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Theorem ablomuldiv 23798
Description: Law for group multiplication and division. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
abldiv.1  |-  X  =  ran  G
abldiv.3  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
ablomuldiv  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) D C )  =  ( ( A D C ) G B ) )

Proof of Theorem ablomuldiv
StepHypRef Expression
1 abldiv.1 . . . . 5  |-  X  =  ran  G
21ablocom 23794 . . . 4  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )
323adant3r3 1198 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A G B )  =  ( B G A ) )
43oveq1d 6127 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) D C )  =  ( ( B G A ) D C ) )
5 3ancoma 972 . . 3  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  <->  ( B  e.  X  /\  A  e.  X  /\  C  e.  X )
)
6 ablogrpo 23793 . . . 4  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
7 abldiv.3 . . . . 5  |-  D  =  (  /g  `  G
)
81, 7grpomuldivass 23758 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  A  e.  X  /\  C  e.  X )
)  ->  ( ( B G A ) D C )  =  ( B G ( A D C ) ) )
96, 8sylan 471 . . 3  |-  ( ( G  e.  AbelOp  /\  ( B  e.  X  /\  A  e.  X  /\  C  e.  X )
)  ->  ( ( B G A ) D C )  =  ( B G ( A D C ) ) )
105, 9sylan2b 475 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( B G A ) D C )  =  ( B G ( A D C ) ) )
11 simpr2 995 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
121, 7grpodivcl 23756 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  C  e.  X )  ->  ( A D C )  e.  X )
136, 12syl3an1 1251 . . . . 5  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  C  e.  X )  ->  ( A D C )  e.  X )
14133adant3r2 1197 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D C )  e.  X
)
1511, 14jca 532 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B  e.  X  /\  ( A D C )  e.  X ) )
161ablocom 23794 . . . 4  |-  ( ( G  e.  AbelOp  /\  B  e.  X  /\  ( A D C )  e.  X )  ->  ( B G ( A D C ) )  =  ( ( A D C ) G B ) )
17163expb 1188 . . 3  |-  ( ( G  e.  AbelOp  /\  ( B  e.  X  /\  ( A D C )  e.  X ) )  ->  ( B G ( A D C ) )  =  ( ( A D C ) G B ) )
1815, 17syldan 470 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B G ( A D C ) )  =  ( ( A D C ) G B ) )
194, 10, 183eqtrd 2479 1  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) D C )  =  ( ( A D C ) G B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ran crn 4862   ` cfv 5439  (class class class)co 6112   GrpOpcgr 23695    /g cgs 23698   AbelOpcablo 23790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-grpo 23700  df-gid 23701  df-ginv 23702  df-gdiv 23703  df-ablo 23791
This theorem is referenced by:  ablodivdiv  23799  nvaddsub  24061  ablo4pnp  28771
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