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Theorem grporinv 21817
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 21815 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( ( N `  A ) G A )  =  U  /\  ( A G ( N `
 A ) )  =  U ) )
54simprd 450 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( N `  A ) )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ran crn 4879   ` cfv 5454  (class class class)co 6081   GrpOpcgr 21774  GIdcgi 21775   invcgn 21776
This theorem is referenced by:  grpoinvid1  21818  grpoinvid2  21819  grpoasscan1  21825  grpo2inv  21827  grpoinvop  21829  grpodivid  21838  grpopnpcan2  21841  subgoinv  21896  vcm  22050  vcrinv  22051  nvrinv  22134  ghomgrpilem2  25097  ghomf1olem  25105  rngoaddneg1  26562
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
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-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-riota 6549  df-grpo 21779  df-gid 21780  df-ginv 21781
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