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Theorem grponpcan 20919
Description: Cancellation law for group division. (npcan 9060 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdivf.1  |-  X  =  ran  G
grpdivf.3  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
grponpcan  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D B ) G B )  =  A )

Proof of Theorem grponpcan
StepHypRef Expression
1 grpdivf.1 . . . 4  |-  X  =  ran  G
2 eqid 2283 . . . 4  |-  ( inv `  G )  =  ( inv `  G )
3 grpdivf.3 . . . 4  |-  D  =  (  /g  `  G
)
41, 2, 3grpodivval 20910 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( ( inv `  G
) `  B )
) )
54oveq1d 5873 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D B ) G B )  =  ( ( A G ( ( inv `  G ) `  B
) ) G B ) )
6 simp1 955 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  G  e.  GrpOp )
7 simp2 956 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
81, 2grpoinvcl 20893 . . . . 5  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  (
( inv `  G
) `  B )  e.  X )
983adant2 974 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( inv `  G
) `  B )  e.  X )
10 simp3 957 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  B  e.  X )
111grpoass 20870 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  ( ( inv `  G
) `  B )  e.  X  /\  B  e.  X ) )  -> 
( ( A G ( ( inv `  G
) `  B )
) G B )  =  ( A G ( ( ( inv `  G ) `  B
) G B ) ) )
126, 7, 9, 10, 11syl13anc 1184 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G ( ( inv `  G
) `  B )
) G B )  =  ( A G ( ( ( inv `  G ) `  B
) G B ) ) )
13 eqid 2283 . . . . . . 7  |-  (GId `  G )  =  (GId
`  G )
141, 13, 2grpolinv 20895 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  (
( ( inv `  G
) `  B ) G B )  =  (GId
`  G ) )
1514oveq2d 5874 . . . . 5  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( A G ( ( ( inv `  G ) `
 B ) G B ) )  =  ( A G (GId
`  G ) ) )
16153adant2 974 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( ( ( inv `  G ) `
 B ) G B ) )  =  ( A G (GId
`  G ) ) )
171, 13grporid 20887 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
18173adant3 975 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G (GId `  G
) )  =  A )
1916, 18eqtrd 2315 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( ( ( inv `  G ) `
 B ) G B ) )  =  A )
2012, 19eqtrd 2315 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G ( ( inv `  G
) `  B )
) G B )  =  A )
215, 20eqtrd 2315 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D B ) G B )  =  A )
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  GIdcgi 20854   invcgn 20855    /g cgs 20856
This theorem is referenced by:  grponpncan  20922  grpodiveq  20923  ablonnncan  20960  ghgrp  21035  trinv  25395  grpoeqdivid  26571  ghomdiv  26574
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
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