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Theorem subgoov 21742
Description: The result of a subgroup operation is the same as the result of its parent operation. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 8-Jul-2014.) (New usage is discouraged.)
Hypothesis
Ref Expression
subgores.1  |-  W  =  ran  H
Assertion
Ref Expression
subgoov  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  ( A  e.  W  /\  B  e.  W )
)  ->  ( A H B )  =  ( A G B ) )

Proof of Theorem subgoov
StepHypRef Expression
1 subgores.1 . . . 4  |-  W  =  ran  H
21subgores 21741 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  =  ( G  |`  ( W  X.  W ) ) )
32oveqd 6038 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( A H B )  =  ( A ( G  |`  ( W  X.  W
) ) B ) )
4 ovres 6153 . 2  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( A ( G  |`  ( W  X.  W
) ) B )  =  ( A G B ) )
53, 4sylan9eq 2440 1  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  ( A  e.  W  /\  B  e.  W )
)  ->  ( A H B )  =  ( A G B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    X. cxp 4817   ran crn 4820    |` cres 4821   ` cfv 5395  (class class class)co 6021   SubGrpOpcsubgo 21738
This theorem is referenced by:  subgoid  21744  subgoinv  21745  subgoablo  21748  ghsubgolem  21807  ghomgsg  24884
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 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fo 5401  df-fv 5403  df-ov 6024  df-grpo 21628  df-subgo 21739
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