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Theorem ghsubgolemOLD 26084
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 5745 . . . . 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 26020 . . . . . 6  |-  ( S  e.  ( SubGrpOp `  G
)  ->  Z  C_  X
)
84, 7syl 17 . . . . 5  |-  ( ph  ->  Z  C_  X )
9 fdm 5747 . . . . . 6  |-  ( F : X --> Y  ->  dom  F  =  X )
101, 9syl 17 . . . . 5  |-  ( ph  ->  dom  F  =  X )
118, 10sseqtr4d 3501 . . . 4  |-  ( ph  ->  Z  C_  dom  F )
12 fores 5816 . . . 4  |-  ( ( Fun  F  /\  Z  C_ 
dom  F )  -> 
( F  |`  Z ) : Z -onto-> ( F
" Z ) )
133, 11, 12syl2anc 665 . . 3  |-  ( ph  ->  ( F  |`  Z ) : Z -onto-> ( F
" Z ) )
14 ssel2 3459 . . . . . . 7  |-  ( ( Z  C_  X  /\  x  e.  Z )  ->  x  e.  X )
15 ssel2 3459 . . . . . . 7  |-  ( ( Z  C_  X  /\  y  e.  Z )  ->  y  e.  X )
1614, 15anim12dan 845 . . . . . 6  |-  ( ( Z  C_  X  /\  ( x  e.  Z  /\  y  e.  Z
) )  ->  (
x  e.  X  /\  y  e.  X )
)
178, 16sylan 473 . . . . 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 472 . . . 4  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
20 issubgo 26017 . . . . . . . . 9  |-  ( S  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  S  e.  GrpOp  /\  S  C_  G )
)
2120simp2bi 1021 . . . . . . . 8  |-  ( S  e.  ( SubGrpOp `  G
)  ->  S  e.  GrpOp
)
224, 21syl 17 . . . . . . 7  |-  ( ph  ->  S  e.  GrpOp )
236grpocl 25914 . . . . . . . 8  |-  ( ( S  e.  GrpOp  /\  x  e.  Z  /\  y  e.  Z )  ->  (
x S y )  e.  Z )
24233expb 1206 . . . . . . 7  |-  ( ( S  e.  GrpOp  /\  (
x  e.  Z  /\  y  e.  Z )
)  ->  ( x S y )  e.  Z )
2522, 24sylan 473 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( x S y )  e.  Z )
26 fvres 5892 . . . . . 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 26019 . . . . . . 7  |-  ( ( S  e.  ( SubGrpOp `  G )  /\  (
x  e.  Z  /\  y  e.  Z )
)  ->  ( x S y )  =  ( x G y ) )
294, 28sylan 473 . . . . . 6  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( x S y )  =  ( x G y ) )
3029fveq2d 5882 . . . . 5  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( F `  (
x S y ) )  =  ( F `
 ( x G y ) ) )
3127, 30eqtrd 2463 . . . 4  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( ( F  |`  Z ) `  (
x S y ) )  =  ( F `
 ( x G y ) ) )
32 fvres 5892 . . . . . 6  |-  ( x  e.  Z  ->  (
( F  |`  Z ) `
 x )  =  ( F `  x
) )
33 fvres 5892 . . . . . 6  |-  ( y  e.  Z  ->  (
( F  |`  Z ) `
 y )  =  ( F `  y
) )
3432, 33oveqan12d 6321 . . . . 5  |-  ( ( x  e.  Z  /\  y  e.  Z )  ->  ( ( ( F  |`  Z ) `  x
) O ( ( F  |`  Z ) `  y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
3534adantl 467 . . . 4  |-  ( (
ph  /\  ( x  e.  Z  /\  y  e.  Z ) )  -> 
( ( ( F  |`  Z ) `  x
) O ( ( F  |`  Z ) `  y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
3619, 31, 353eqtr4d 2473 . . 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 4872 . . . . 5  |-  ( W  X.  W )  =  ( ( F " Z )  X.  ( F " Z ) )
4039reseq2i 5118 . . . 4  |-  ( O  |`  ( W  X.  W
) )  =  ( O  |`  ( ( F " Z )  X.  ( F " Z
) ) )
4137, 40eqtri 2451 . . 3  |-  H  =  ( O  |`  (
( F " Z
)  X.  ( F
" Z ) ) )
42 imassrn 5195 . . . . 5  |-  ( F
" Z )  C_  ran  F
43 frn 5749 . . . . . 6  |-  ( F : X --> Y  ->  ran  F  C_  Y )
441, 43syl 17 . . . . 5  |-  ( ph  ->  ran  F  C_  Y
)
4542, 44syl5ss 3475 . . . 4  |-  ( ph  ->  ( F " Z
)  C_  Y )
46 ghsubgoOLD.4 . . . 4  |-  ( ph  ->  Y  C_  A )
4745, 46sstrd 3474 . . 3  |-  ( ph  ->  ( F " Z
)  C_  A )
48 ghsubgoOLD.5 . . 3  |-  ( ph  ->  O  Fn  ( A  X.  A ) )
4913, 36, 41, 6, 47, 48, 22ghgrpOLD 26082 . 2  |-  ( ph  ->  H  e.  GrpOp )
5013adantr 466 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  ( F  |`  Z ) : Z -onto->
( F " Z
) )
5136adantlr 719 . . . 4  |-  ( ( ( ph  /\  S  e.  AbelOp )  /\  (
x  e.  Z  /\  y  e.  Z )
)  ->  ( ( F  |`  Z ) `  ( x S y ) )  =  ( ( ( F  |`  Z ) `  x
) O ( ( F  |`  Z ) `  y ) ) )
5247adantr 466 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  ( F " Z )  C_  A
)
5348adantr 466 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  O  Fn  ( A  X.  A ) )
54 simpr 462 . . . 4  |-  ( (
ph  /\  S  e.  AbelOp )  ->  S  e.  AbelOp )
5550, 51, 41, 6, 52, 53, 54ghabloOLD 26083 . . 3  |-  ( (
ph  /\  S  e.  AbelOp )  ->  H  e.  AbelOp )
5655ex 435 . 2  |-  ( ph  ->  ( S  e.  AbelOp  ->  H  e.  AbelOp ) )
5749, 56jca 534 1  |-  ( ph  ->  ( H  e.  GrpOp  /\  ( S  e.  AbelOp  ->  H  e.  AbelOp ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868    C_ wss 3436    X. cxp 4848   dom cdm 4850   ran crn 4851    |` cres 4852   "cima 4853   Fun wfun 5592    Fn wfn 5593   -->wf 5594   -onto->wfo 5596   ` cfv 5598  (class class class)co 6302   GrpOpcgr 25900   AbelOpcablo 25995   SubGrpOpcsubgo 26015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-1st 6804  df-2nd 6805  df-grpo 25905  df-gid 25906  df-ginv 25907  df-gdiv 25908  df-ablo 25996  df-subgo 26016
This theorem is referenced by:  ghsubgoOLD  26085  ghsubabloOLD  26086
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