MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ablodivdiv4 Unicode version

Theorem ablodivdiv4 21862
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 21855 . . 3  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
2 simpl 444 . . . 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 21818 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e.  X )
653adant3r3 1164 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D B )  e.  X
)
7 simpr3 965 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
8 eqid 2430 . . . . 5  |-  ( inv `  G )  =  ( inv `  G )
93, 8, 4grpodivval 21814 . . . 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 1184 . . 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 458 . 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 963 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
13 simpr2 964 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
14 simp3 959 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  C  e.  X )
153, 8grpoinvcl 21797 . . . . 5  |-  ( ( G  e.  GrpOp  /\  C  e.  X )  ->  (
( inv `  G
) `  C )  e.  X )
161, 14, 15syl2an 464 . . . 4  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  C )  e.  X
)
1712, 13, 163jca 1134 . . 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 21861 . . 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 457 . 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 21815 . . . . 5  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B D ( ( inv `  G ) `  C
) )  =  ( B G C ) )
211, 20syl3an1 1217 . . . 4  |-  ( ( G  e.  AbelOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B D ( ( inv `  G ) `  C
) )  =  ( B G C ) )
22213adant3r1 1162 . . 3  |-  ( ( G  e.  AbelOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B D ( ( inv `  G ) `  C
) )  =  ( B G C ) )
2322oveq2d 6083 . 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 2468 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   ran crn 4865   ` cfv 5440  (class class class)co 6067   GrpOpcgr 21757   invcgn 21759    /g cgs 21760   AbelOpcablo 21852
This theorem is referenced by:  ablodiv32  21863  ablonnncan  21864  ablo4pnp  26487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-reu 2699  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-riota 6535  df-grpo 21762  df-gid 21763  df-ginv 21764  df-gdiv 21765  df-ablo 21853
  Copyright terms: Public domain W3C validator