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Theorem ghsubgolem 21143
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 5471 . . . . 5  |-  ( F : X --> Y  ->  Fun  F )
31, 2syl 15 . . . 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 21079 . . . . . 6  |-  ( S  e.  ( SubGrpOp `  G
)  ->  Z  C_  X
)
84, 7syl 15 . . . . 5  |-  ( ph  ->  Z  C_  X )
9 fdm 5473 . . . . . 6  |-  ( F : X --> Y  ->  dom  F  =  X )
101, 9syl 15 . . . . 5  |-  ( ph  ->  dom  F  =  X )
118, 10sseqtr4d 3291 . . . 4  |-  ( ph  ->  Z  C_  dom  F )
12 fores 5540 . . . 4  |-  ( ( Fun  F  /\  Z  C_ 
dom  F )  -> 
( F  |`  Z ) : Z -onto-> ( F
" Z ) )
133, 11, 12syl2anc 642 . . 3  |-  ( ph  ->  ( F  |`  Z ) : Z -onto-> ( F
" Z ) )
14 ssel2 3251 . . . . . . 7  |-  ( ( Z  C_  X  /\  x  e.  Z )  ->  x  e.  X )
15 ssel2 3251 . . . . . . 7  |-  ( ( Z  C_  X  /\  y  e.  Z )  ->  y  e.  X )
1614, 15anim12dan 810 . . . . . 6  |-  ( ( Z  C_  X  /\  ( x  e.  Z  /\  y  e.  Z
) )  ->  (
x  e.  X  /\  y  e.  X )
)
178, 16sylan 457 . . . . 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 456 . . . 4  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
20 issubgo 21076 . . . . . . . . 9  |-  ( S  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  G )
)
2120simp2bi 971 . . . . . . . 8  |-  ( S  e.  ( SubGrpOp `  G
)  ->  S  e.  GrpOp
)
224, 21syl 15 . . . . . . 7  |-  ( ph  ->  S  e.  GrpOp )
236grpocl 20973 . . . . . . . 8  |-  ( ( S  e.  GrpOp  /\  x  e.  Z  /\  y  e.  Z )  ->  (
x S y )  e.  Z )
24233expb 1152 . . . . . . 7  |-  ( ( S  e.  GrpOp  /\  (
x  e.  Z  /\  y  e.  Z )
)  ->  ( x S y )  e.  Z )
2522, 24sylan 457 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( x S y )  e.  Z )
26 fvres 5622 . . . . . 6  |-  ( ( x S y )  e.  Z  ->  (
( F  |`  Z ) `
 ( x S y ) )  =  ( F `  (
x S y ) ) )
2725, 26syl 15 . . . . 5  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( ( F  |`  Z ) `  (
x S y ) )  =  ( F `
 ( x S y ) ) )
286subgoov 21078 . . . . . . 7  |-  ( ( S  e.  ( SubGrpOp `  G )  /\  (
x  e.  Z  /\  y  e.  Z )
)  ->  ( x S y )  =  ( x G y ) )
294, 28sylan 457 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( x S y )  =  ( x G y ) )
3029fveq2d 5609 . . . . 5  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( F `  (
x S y ) )  =  ( F `
 ( x G y ) ) )
3127, 30eqtrd 2390 . . . 4  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( ( F  |`  Z ) `  (
x S y ) )  =  ( F `
 ( x G y ) ) )
32 fvres 5622 . . . . . 6  |-  ( x  e.  Z  ->  (
( F  |`  Z ) `
 x )  =  ( F `  x
) )
33 fvres 5622 . . . . . 6  |-  ( y  e.  Z  ->  (
( F  |`  Z ) `
 y )  =  ( F `  y
) )
3432, 33oveqan12d 5961 . . . . 5  |-  ( ( x  e.  Z  /\  y  e.  Z )  ->  ( ( ( F  |`  Z ) `  x
) O ( ( F  |`  Z ) `  y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
3534adantl 452 . . . 4  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( ( ( F  |`  Z ) `  x
) O ( ( F  |`  Z ) `  y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
3619, 31, 353eqtr4d 2400 . . 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 4790 . . . . 5  |-  ( W  X.  W )  =  ( ( F " Z )  X.  ( F " Z ) )
4039reseq2i 5031 . . . 4  |-  ( O  |`  ( W  X.  W
) )  =  ( O  |`  ( ( F " Z )  X.  ( F " Z
) ) )
4137, 40eqtri 2378 . . 3  |-  H  =  ( O  |`  (
( F " Z
)  X.  ( F
" Z ) ) )
42 imassrn 5104 . . . . 5  |-  ( F
" Z )  C_  ran  F
43 frn 5475 . . . . . 6  |-  ( F : X --> Y  ->  ran  F  C_  Y )
441, 43syl 15 . . . . 5  |-  ( ph  ->  ran  F  C_  Y
)
4542, 44syl5ss 3266 . . . 4  |-  ( ph  ->  ( F " Z
)  C_  Y )
46 ghsubgo.4 . . . 4  |-  ( ph  ->  Y  C_  A )
4745, 46sstrd 3265 . . 3  |-  ( ph  ->  ( F " Z
)  C_  A )
48 ghsubgo.5 . . 3  |-  ( ph  ->  O  Fn  ( A  X.  A ) )
4913, 36, 41, 6, 47, 48, 22ghgrp 21141 . 2  |-  ( ph  ->  H  e.  GrpOp )
5013adantr 451 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  ( F  |`  Z ) : Z -onto->
( F " Z
) )
5136adantlr 695 . . . 4  |-  ( ( ( ph  /\  S  e.  AbelOp )  /\  (
x  e.  Z  /\  y  e.  Z )
)  ->  ( ( F  |`  Z ) `  ( x S y ) )  =  ( ( ( F  |`  Z ) `  x
) O ( ( F  |`  Z ) `  y ) ) )
5247adantr 451 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  ( F " Z )  C_  A
)
5348adantr 451 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  O  Fn  ( A  X.  A ) )
54 simpr 447 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  S  e.  AbelOp )
5550, 51, 41, 6, 52, 53, 54ghablo 21142 . . 3  |-  ( (
ph  /\  S  e.  AbelOp )  ->  H  e.  AbelOp )
5655ex 423 . 2  |-  ( ph  ->  ( S  e.  AbelOp  ->  H  e.  AbelOp ) )
5749, 56jca 518 1  |-  ( ph  ->  ( H  e.  GrpOp  /\  ( S  e.  AbelOp  ->  H  e.  AbelOp ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    C_ wss 3228    X. cxp 4766   dom cdm 4768   ran crn 4769    |` cres 4770   "cima 4771   Fun wfun 5328    Fn wfn 5329   -->wf 5330   -onto->wfo 5332   ` cfv 5334  (class class class)co 5942   GrpOpcgr 20959   AbelOpcablo 21054   SubGrpOpcsubgo 21074
This theorem is referenced by:  ghsubgo  21144  ghsubablo  21145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-grpo 20964  df-gid 20965  df-ginv 20966  df-gdiv 20967  df-ablo 21055  df-subgo 21075
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