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Theorem subgoov 20972
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 20971 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  =  ( G  |`  ( W  X.  W ) ) )
32oveqd 5875 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( A H B )  =  ( A ( G  |`  ( W  X.  W
) ) B ) )
4 ovres 5987 . 2  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( A ( G  |`  ( W  X.  W
) ) B )  =  ( A G B ) )
53, 4sylan9eq 2335 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 358    = wceq 1623    e. wcel 1684    X. cxp 4687   ran crn 4690    |` cres 4691   ` cfv 5255  (class class class)co 5858   SubGrpOpcsubgo 20968
This theorem is referenced by:  subgoid  20974  subgoinv  20975  subgoablo  20978  ghsubgolem  21037  ghomgsg  24000
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-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-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-fo 5261  df-fv 5263  df-ov 5861  df-grpo 20858  df-subgo 20969
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