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Theorem subgoov 25449
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 25448 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  =  ( G  |`  ( W  X.  W ) ) )
32oveqd 6235 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( A H B )  =  ( A ( G  |`  ( W  X.  W
) ) B ) )
4 ovres 6363 . 2  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( A ( G  |`  ( W  X.  W
) ) B )  =  ( A G B ) )
53, 4sylan9eq 2457 1  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  ( A  e.  W  /\  B  e.  W )
)  ->  ( A H B )  =  ( A G B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1836    X. cxp 4928   ran crn 4931    |` cres 4932   ` cfv 5513  (class class class)co 6218   SubGrpOpcsubgo 25445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-fo 5519  df-fv 5521  df-ov 6221  df-grpo 25335  df-subgo 25446
This theorem is referenced by:  subgoid  25451  subgoinv  25452  subgoablo  25455  ghsubgolemOLD  25514  ghomgsg  29262
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