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Theorem ablomuldiv 24953
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 24949 . . . 4  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  =  ( B G A ) )
323adant3r3 1202 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A G B )  =  ( B G A ) )
43oveq1d 6290 . 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 975 . . 3  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  <->  ( B  e.  X  /\  A  e.  X  /\  C  e.  X )
)
6 ablogrpo 24948 . . . 4  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
7 abldiv.3 . . . . 5  |-  D  =  (  /g  `  G
)
81, 7grpomuldivass 24913 . . . 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 998 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
121, 7grpodivcl 24911 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  C  e.  X )  ->  ( A D C )  e.  X )
136, 12syl3an1 1256 . . . . 5  |-  ( ( G  e.  AbelOp  /\  A  e.  X  /\  C  e.  X )  ->  ( A D C )  e.  X )
14133adant3r2 1201 . . . 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 24949 . . . 4  |-  ( ( G  e.  AbelOp  /\  B  e.  X  /\  ( A D C )  e.  X )  ->  ( B G ( A D C ) )  =  ( ( A D C ) G B ) )
17163expb 1192 . . 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 2505 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 968    = wceq 1374    e. wcel 1762   ran crn 4993   ` cfv 5579  (class class class)co 6275   GrpOpcgr 24850    /g cgs 24853   AbelOpcablo 24945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-grpo 24855  df-gid 24856  df-ginv 24857  df-gdiv 24858  df-ablo 24946
This theorem is referenced by:  ablodivdiv  24954  nvaddsub  25216  ablo4pnp  29932
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