MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpinvex Unicode version

Theorem grpinvex 14513
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 14509 . . 3  |-  ( G  e.  Grp  <->  ( G  e.  Mnd  /\  A. x  e.  B  E. y  e.  B  ( y  .+  x )  =  .0.  ) )
54simprbi 450 . 2  |-  ( G  e.  Grp  ->  A. x  e.  B  E. y  e.  B  ( y  .+  x )  =  .0.  )
6 oveq2 5882 . . . . 5  |-  ( x  =  X  ->  (
y  .+  x )  =  ( y  .+  X ) )
76eqeq1d 2304 . . . 4  |-  ( x  =  X  ->  (
( y  .+  x
)  =  .0.  <->  ( y  .+  X )  =  .0.  ) )
87rexbidv 2577 . . 3  |-  ( x  =  X  ->  ( E. y  e.  B  ( y  .+  x
)  =  .0.  <->  E. y  e.  B  ( y  .+  X )  =  .0.  ) )
98rspccva 2896 . 2  |-  ( ( A. x  e.  B  E. y  e.  B  ( y  .+  x
)  =  .0.  /\  X  e.  B )  ->  E. y  e.  B  ( y  .+  X
)  =  .0.  )
105, 9sylan 457 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 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Mndcmnd 14377   Grpcgrp 14378
This theorem is referenced by:  grprcan  14531  grpinveu  14532  grprinv  14545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-grp 14505
  Copyright terms: Public domain W3C validator