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Theorem ghabloOLD 26095
Description: Obsolete version of ghmabl 17472 as of 14-Mar-2020. The image of an Abelian group  G under a group homomorphism  F is an Abelian group (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
ghgrpOLD.1  |-  ( ph  ->  F : X -onto-> Y
)
ghgrpOLD.2  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
ghgrpOLD.3  |-  H  =  ( O  |`  ( Y  X.  Y ) )
ghgrpOLD.4  |-  X  =  ran  G
ghgrpOLD.5  |-  ( ph  ->  Y  C_  A )
ghgrpOLD.6  |-  ( ph  ->  O  Fn  ( A  X.  A ) )
ghabloOLD.7  |-  ( ph  ->  G  e.  AbelOp )
Assertion
Ref Expression
ghabloOLD  |-  ( ph  ->  H  e.  AbelOp )
Distinct variable groups:    x, y, F    x, G, y    ph, x, y    x, H, y    x, X, y    x, Y, y   
x, O, y
Allowed substitution hints:    A( x, y)

Proof of Theorem ghabloOLD
Dummy variables  b 
a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghgrpOLD.1 . . 3  |-  ( ph  ->  F : X -onto-> Y
)
2 ghgrpOLD.2 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
3 ghgrpOLD.3 . . 3  |-  H  =  ( O  |`  ( Y  X.  Y ) )
4 ghgrpOLD.4 . . 3  |-  X  =  ran  G
5 ghgrpOLD.5 . . 3  |-  ( ph  ->  Y  C_  A )
6 ghgrpOLD.6 . . 3  |-  ( ph  ->  O  Fn  ( A  X.  A ) )
7 ghabloOLD.7 . . . 4  |-  ( ph  ->  G  e.  AbelOp )
8 ablogrpo 26010 . . . 4  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
97, 8syl 17 . . 3  |-  ( ph  ->  G  e.  GrpOp )
101, 2, 3, 4, 5, 6, 9ghgrpOLD 26094 . 2  |-  ( ph  ->  H  e.  GrpOp )
11 fndm 5693 . . . . . . . . 9  |-  ( O  Fn  ( A  X.  A )  ->  dom  O  =  ( A  X.  A ) )
126, 11syl 17 . . . . . . . 8  |-  ( ph  ->  dom  O  =  ( A  X.  A ) )
133resgrprn 26006 . . . . . . . 8  |-  ( ( dom  O  =  ( A  X.  A )  /\  H  e.  GrpOp  /\  Y  C_  A )  ->  Y  =  ran  H
)
1412, 10, 5, 13syl3anc 1264 . . . . . . 7  |-  ( ph  ->  Y  =  ran  H
)
1514eleq2d 2492 . . . . . 6  |-  ( ph  ->  ( a  e.  Y  <->  a  e.  ran  H ) )
1614eleq2d 2492 . . . . . 6  |-  ( ph  ->  ( b  e.  Y  <->  b  e.  ran  H ) )
1715, 16anbi12d 715 . . . . 5  |-  ( ph  ->  ( ( a  e.  Y  /\  b  e.  Y )  <->  ( a  e.  ran  H  /\  b  e.  ran  H ) ) )
1817biimpar 487 . . . 4  |-  ( (
ph  /\  ( a  e.  ran  H  /\  b  e.  ran  H ) )  ->  ( a  e.  Y  /\  b  e.  Y ) )
197adantr 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  G  e.  AbelOp )
20 simprl 762 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  x  e.  X )
21 simprr 764 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
y  e.  X )
224ablocom 26011 . . . . . . . . . . . . 13  |-  ( ( G  e.  AbelOp  /\  x  e.  X  /\  y  e.  X )  ->  (
x G y )  =  ( y G x ) )
2319, 20, 21, 22syl3anc 1264 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x G y )  =  ( y G x ) )
2423fveq2d 5885 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( F `
 ( y G x ) ) )
251, 2, 3ghgrplem2OLD 26093 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) H ( F `  y ) ) )
261, 2, 3ghgrplem2OLD 26093 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( y  e.  X  /\  x  e.  X ) )  -> 
( F `  (
y G x ) )  =  ( ( F `  y ) H ( F `  x ) ) )
2726ancom2s 809 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
y G x ) )  =  ( ( F `  y ) H ( F `  x ) ) )
2824, 25, 273eqtr3d 2471 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( ( F `  x ) H ( F `  y ) )  =  ( ( F `  y ) H ( F `  x ) ) )
2928ancom2s 809 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  X  /\  x  e.  X ) )  -> 
( ( F `  x ) H ( F `  y ) )  =  ( ( F `  y ) H ( F `  x ) ) )
3029expr 618 . . . . . . . 8  |-  ( (
ph  /\  y  e.  X )  ->  (
x  e.  X  -> 
( ( F `  x ) H ( F `  y ) )  =  ( ( F `  y ) H ( F `  x ) ) ) )
31 oveq2 6313 . . . . . . . . . 10  |-  ( b  =  ( F `  y )  ->  (
( F `  x
) H b )  =  ( ( F `
 x ) H ( F `  y
) ) )
32 oveq1 6312 . . . . . . . . . 10  |-  ( b  =  ( F `  y )  ->  (
b H ( F `
 x ) )  =  ( ( F `
 y ) H ( F `  x
) ) )
3331, 32eqeq12d 2444 . . . . . . . . 9  |-  ( b  =  ( F `  y )  ->  (
( ( F `  x ) H b )  =  ( b H ( F `  x ) )  <->  ( ( F `  x ) H ( F `  y ) )  =  ( ( F `  y ) H ( F `  x ) ) ) )
3433imbi2d 317 . . . . . . . 8  |-  ( b  =  ( F `  y )  ->  (
( x  e.  X  ->  ( ( F `  x ) H b )  =  ( b H ( F `  x ) ) )  <-> 
( x  e.  X  ->  ( ( F `  x ) H ( F `  y ) )  =  ( ( F `  y ) H ( F `  x ) ) ) ) )
351, 30, 34ghgrplem1OLD 26092 . . . . . . 7  |-  ( (
ph  /\  b  e.  Y )  ->  (
x  e.  X  -> 
( ( F `  x ) H b )  =  ( b H ( F `  x ) ) ) )
3635impancom 441 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
b  e.  Y  -> 
( ( F `  x ) H b )  =  ( b H ( F `  x ) ) ) )
37 oveq1 6312 . . . . . . . 8  |-  ( a  =  ( F `  x )  ->  (
a H b )  =  ( ( F `
 x ) H b ) )
38 oveq2 6313 . . . . . . . 8  |-  ( a  =  ( F `  x )  ->  (
b H a )  =  ( b H ( F `  x
) ) )
3937, 38eqeq12d 2444 . . . . . . 7  |-  ( a  =  ( F `  x )  ->  (
( a H b )  =  ( b H a )  <->  ( ( F `  x ) H b )  =  ( b H ( F `  x ) ) ) )
4039imbi2d 317 . . . . . 6  |-  ( a  =  ( F `  x )  ->  (
( b  e.  Y  ->  ( a H b )  =  ( b H a ) )  <-> 
( b  e.  Y  ->  ( ( F `  x ) H b )  =  ( b H ( F `  x ) ) ) ) )
411, 36, 40ghgrplem1OLD 26092 . . . . 5  |-  ( (
ph  /\  a  e.  Y )  ->  (
b  e.  Y  -> 
( a H b )  =  ( b H a ) ) )
4241impr 623 . . . 4  |-  ( (
ph  /\  ( a  e.  Y  /\  b  e.  Y ) )  -> 
( a H b )  =  ( b H a ) )
4318, 42syldan 472 . . 3  |-  ( (
ph  /\  ( a  e.  ran  H  /\  b  e.  ran  H ) )  ->  ( a H b )  =  ( b H a ) )
4443ralrimivva 2843 . 2  |-  ( ph  ->  A. a  e.  ran  H A. b  e.  ran  H ( a H b )  =  ( b H a ) )
45 eqid 2422 . . 3  |-  ran  H  =  ran  H
4645isablo 26009 . 2  |-  ( H  e.  AbelOp 
<->  ( H  e.  GrpOp  /\ 
A. a  e.  ran  H A. b  e.  ran  H ( a H b )  =  ( b H a ) ) )
4710, 44, 46sylanbrc 668 1  |-  ( ph  ->  H  e.  AbelOp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2771    C_ wss 3436    X. cxp 4851   dom cdm 4853   ran crn 4854    |` cres 4855    Fn wfn 5596   -onto->wfo 5599   ` cfv 5601  (class class class)co 6305   GrpOpcgr 25912   AbelOpcablo 26007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-grpo 25917  df-gid 25918  df-ginv 25919  df-gdiv 25920  df-ablo 26008
This theorem is referenced by:  ghsubgolemOLD  26096
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