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Theorem ghsubgo 21038
Description: The image of a subgroup  S of group  G under a group homomorphism  F on  G is a group. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
ghsubgo.1  |-  ( ph  ->  S  e.  ( SubGrpOp `  G ) )
ghsubgo.2  |-  X  =  ran  G
ghsubgo.3  |-  ( ph  ->  F : X --> Y )
ghsubgo.4  |-  ( ph  ->  Y  C_  A )
ghsubgo.5  |-  ( ph  ->  O  Fn  ( A  X.  A ) )
ghsubgo.6  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
ghsubgo.7  |-  Z  =  ran  S
ghsubgo.8  |-  W  =  ( F " Z
)
ghsubgo.9  |-  H  =  ( O  |`  ( W  X.  W ) )
Assertion
Ref Expression
ghsubgo  |-  ( ph  ->  H  e.  GrpOp )
Distinct variable groups:    x, y, F    x, H, y    x, O, y    x, S, y   
x, W, y    x, Z, y    ph, x, y
Allowed substitution hints:    A( x, y)    G( x, y)    X( x, y)    Y( x, y)

Proof of Theorem ghsubgo
StepHypRef Expression
1 ghsubgo.1 . . 3  |-  ( ph  ->  S  e.  ( SubGrpOp `  G ) )
2 ghsubgo.2 . . 3  |-  X  =  ran  G
3 ghsubgo.3 . . 3  |-  ( ph  ->  F : X --> Y )
4 ghsubgo.4 . . 3  |-  ( ph  ->  Y  C_  A )
5 ghsubgo.5 . . 3  |-  ( ph  ->  O  Fn  ( A  X.  A ) )
6 ghsubgo.6 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
7 ghsubgo.7 . . 3  |-  Z  =  ran  S
8 ghsubgo.8 . . 3  |-  W  =  ( F " Z
)
9 ghsubgo.9 . . 3  |-  H  =  ( O  |`  ( W  X.  W ) )
101, 2, 3, 4, 5, 6, 7, 8, 9ghsubgolem 21037 . 2  |-  ( ph  ->  ( H  e.  GrpOp  /\  ( S  e.  AbelOp  ->  H  e.  AbelOp ) ) )
1110simpld 445 1  |-  ( ph  ->  H  e.  GrpOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152    X. cxp 4687   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853   AbelOpcablo 20948   SubGrpOpcsubgo 20968
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-rep 4131  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-reu 2550  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-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-subgo 20969
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