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Theorem ghsubablo 21921
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 21919 . . 3  |-  ( ph  ->  ( H  e.  GrpOp  /\  ( S  e.  AbelOp  ->  H  e.  AbelOp ) ) )
1211simprd 450 . 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 359    = wceq 1649    e. wcel 1721    C_ wss 3288    X. cxp 4843   ran crn 4846    |` cres 4847   "cima 4848    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6048   GrpOpcgr 21735   AbelOpcablo 21830   SubGrpOpcsubgo 21850
This theorem is referenced by:  efghgrp  21922
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-grpo 21740  df-gid 21741  df-ginv 21742  df-gdiv 21743  df-ablo 21831  df-subgo 21851
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