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Theorem subgoov 20988
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 20987 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  =  ( G  |`  ( W  X.  W ) ) )
32oveqd 5891 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( A H B )  =  ( A ( G  |`  ( W  X.  W
) ) B ) )
4 ovres 6003 . 2  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( A ( G  |`  ( W  X.  W
) ) B )  =  ( A G B ) )
53, 4sylan9eq 2348 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 1632    e. wcel 1696    X. cxp 4703   ran crn 4706    |` cres 4707   ` cfv 5271  (class class class)co 5874   SubGrpOpcsubgo 20984
This theorem is referenced by:  subgoid  20990  subgoinv  20991  subgoablo  20994  ghsubgolem  21053  ghomgsg  24015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-grpo 20874  df-subgo 20985
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