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Theorem ghsubablo 21965
Description: The image of an Abelian subgroup  S of group  G under a group homomorphism  F on  G is an Abelian group. (Contributed 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 ) )
ghsubablo.10  |-  ( ph  ->  S  e.  AbelOp )
Assertion
Ref Expression
ghsubablo  |-  ( ph  ->  H  e.  AbelOp )
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 ghsubablo
StepHypRef Expression
1 ghsubablo.10 . 2  |-  ( ph  ->  S  e.  AbelOp )
2 ghsubgo.1 . . . 4  |-  ( ph  ->  S  e.  ( SubGrpOp `  G ) )
3 ghsubgo.2 . . . 4  |-  X  =  ran  G
4 ghsubgo.3 . . . 4  |-  ( ph  ->  F : X --> Y )
5 ghsubgo.4 . . . 4  |-  ( ph  ->  Y  C_  A )
6 ghsubgo.5 . . . 4  |-  ( ph  ->  O  Fn  ( A  X.  A ) )
7 ghsubgo.6 . . . 4  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
8 ghsubgo.7 . . . 4  |-  Z  =  ran  S
9 ghsubgo.8 . . . 4  |-  W  =  ( F " Z
)
10 ghsubgo.9 . . . 4  |-  H  =  ( O  |`  ( W  X.  W ) )
112, 3, 4, 5, 6, 7, 8, 9, 10ghsubgolem 21963 . . 3  |-  ( ph  ->  ( H  e.  GrpOp  /\  ( S  e.  AbelOp  ->  H  e.  AbelOp ) ) )
1211simprd 451 . 2  |-  ( ph  ->  ( S  e.  AbelOp  ->  H  e.  AbelOp ) )
131, 12mpd 15 1  |-  ( ph  ->  H  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3322    X. cxp 4879   ran crn 4882    |` cres 4883   "cima 4884    Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084   GrpOpcgr 21779   AbelOpcablo 21874   SubGrpOpcsubgo 21894
This theorem is referenced by:  efghgrp  21966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-grpo 21784  df-gid 21785  df-ginv 21786  df-gdiv 21787  df-ablo 21875  df-subgo 21895
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