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Theorem ghablo 23856
Description: 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.)
Hypotheses
Ref Expression
ghgrp.1  |-  ( ph  ->  F : X -onto-> Y
)
ghgrp.2  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
ghgrp.3  |-  H  =  ( O  |`  ( Y  X.  Y ) )
ghgrp.4  |-  X  =  ran  G
ghgrp.5  |-  ( ph  ->  Y  C_  A )
ghgrp.6  |-  ( ph  ->  O  Fn  ( A  X.  A ) )
ghablo.7  |-  ( ph  ->  G  e.  AbelOp )
Assertion
Ref Expression
ghablo  |-  ( 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 ghablo
Dummy variables  b 
a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghgrp.1 . . 3  |-  ( ph  ->  F : X -onto-> Y
)
2 ghgrp.2 . . 3  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) O ( F `  y ) ) )
3 ghgrp.3 . . 3  |-  H  =  ( O  |`  ( Y  X.  Y ) )
4 ghgrp.4 . . 3  |-  X  =  ran  G
5 ghgrp.5 . . 3  |-  ( ph  ->  Y  C_  A )
6 ghgrp.6 . . 3  |-  ( ph  ->  O  Fn  ( A  X.  A ) )
7 ghablo.7 . . . 4  |-  ( ph  ->  G  e.  AbelOp )
8 ablogrpo 23771 . . . 4  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
97, 8syl 16 . . 3  |-  ( ph  ->  G  e.  GrpOp )
101, 2, 3, 4, 5, 6, 9ghgrp 23855 . 2  |-  ( ph  ->  H  e.  GrpOp )
11 fndm 5510 . . . . . . . . 9  |-  ( O  Fn  ( A  X.  A )  ->  dom  O  =  ( A  X.  A ) )
126, 11syl 16 . . . . . . . 8  |-  ( ph  ->  dom  O  =  ( A  X.  A ) )
133resgrprn 23767 . . . . . . . 8  |-  ( ( dom  O  =  ( A  X.  A )  /\  H  e.  GrpOp  /\  Y  C_  A )  ->  Y  =  ran  H
)
1412, 10, 5, 13syl3anc 1218 . . . . . . 7  |-  ( ph  ->  Y  =  ran  H
)
1514eleq2d 2510 . . . . . 6  |-  ( ph  ->  ( a  e.  Y  <->  a  e.  ran  H ) )
1614eleq2d 2510 . . . . . 6  |-  ( ph  ->  ( b  e.  Y  <->  b  e.  ran  H ) )
1715, 16anbi12d 710 . . . . 5  |-  ( ph  ->  ( ( a  e.  Y  /\  b  e.  Y )  <->  ( a  e.  ran  H  /\  b  e.  ran  H ) ) )
1817biimpar 485 . . . 4  |-  ( (
ph  /\  ( a  e.  ran  H  /\  b  e.  ran  H ) )  ->  ( a  e.  Y  /\  b  e.  Y ) )
197adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  G  e.  AbelOp )
20 simprl 755 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  x  e.  X )
21 simprr 756 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
y  e.  X )
224ablocom 23772 . . . . . . . . . . . . 13  |-  ( ( G  e.  AbelOp  /\  x  e.  X  /\  y  e.  X )  ->  (
x G y )  =  ( y G x ) )
2319, 20, 21, 22syl3anc 1218 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x G y )  =  ( y G x ) )
2423fveq2d 5695 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( F `
 ( y G x ) ) )
251, 2, 3ghgrplem2 23854 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) H ( F `  y ) ) )
261, 2, 3ghgrplem2 23854 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( y  e.  X  /\  x  e.  X ) )  -> 
( F `  (
y G x ) )  =  ( ( F `  y ) H ( F `  x ) ) )
2726ancom2s 800 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
y G x ) )  =  ( ( F `  y ) H ( F `  x ) ) )
2824, 25, 273eqtr3d 2483 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( ( F `  x ) H ( F `  y ) )  =  ( ( F `  y ) H ( F `  x ) ) )
2928ancom2s 800 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  X  /\  x  e.  X ) )  -> 
( ( F `  x ) H ( F `  y ) )  =  ( ( F `  y ) H ( F `  x ) ) )
3029expr 615 . . . . . . . 8  |-  ( (
ph  /\  y  e.  X )  ->  (
x  e.  X  -> 
( ( F `  x ) H ( F `  y ) )  =  ( ( F `  y ) H ( F `  x ) ) ) )
31 oveq2 6099 . . . . . . . . . 10  |-  ( b  =  ( F `  y )  ->  (
( F `  x
) H b )  =  ( ( F `
 x ) H ( F `  y
) ) )
32 oveq1 6098 . . . . . . . . . 10  |-  ( b  =  ( F `  y )  ->  (
b H ( F `
 x ) )  =  ( ( F `
 y ) H ( F `  x
) ) )
3331, 32eqeq12d 2457 . . . . . . . . 9  |-  ( b  =  ( F `  y )  ->  (
( ( F `  x ) H b )  =  ( b H ( F `  x ) )  <->  ( ( F `  x ) H ( F `  y ) )  =  ( ( F `  y ) H ( F `  x ) ) ) )
3433imbi2d 316 . . . . . . . 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, 34ghgrplem1 23853 . . . . . . 7  |-  ( (
ph  /\  b  e.  Y )  ->  (
x  e.  X  -> 
( ( F `  x ) H b )  =  ( b H ( F `  x ) ) ) )
3635impancom 440 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
b  e.  Y  -> 
( ( F `  x ) H b )  =  ( b H ( F `  x ) ) ) )
37 oveq1 6098 . . . . . . . 8  |-  ( a  =  ( F `  x )  ->  (
a H b )  =  ( ( F `
 x ) H b ) )
38 oveq2 6099 . . . . . . . 8  |-  ( a  =  ( F `  x )  ->  (
b H a )  =  ( b H ( F `  x
) ) )
3937, 38eqeq12d 2457 . . . . . . 7  |-  ( a  =  ( F `  x )  ->  (
( a H b )  =  ( b H a )  <->  ( ( F `  x ) H b )  =  ( b H ( F `  x ) ) ) )
4039imbi2d 316 . . . . . 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, 40ghgrplem1 23853 . . . . 5  |-  ( (
ph  /\  a  e.  Y )  ->  (
b  e.  Y  -> 
( a H b )  =  ( b H a ) ) )
4241impr 619 . . . 4  |-  ( (
ph  /\  ( a  e.  Y  /\  b  e.  Y ) )  -> 
( a H b )  =  ( b H a ) )
4318, 42syldan 470 . . 3  |-  ( (
ph  /\  ( a  e.  ran  H  /\  b  e.  ran  H ) )  ->  ( a H b )  =  ( b H a ) )
4443ralrimivva 2808 . 2  |-  ( ph  ->  A. a  e.  ran  H A. b  e.  ran  H ( a H b )  =  ( b H a ) )
45 eqid 2443 . . 3  |-  ran  H  =  ran  H
4645isablo 23770 . 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 664 1  |-  ( ph  ->  H  e.  AbelOp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715    C_ wss 3328    X. cxp 4838   dom cdm 4840   ran crn 4841    |` cres 4842    Fn wfn 5413   -onto->wfo 5416   ` cfv 5418  (class class class)co 6091   GrpOpcgr 23673   AbelOpcablo 23768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-grpo 23678  df-gid 23679  df-ginv 23680  df-gdiv 23681  df-ablo 23769
This theorem is referenced by:  ghsubgolem  23857
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