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Theorem ghsubgolemOLD 26179
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 5742 . . . . 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 26115 . . . . . 6  |-  ( S  e.  ( SubGrpOp `  G
)  ->  Z  C_  X
)
84, 7syl 17 . . . . 5  |-  ( ph  ->  Z  C_  X )
9 fdm 5745 . . . . . 6  |-  ( F : X --> Y  ->  dom  F  =  X )
101, 9syl 17 . . . . 5  |-  ( ph  ->  dom  F  =  X )
118, 10sseqtr4d 3455 . . . 4  |-  ( ph  ->  Z  C_  dom  F )
12 fores 5815 . . . 4  |-  ( ( Fun  F  /\  Z  C_ 
dom  F )  -> 
( F  |`  Z ) : Z -onto-> ( F
" Z ) )
133, 11, 12syl2anc 673 . . 3  |-  ( ph  ->  ( F  |`  Z ) : Z -onto-> ( F
" Z ) )
14 ssel2 3413 . . . . . . 7  |-  ( ( Z  C_  X  /\  x  e.  Z )  ->  x  e.  X )
15 ssel2 3413 . . . . . . 7  |-  ( ( Z  C_  X  /\  y  e.  Z )  ->  y  e.  X )
1614, 15anim12dan 855 . . . . . 6  |-  ( ( Z  C_  X  /\  ( x  e.  Z  /\  y  e.  Z
) )  ->  (
x  e.  X  /\  y  e.  X )
)
178, 16sylan 479 . . . . 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 478 . . . 4  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
20 issubgo 26112 . . . . . . . . 9  |-  ( S  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  G )
)
2120simp2bi 1046 . . . . . . . 8  |-  ( S  e.  ( SubGrpOp `  G
)  ->  S  e.  GrpOp
)
224, 21syl 17 . . . . . . 7  |-  ( ph  ->  S  e.  GrpOp )
236grpocl 26009 . . . . . . . 8  |-  ( ( S  e.  GrpOp  /\  x  e.  Z  /\  y  e.  Z )  ->  (
x S y )  e.  Z )
24233expb 1232 . . . . . . 7  |-  ( ( S  e.  GrpOp  /\  (
x  e.  Z  /\  y  e.  Z )
)  ->  ( x S y )  e.  Z )
2522, 24sylan 479 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( x S y )  e.  Z )
26 fvres 5893 . . . . . 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 26114 . . . . . . 7  |-  ( ( S  e.  ( SubGrpOp `  G )  /\  (
x  e.  Z  /\  y  e.  Z )
)  ->  ( x S y )  =  ( x G y ) )
294, 28sylan 479 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( x S y )  =  ( x G y ) )
3029fveq2d 5883 . . . . 5  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( F `  (
x S y ) )  =  ( F `
 ( x G y ) ) )
3127, 30eqtrd 2505 . . . 4  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( ( F  |`  Z ) `  (
x S y ) )  =  ( F `
 ( x G y ) ) )
32 fvres 5893 . . . . . 6  |-  ( x  e.  Z  ->  (
( F  |`  Z ) `
 x )  =  ( F `  x
) )
33 fvres 5893 . . . . . 6  |-  ( y  e.  Z  ->  (
( F  |`  Z ) `
 y )  =  ( F `  y
) )
3432, 33oveqan12d 6327 . . . . 5  |-  ( ( x  e.  Z  /\  y  e.  Z )  ->  ( ( ( F  |`  Z ) `  x
) O ( ( F  |`  Z ) `  y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
3534adantl 473 . . . 4  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( ( ( F  |`  Z ) `  x
) O ( ( F  |`  Z ) `  y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
3619, 31, 353eqtr4d 2515 . . 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 4861 . . . . 5  |-  ( W  X.  W )  =  ( ( F " Z )  X.  ( F " Z ) )
4039reseq2i 5108 . . . 4  |-  ( O  |`  ( W  X.  W
) )  =  ( O  |`  ( ( F " Z )  X.  ( F " Z
) ) )
4137, 40eqtri 2493 . . 3  |-  H  =  ( O  |`  (
( F " Z
)  X.  ( F
" Z ) ) )
42 imassrn 5185 . . . . 5  |-  ( F
" Z )  C_  ran  F
43 frn 5747 . . . . . 6  |-  ( F : X --> Y  ->  ran  F  C_  Y )
441, 43syl 17 . . . . 5  |-  ( ph  ->  ran  F  C_  Y
)
4542, 44syl5ss 3429 . . . 4  |-  ( ph  ->  ( F " Z
)  C_  Y )
46 ghsubgoOLD.4 . . . 4  |-  ( ph  ->  Y  C_  A )
4745, 46sstrd 3428 . . 3  |-  ( ph  ->  ( F " Z
)  C_  A )
48 ghsubgoOLD.5 . . 3  |-  ( ph  ->  O  Fn  ( A  X.  A ) )
4913, 36, 41, 6, 47, 48, 22ghgrpOLD 26177 . 2  |-  ( ph  ->  H  e.  GrpOp )
5013adantr 472 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  ( F  |`  Z ) : Z -onto->
( F " Z
) )
5136adantlr 729 . . . 4  |-  ( ( ( ph  /\  S  e.  AbelOp )  /\  (
x  e.  Z  /\  y  e.  Z )
)  ->  ( ( F  |`  Z ) `  ( x S y ) )  =  ( ( ( F  |`  Z ) `  x
) O ( ( F  |`  Z ) `  y ) ) )
5247adantr 472 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  ( F " Z )  C_  A
)
5348adantr 472 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  O  Fn  ( A  X.  A ) )
54 simpr 468 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  S  e.  AbelOp )
5550, 51, 41, 6, 52, 53, 54ghabloOLD 26178 . . 3  |-  ( (
ph  /\  S  e.  AbelOp )  ->  H  e.  AbelOp )
5655ex 441 . 2  |-  ( ph  ->  ( S  e.  AbelOp  ->  H  e.  AbelOp ) )
5749, 56jca 541 1  |-  ( ph  ->  ( H  e.  GrpOp  /\  ( S  e.  AbelOp  ->  H  e.  AbelOp ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    C_ wss 3390    X. cxp 4837   dom cdm 4839   ran crn 4840    |` cres 4841   "cima 4842   Fun wfun 5583    Fn wfn 5584   -->wf 5585   -onto->wfo 5587   ` cfv 5589  (class class class)co 6308   GrpOpcgr 25995   AbelOpcablo 26090   SubGrpOpcsubgo 26110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-grpo 26000  df-gid 26001  df-ginv 26002  df-gdiv 26003  df-ablo 26091  df-subgo 26111
This theorem is referenced by:  ghsubgoOLD  26180  ghsubabloOLD  26181
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