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Theorem grporinv 20912
Description: The right inverse of a group element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinv.1  |-  X  =  ran  G
grpinv.2  |-  U  =  (GId `  G )
grpinv.3  |-  N  =  ( inv `  G
)
Assertion
Ref Expression
grporinv  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( N `  A ) )  =  U )

Proof of Theorem grporinv
StepHypRef Expression
1 grpinv.1 . . 3  |-  X  =  ran  G
2 grpinv.2 . . 3  |-  U  =  (GId `  G )
3 grpinv.3 . . 3  |-  N  =  ( inv `  G
)
41, 2, 3grpoinv 20910 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( ( N `  A ) G A )  =  U  /\  ( A G ( N `
 A ) )  =  U ) )
54simprd 449 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( N `  A ) )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   ran crn 4706   ` cfv 5271  (class class class)co 5874   GrpOpcgr 20869  GIdcgi 20870   invcgn 20871
This theorem is referenced by:  grpoinvid1  20913  grpoinvid2  20914  grpoasscan1  20920  grpo2inv  20922  grpoinvop  20924  grpodivid  20933  grpopnpcan2  20936  subgoinv  20991  vcm  21143  vcrinv  21144  nvrinv  21227  ghomgrpilem2  24008  ghomf1olem  24016  ltrran2  25506  ltrinvlem  25509  rltrran  25517  multinv  25525  multinvb  25526  mulinvsca  25583  rngoaddneg1  26680
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
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-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320  df-grpo 20874  df-gid 20875  df-ginv 20876
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