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Theorem ablodiv32 21721
Description: Swap the second and third terms in a double 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
ablodiv32  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D B ) D C )  =  ( ( A D C ) D B ) )

Proof of Theorem ablodiv32
StepHypRef Expression
1 abldiv.1 . . . . 5  |-  X  =  ran  G
21ablocom 21714 . . . 4  |-  ( ( G  e.  AbelOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B G C )  =  ( C G B ) )
323adant3r1 1162 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B G C )  =  ( C G B ) )
43oveq2d 6029 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D ( B G C ) )  =  ( A D ( C G B ) ) )
5 abldiv.3 . . 3  |-  D  =  (  /g  `  G
)
61, 5ablodivdiv4 21720 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D B ) D C )  =  ( A D ( B G C ) ) )
7 3ancomb 945 . . 3  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  <->  ( A  e.  X  /\  C  e.  X  /\  B  e.  X )
)
81, 5ablodivdiv4 21720 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  C  e.  X  /\  B  e.  X )
)  ->  ( ( A D C ) D B )  =  ( A D ( C G B ) ) )
97, 8sylan2b 462 . 2  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D C ) D B )  =  ( A D ( C G B ) ) )
104, 6, 93eqtr4d 2422 1  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D B ) D C )  =  ( ( A D C ) D B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ran crn 4812   ` cfv 5387  (class class class)co 6013    /g cgs 21618   AbelOpcablo 21710
This theorem is referenced by:  ablonnncan1  21724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-grpo 21620  df-gid 21621  df-ginv 21622  df-gdiv 21623  df-ablo 21711
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