MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subgores Unicode version

Theorem subgores 20971
Description: A subgroup operation is the restriction of its parent group operation to its underlying set. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
subgores.1  |-  W  =  ran  H
Assertion
Ref Expression
subgores  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  =  ( G  |`  ( W  X.  W ) ) )

Proof of Theorem subgores
StepHypRef Expression
1 issubgo 20970 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)
21simp1bi 970 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  GrpOp
)
3 eqid 2283 . . . . 5  |-  ran  G  =  ran  G
43grpofo 20866 . . . 4  |-  ( G  e.  GrpOp  ->  G :
( ran  G  X.  ran  G ) -onto-> ran  G
)
5 fofun 5452 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) -onto-> ran 
G  ->  Fun  G )
62, 4, 53syl 18 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  Fun  G )
71simp3bi 972 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  C_  G
)
81simp2bi 971 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  e.  GrpOp
)
9 subgores.1 . . . . . 6  |-  W  =  ran  H
109grpofo 20866 . . . . 5  |-  ( H  e.  GrpOp  ->  H :
( W  X.  W
) -onto-> W )
11 fof 5451 . . . . 5  |-  ( H : ( W  X.  W ) -onto-> W  ->  H : ( W  X.  W ) --> W )
128, 10, 113syl 18 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H :
( W  X.  W
) --> W )
13 fdm 5393 . . . 4  |-  ( H : ( W  X.  W ) --> W  ->  dom  H  =  ( W  X.  W ) )
14 eqimss2 3231 . . . 4  |-  ( dom 
H  =  ( W  X.  W )  -> 
( W  X.  W
)  C_  dom  H )
1512, 13, 143syl 18 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( W  X.  W )  C_  dom  H )
16 fun2ssres 5295 . . 3  |-  ( ( Fun  G  /\  H  C_  G  /\  ( W  X.  W )  C_  dom  H )  ->  ( G  |`  ( W  X.  W ) )  =  ( H  |`  ( W  X.  W ) ) )
176, 7, 15, 16syl3anc 1182 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( G  |`  ( W  X.  W
) )  =  ( H  |`  ( W  X.  W ) ) )
18 fofn 5453 . . . 4  |-  ( H : ( W  X.  W ) -onto-> W  ->  H  Fn  ( W  X.  W ) )
19 fnresdm 5353 . . . 4  |-  ( H  Fn  ( W  X.  W )  ->  ( H  |`  ( W  X.  W ) )  =  H )
2018, 19syl 15 . . 3  |-  ( H : ( W  X.  W ) -onto-> W  -> 
( H  |`  ( W  X.  W ) )  =  H )
218, 10, 203syl 18 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( H  |`  ( W  X.  W
) )  =  H )
2217, 21eqtr2d 2316 1  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  =  ( G  |`  ( W  X.  W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    C_ wss 3152    X. cxp 4687   dom cdm 4689   ran crn 4690    |` cres 4691   Fun wfun 5249    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   ` cfv 5255   GrpOpcgr 20853   SubGrpOpcsubgo 20968
This theorem is referenced by:  subgoov  20972  subgornss  20973
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
  Copyright terms: Public domain W3C validator