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Theorem ghsubgolemOLD 26098
Description: Obsolete as of 14-Mar-2020. 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.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ghsubgoOLD.1  |-  ( ph  ->  S  e.  ( SubGrpOp `  G ) )
ghsubgoOLD.2  |-  X  =  ran  G
ghsubgoOLD.3  |-  ( ph  ->  F : X --> Y )
ghsubgoOLD.4  |-  ( ph  ->  Y  C_  A )
ghsubgoOLD.5  |-  ( ph  ->  O  Fn  ( A  X.  A ) )
ghsubgoOLD.6  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
ghsubgoOLD.7  |-  Z  =  ran  S
ghsubgoOLD.8  |-  W  =  ( F " Z
)
ghsubgoOLD.9  |-  H  =  ( O  |`  ( W  X.  W ) )
Assertion
Ref Expression
ghsubgolemOLD  |-  ( 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 ghsubgolemOLD
StepHypRef Expression
1 ghsubgoOLD.3 . . . . 5  |-  ( ph  ->  F : X --> Y )
2 ffun 5731 . . . . 5  |-  ( F : X --> Y  ->  Fun  F )
31, 2syl 17 . . . 4  |-  ( ph  ->  Fun  F )
4 ghsubgoOLD.1 . . . . . 6  |-  ( ph  ->  S  e.  ( SubGrpOp `  G ) )
5 ghsubgoOLD.2 . . . . . . 7  |-  X  =  ran  G
6 ghsubgoOLD.7 . . . . . . 7  |-  Z  =  ran  S
75, 6subgornss 26034 . . . . . 6  |-  ( S  e.  ( SubGrpOp `  G
)  ->  Z  C_  X
)
84, 7syl 17 . . . . 5  |-  ( ph  ->  Z  C_  X )
9 fdm 5733 . . . . . 6  |-  ( F : X --> Y  ->  dom  F  =  X )
101, 9syl 17 . . . . 5  |-  ( ph  ->  dom  F  =  X )
118, 10sseqtr4d 3469 . . . 4  |-  ( ph  ->  Z  C_  dom  F )
12 fores 5802 . . . 4  |-  ( ( Fun  F  /\  Z  C_ 
dom  F )  -> 
( F  |`  Z ) : Z -onto-> ( F
" Z ) )
133, 11, 12syl2anc 667 . . 3  |-  ( ph  ->  ( F  |`  Z ) : Z -onto-> ( F
" Z ) )
14 ssel2 3427 . . . . . . 7  |-  ( ( Z  C_  X  /\  x  e.  Z )  ->  x  e.  X )
15 ssel2 3427 . . . . . . 7  |-  ( ( Z  C_  X  /\  y  e.  Z )  ->  y  e.  X )
1614, 15anim12dan 848 . . . . . 6  |-  ( ( Z  C_  X  /\  ( x  e.  Z  /\  y  e.  Z
) )  ->  (
x  e.  X  /\  y  e.  X )
)
178, 16sylan 474 . . . . 5  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( x  e.  X  /\  y  e.  X
) )
18 ghsubgoOLD.6 . . . . 5  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
1917, 18syldan 473 . . . 4  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
20 issubgo 26031 . . . . . . . . 9  |-  ( S  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  G )
)
2120simp2bi 1024 . . . . . . . 8  |-  ( S  e.  ( SubGrpOp `  G
)  ->  S  e.  GrpOp
)
224, 21syl 17 . . . . . . 7  |-  ( ph  ->  S  e.  GrpOp )
236grpocl 25928 . . . . . . . 8  |-  ( ( S  e.  GrpOp  /\  x  e.  Z  /\  y  e.  Z )  ->  (
x S y )  e.  Z )
24233expb 1209 . . . . . . 7  |-  ( ( S  e.  GrpOp  /\  (
x  e.  Z  /\  y  e.  Z )
)  ->  ( x S y )  e.  Z )
2522, 24sylan 474 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( x S y )  e.  Z )
26 fvres 5879 . . . . . 6  |-  ( ( x S y )  e.  Z  ->  (
( F  |`  Z ) `
 ( x S y ) )  =  ( F `  (
x S y ) ) )
2725, 26syl 17 . . . . 5  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( ( F  |`  Z ) `  (
x S y ) )  =  ( F `
 ( x S y ) ) )
286subgoov 26033 . . . . . . 7  |-  ( ( S  e.  ( SubGrpOp `  G )  /\  (
x  e.  Z  /\  y  e.  Z )
)  ->  ( x S y )  =  ( x G y ) )
294, 28sylan 474 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( x S y )  =  ( x G y ) )
3029fveq2d 5869 . . . . 5  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( F `  (
x S y ) )  =  ( F `
 ( x G y ) ) )
3127, 30eqtrd 2485 . . . 4  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( ( F  |`  Z ) `  (
x S y ) )  =  ( F `
 ( x G y ) ) )
32 fvres 5879 . . . . . 6  |-  ( x  e.  Z  ->  (
( F  |`  Z ) `
 x )  =  ( F `  x
) )
33 fvres 5879 . . . . . 6  |-  ( y  e.  Z  ->  (
( F  |`  Z ) `
 y )  =  ( F `  y
) )
3432, 33oveqan12d 6309 . . . . 5  |-  ( ( x  e.  Z  /\  y  e.  Z )  ->  ( ( ( F  |`  Z ) `  x
) O ( ( F  |`  Z ) `  y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
3534adantl 468 . . . 4  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( ( ( F  |`  Z ) `  x
) O ( ( F  |`  Z ) `  y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
3619, 31, 353eqtr4d 2495 . . 3  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( ( F  |`  Z ) `  (
x S y ) )  =  ( ( ( F  |`  Z ) `
 x ) O ( ( F  |`  Z ) `  y
) ) )
37 ghsubgoOLD.9 . . . 4  |-  H  =  ( O  |`  ( W  X.  W ) )
38 ghsubgoOLD.8 . . . . . 6  |-  W  =  ( F " Z
)
3938, 38xpeq12i 4856 . . . . 5  |-  ( W  X.  W )  =  ( ( F " Z )  X.  ( F " Z ) )
4039reseq2i 5102 . . . 4  |-  ( O  |`  ( W  X.  W
) )  =  ( O  |`  ( ( F " Z )  X.  ( F " Z
) ) )
4137, 40eqtri 2473 . . 3  |-  H  =  ( O  |`  (
( F " Z
)  X.  ( F
" Z ) ) )
42 imassrn 5179 . . . . 5  |-  ( F
" Z )  C_  ran  F
43 frn 5735 . . . . . 6  |-  ( F : X --> Y  ->  ran  F  C_  Y )
441, 43syl 17 . . . . 5  |-  ( ph  ->  ran  F  C_  Y
)
4542, 44syl5ss 3443 . . . 4  |-  ( ph  ->  ( F " Z
)  C_  Y )
46 ghsubgoOLD.4 . . . 4  |-  ( ph  ->  Y  C_  A )
4745, 46sstrd 3442 . . 3  |-  ( ph  ->  ( F " Z
)  C_  A )
48 ghsubgoOLD.5 . . 3  |-  ( ph  ->  O  Fn  ( A  X.  A ) )
4913, 36, 41, 6, 47, 48, 22ghgrpOLD 26096 . 2  |-  ( ph  ->  H  e.  GrpOp )
5013adantr 467 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  ( F  |`  Z ) : Z -onto->
( F " Z
) )
5136adantlr 721 . . . 4  |-  ( ( ( ph  /\  S  e.  AbelOp )  /\  (
x  e.  Z  /\  y  e.  Z )
)  ->  ( ( F  |`  Z ) `  ( x S y ) )  =  ( ( ( F  |`  Z ) `  x
) O ( ( F  |`  Z ) `  y ) ) )
5247adantr 467 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  ( F " Z )  C_  A
)
5348adantr 467 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  O  Fn  ( A  X.  A ) )
54 simpr 463 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  S  e.  AbelOp )
5550, 51, 41, 6, 52, 53, 54ghabloOLD 26097 . . 3  |-  ( (
ph  /\  S  e.  AbelOp )  ->  H  e.  AbelOp )
5655ex 436 . 2  |-  ( ph  ->  ( S  e.  AbelOp  ->  H  e.  AbelOp ) )
5749, 56jca 535 1  |-  ( ph  ->  ( H  e.  GrpOp  /\  ( S  e.  AbelOp  ->  H  e.  AbelOp ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887    C_ wss 3404    X. cxp 4832   dom cdm 4834   ran crn 4835    |` cres 4836   "cima 4837   Fun wfun 5576    Fn wfn 5577   -->wf 5578   -onto->wfo 5580   ` cfv 5582  (class class class)co 6290   GrpOpcgr 25914   AbelOpcablo 26009   SubGrpOpcsubgo 26029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-grpo 25919  df-gid 25920  df-ginv 25921  df-gdiv 25922  df-ablo 26010  df-subgo 26030
This theorem is referenced by:  ghsubgoOLD  26099  ghsubabloOLD  26100
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