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Theorem ghsubgolem 21911
Description: The image of a subgroup  S of group  G under a group homomorphism  F on  G is a group, and furthermore is Abelian if  S is Abelian. (Contributed by Paul Chapman, 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
ghsubgolem  |-  ( ph  ->  ( H  e.  GrpOp  /\  ( S  e.  AbelOp  ->  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 ghsubgolem
StepHypRef Expression
1 ghsubgo.3 . . . . 5  |-  ( ph  ->  F : X --> Y )
2 ffun 5552 . . . . 5  |-  ( F : X --> Y  ->  Fun  F )
31, 2syl 16 . . . 4  |-  ( ph  ->  Fun  F )
4 ghsubgo.1 . . . . . 6  |-  ( ph  ->  S  e.  ( SubGrpOp `  G ) )
5 ghsubgo.2 . . . . . . 7  |-  X  =  ran  G
6 ghsubgo.7 . . . . . . 7  |-  Z  =  ran  S
75, 6subgornss 21847 . . . . . 6  |-  ( S  e.  ( SubGrpOp `  G
)  ->  Z  C_  X
)
84, 7syl 16 . . . . 5  |-  ( ph  ->  Z  C_  X )
9 fdm 5554 . . . . . 6  |-  ( F : X --> Y  ->  dom  F  =  X )
101, 9syl 16 . . . . 5  |-  ( ph  ->  dom  F  =  X )
118, 10sseqtr4d 3345 . . . 4  |-  ( ph  ->  Z  C_  dom  F )
12 fores 5621 . . . 4  |-  ( ( Fun  F  /\  Z  C_ 
dom  F )  -> 
( F  |`  Z ) : Z -onto-> ( F
" Z ) )
133, 11, 12syl2anc 643 . . 3  |-  ( ph  ->  ( F  |`  Z ) : Z -onto-> ( F
" Z ) )
14 ssel2 3303 . . . . . . 7  |-  ( ( Z  C_  X  /\  x  e.  Z )  ->  x  e.  X )
15 ssel2 3303 . . . . . . 7  |-  ( ( Z  C_  X  /\  y  e.  Z )  ->  y  e.  X )
1614, 15anim12dan 811 . . . . . 6  |-  ( ( Z  C_  X  /\  ( x  e.  Z  /\  y  e.  Z
) )  ->  (
x  e.  X  /\  y  e.  X )
)
178, 16sylan 458 . . . . 5  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( x  e.  X  /\  y  e.  X
) )
18 ghsubgo.6 . . . . 5  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
1917, 18syldan 457 . . . 4  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
20 issubgo 21844 . . . . . . . . 9  |-  ( S  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  G )
)
2120simp2bi 973 . . . . . . . 8  |-  ( S  e.  ( SubGrpOp `  G
)  ->  S  e.  GrpOp
)
224, 21syl 16 . . . . . . 7  |-  ( ph  ->  S  e.  GrpOp )
236grpocl 21741 . . . . . . . 8  |-  ( ( S  e.  GrpOp  /\  x  e.  Z  /\  y  e.  Z )  ->  (
x S y )  e.  Z )
24233expb 1154 . . . . . . 7  |-  ( ( S  e.  GrpOp  /\  (
x  e.  Z  /\  y  e.  Z )
)  ->  ( x S y )  e.  Z )
2522, 24sylan 458 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( x S y )  e.  Z )
26 fvres 5704 . . . . . 6  |-  ( ( x S y )  e.  Z  ->  (
( F  |`  Z ) `
 ( x S y ) )  =  ( F `  (
x S y ) ) )
2725, 26syl 16 . . . . 5  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( ( F  |`  Z ) `  (
x S y ) )  =  ( F `
 ( x S y ) ) )
286subgoov 21846 . . . . . . 7  |-  ( ( S  e.  ( SubGrpOp `  G )  /\  (
x  e.  Z  /\  y  e.  Z )
)  ->  ( x S y )  =  ( x G y ) )
294, 28sylan 458 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( x S y )  =  ( x G y ) )
3029fveq2d 5691 . . . . 5  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( F `  (
x S y ) )  =  ( F `
 ( x G y ) ) )
3127, 30eqtrd 2436 . . . 4  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( ( F  |`  Z ) `  (
x S y ) )  =  ( F `
 ( x G y ) ) )
32 fvres 5704 . . . . . 6  |-  ( x  e.  Z  ->  (
( F  |`  Z ) `
 x )  =  ( F `  x
) )
33 fvres 5704 . . . . . 6  |-  ( y  e.  Z  ->  (
( F  |`  Z ) `
 y )  =  ( F `  y
) )
3432, 33oveqan12d 6059 . . . . 5  |-  ( ( x  e.  Z  /\  y  e.  Z )  ->  ( ( ( F  |`  Z ) `  x
) O ( ( F  |`  Z ) `  y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
3534adantl 453 . . . 4  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( ( ( F  |`  Z ) `  x
) O ( ( F  |`  Z ) `  y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
3619, 31, 353eqtr4d 2446 . . 3  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( ( F  |`  Z ) `  (
x S y ) )  =  ( ( ( F  |`  Z ) `
 x ) O ( ( F  |`  Z ) `  y
) ) )
37 ghsubgo.9 . . . 4  |-  H  =  ( O  |`  ( W  X.  W ) )
38 ghsubgo.8 . . . . . 6  |-  W  =  ( F " Z
)
3938, 38xpeq12i 4859 . . . . 5  |-  ( W  X.  W )  =  ( ( F " Z )  X.  ( F " Z ) )
4039reseq2i 5102 . . . 4  |-  ( O  |`  ( W  X.  W
) )  =  ( O  |`  ( ( F " Z )  X.  ( F " Z
) ) )
4137, 40eqtri 2424 . . 3  |-  H  =  ( O  |`  (
( F " Z
)  X.  ( F
" Z ) ) )
42 imassrn 5175 . . . . 5  |-  ( F
" Z )  C_  ran  F
43 frn 5556 . . . . . 6  |-  ( F : X --> Y  ->  ran  F  C_  Y )
441, 43syl 16 . . . . 5  |-  ( ph  ->  ran  F  C_  Y
)
4542, 44syl5ss 3319 . . . 4  |-  ( ph  ->  ( F " Z
)  C_  Y )
46 ghsubgo.4 . . . 4  |-  ( ph  ->  Y  C_  A )
4745, 46sstrd 3318 . . 3  |-  ( ph  ->  ( F " Z
)  C_  A )
48 ghsubgo.5 . . 3  |-  ( ph  ->  O  Fn  ( A  X.  A ) )
4913, 36, 41, 6, 47, 48, 22ghgrp 21909 . 2  |-  ( ph  ->  H  e.  GrpOp )
5013adantr 452 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  ( F  |`  Z ) : Z -onto->
( F " Z
) )
5136adantlr 696 . . . 4  |-  ( ( ( ph  /\  S  e.  AbelOp )  /\  (
x  e.  Z  /\  y  e.  Z )
)  ->  ( ( F  |`  Z ) `  ( x S y ) )  =  ( ( ( F  |`  Z ) `  x
) O ( ( F  |`  Z ) `  y ) ) )
5247adantr 452 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  ( F " Z )  C_  A
)
5348adantr 452 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  O  Fn  ( A  X.  A ) )
54 simpr 448 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  S  e.  AbelOp )
5550, 51, 41, 6, 52, 53, 54ghablo 21910 . . 3  |-  ( (
ph  /\  S  e.  AbelOp )  ->  H  e.  AbelOp )
5655ex 424 . 2  |-  ( ph  ->  ( S  e.  AbelOp  ->  H  e.  AbelOp ) )
5749, 56jca 519 1  |-  ( ph  ->  ( H  e.  GrpOp  /\  ( S  e.  AbelOp  ->  H  e.  AbelOp ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    C_ wss 3280    X. cxp 4835   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   Fun wfun 5407    Fn wfn 5408   -->wf 5409   -onto->wfo 5411   ` cfv 5413  (class class class)co 6040   GrpOpcgr 21727   AbelOpcablo 21822   SubGrpOpcsubgo 21842
This theorem is referenced by:  ghsubgo  21912  ghsubablo  21913
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-grpo 21732  df-gid 21733  df-ginv 21734  df-gdiv 21735  df-ablo 21823  df-subgo 21843
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