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Theorem subgoov 21883
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 21882 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  =  ( G  |`  ( W  X.  W ) ) )
32oveqd 6090 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( A H B )  =  ( A ( G  |`  ( W  X.  W
) ) B ) )
4 ovres 6205 . 2  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( A ( G  |`  ( W  X.  W
) ) B )  =  ( A G B ) )
53, 4sylan9eq 2487 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 1652    e. wcel 1725    X. cxp 4868   ran crn 4871    |` cres 4872   ` cfv 5446  (class class class)co 6073   SubGrpOpcsubgo 21879
This theorem is referenced by:  subgoid  21885  subgoinv  21886  subgoablo  21889  ghsubgolem  21948  ghomgsg  25094
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-grpo 21769  df-subgo 21880
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