( y  .+  x
)  =  .0.  <->  ( y  .+  X )  =  .0.  ) )
87rexbidv 2718 . . 3  |-  ( x  =  X  ->  ( E. y  e.  B  ( y  .+  x
)  =  .0.  <->  E. y  e.  B  ( y  .+  X )  =  .0.  ) )
98rspccva 3043 . 2  |-  ( ( A. x  e.  B  E. y  e.  B  ( y  .+  x
)  =  .0.  /\  X  e.  B )  ->  E. y  e.  B  ( y  .+  X
)  =  .0.  )
105, 9sylan 458 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  E. y  e.  B  ( y  .+  X
)  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   0gc0g 13715   Mndcmnd 14676   Grpcgrp 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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