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Theorem abloinvop 24765
Description: The inverse of the abelian group operation doesn't reverse the arguments. cf grpoinvop 20908. (Contributed by FL, 14-Sep-2010.)
Hypotheses
Ref Expression
ablinvop.1  |-  X  =  ran  G
ablinvop.2  |-  N  =  ( inv `  G
)
Assertion
Ref Expression
abloinvop  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G B ) )  =  ( ( N `  A ) G ( N `  B ) ) )

Proof of Theorem abloinvop
StepHypRef Expression
1 ablogrpo 20951 . . 3  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
2 ablinvop.1 . . . 4  |-  X  =  ran  G
3 ablinvop.2 . . . 4  |-  N  =  ( inv `  G
)
42, 3grpoinvop 20908 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G B ) )  =  ( ( N `  B ) G ( N `  A ) ) )
51, 4syl3an1 1215 . 2  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G B ) )  =  ( ( N `  B ) G ( N `  A ) ) )
6 simp1 955 . . 3  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  G  e.  AbelOp )
72, 3grpoinvcl 20893 . . . . 5  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( N `  B )  e.  X )
873adant2 974 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  B )  e.  X )
91, 8syl3an1 1215 . . 3  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  B )  e.  X )
102, 3grpoinvcl 20893 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
11103adant3 975 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  A )  e.  X )
121, 11syl3an1 1215 . . 3  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  A )  e.  X )
132ablocom 20952 . . 3  |-  ( ( G  e.  AbelOp  /\  ( N `  B )  e.  X  /\  ( N `  A )  e.  X )  ->  (
( N `  B
) G ( N `
 A ) )  =  ( ( N `
 A ) G ( N `  B
) ) )
146, 9, 12, 13syl3anc 1182 . 2  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  B
) G ( N `
 A ) )  =  ( ( N `
 A ) G ( N `  B
) ) )
155, 14eqtrd 2315 1  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A G B ) )  =  ( ( N `  A ) G ( N `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853   invcgn 20855   AbelOpcablo 20948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949
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