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Theorem grpinvex 14748
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 14744 . . 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 6029 . . . . 5  |-  ( x  =  X  ->  (
y  .+  x )  =  ( y  .+  X ) )
76eqeq1d 2396 . . . 4  |-  ( x  =  X  ->  (
( y  .+  x
)  =  .0.  <->  ( y  .+  X )  =  .0.  ) )
87rexbidv 2671 . . 3  |-  ( x  =  X  ->  ( E. y  e.  B  ( y  .+  x
)  =  .0.  <->  E. y  e.  B  ( y  .+  X )  =  .0.  ) )
98rspccva 2995 . 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 1649    e. wcel 1717   A.wral 2650   E.wrex 2651   ` cfv 5395  (class class class)co 6021   Basecbs 13397   +g cplusg 13457   0gc0g 13651   Mndcmnd 14612   Grpcgrp 14613
This theorem is referenced by:  grprcan  14766  grpinveu  14767  grprinv  14780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-iota 5359  df-fv 5403  df-ov 6024  df-grp 14740
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