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Theorem grpinvex 12777
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 12773 . . 3  |-  ( G  e.  Grp  <->  ( G  e.  Mnd  /\  A. x  e.  B  E. y  e.  B  ( y  .+  x )  =  .0.  ) )
54simprbi 275 . 2  |-  ( G  e.  Grp  ->  A. x  e.  B  E. y  e.  B  ( y  .+  x )  =  .0.  )
6 oveq2 5877 . . . . 5  |-  ( x  =  X  ->  (
y  .+  x )  =  ( y  .+  X ) )
76eqeq1d 2186 . . . 4  |-  ( x  =  X  ->  (
( y  .+  x
)  =  .0.  <->  ( y  .+  X )  =  .0.  ) )
87rexbidv 2478 . . 3  |-  ( x  =  X  ->  ( E. y  e.  B  ( y  .+  x
)  =  .0.  <->  E. y  e.  B  ( y  .+  X )  =  .0.  ) )
98rspccva 2840 . 2  |-  ( ( A. x  e.  B  E. y  e.  B  ( y  .+  x
)  =  .0.  /\  X  e.  B )  ->  E. y  e.  B  ( y  .+  X
)  =  .0.  )
105, 9sylan 283 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 104    = wceq 1353    e. wcel 2148   A.wral 2455   E.wrex 2456   ` cfv 5212  (class class class)co 5869   Basecbs 12445   +g cplusg 12518   0gc0g 12653   Mndcmnd 12709   Grpcgrp 12767
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-iota 5174  df-fv 5220  df-ov 5872  df-grp 12770
This theorem is referenced by:  dfgrp2  12792  grprcan  12800  grpinveu  12801  grprinv  12813
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