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Theorem ablodivdiv4 20958
Description: Law for double group division. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
abldiv.1  |-  X  =  ran  G
abldiv.3  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
ablodivdiv4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D B ) D C )  =  ( A D ( B G C ) ) )

Proof of Theorem ablodivdiv4
StepHypRef Expression
1 ablogrpo 20951 . . 3  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
2 simpl 443 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  G  e.  GrpOp
)
3 abldiv.1 . . . . . 6  |-  X  =  ran  G
4 abldiv.3 . . . . . 6  |-  D  =  (  /g  `  G
)
53, 4grpodivcl 20914 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  X )
653adant3r3 1162 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D B )  e.  X
)
7 simpr3 963 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
8 eqid 2283 . . . . 5  |-  ( inv `  G )  =  ( inv `  G )
93, 8, 4grpodivval 20910 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A D B )  e.  X  /\  C  e.  X )  ->  (
( A D B ) D C )  =  ( ( A D B ) G ( ( inv `  G
) `  C )
) )
102, 6, 7, 9syl3anc 1182 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D B ) D C )  =  ( ( A D B ) G ( ( inv `  G ) `
 C ) ) )
111, 10sylan 457 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D B ) D C )  =  ( ( A D B ) G ( ( inv `  G ) `
 C ) ) )
12 simpr1 961 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
13 simpr2 962 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
14 simp3 957 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  C  e.  X )
153, 8grpoinvcl 20893 . . . . 5  |-  ( ( G  e.  GrpOp  /\  C  e.  X )  ->  (
( inv `  G
) `  C )  e.  X )
161, 14, 15syl2an 463 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  C )  e.  X
)
1712, 13, 163jca 1132 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A  e.  X  /\  B  e.  X  /\  ( ( inv `  G ) `
 C )  e.  X ) )
183, 4ablodivdiv 20957 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( inv `  G
) `  C )  e.  X ) )  -> 
( A D ( B D ( ( inv `  G ) `
 C ) ) )  =  ( ( A D B ) G ( ( inv `  G ) `  C
) ) )
1917, 18syldan 456 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D ( B D ( ( inv `  G
) `  C )
) )  =  ( ( A D B ) G ( ( inv `  G ) `
 C ) ) )
203, 8, 4grpodivinv 20911 . . . . 5  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B D ( ( inv `  G ) `  C
) )  =  ( B G C ) )
211, 20syl3an1 1215 . . . 4  |-  ( ( G  e.  AbelOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B D ( ( inv `  G ) `  C
) )  =  ( B G C ) )
22213adant3r1 1160 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B D ( ( inv `  G ) `  C
) )  =  ( B G C ) )
2322oveq2d 5874 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D ( B D ( ( inv `  G
) `  C )
) )  =  ( A D ( B G C ) ) )
2411, 19, 233eqtr2d 2321 1  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D B ) D C )  =  ( A D ( B G C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853   invcgn 20855    /g cgs 20856   AbelOpcablo 20948
This theorem is referenced by:  ablodiv32  20959  ablonnncan  20960  fprodsub  25379  ablo4pnp  26570
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-pow 4188  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-pw 3627  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-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949
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