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Theorem ghablo 21052
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 20967 . . . 4  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
97, 8syl 15 . . 3  |-  ( ph  ->  G  e.  GrpOp )
101, 2, 3, 4, 5, 6, 9ghgrp 21051 . 2  |-  ( ph  ->  H  e.  GrpOp )
11 fndm 5359 . . . . . . . . 9  |-  ( O  Fn  ( A  X.  A )  ->  dom  O  =  ( A  X.  A ) )
126, 11syl 15 . . . . . . . 8  |-  ( ph  ->  dom  O  =  ( A  X.  A ) )
133resgrprn 20963 . . . . . . . 8  |-  ( ( dom  O  =  ( A  X.  A )  /\  H  e.  GrpOp  /\  Y  C_  A )  ->  Y  =  ran  H
)
1412, 10, 5, 13syl3anc 1182 . . . . . . 7  |-  ( ph  ->  Y  =  ran  H
)
1514eleq2d 2363 . . . . . 6  |-  ( ph  ->  ( a  e.  Y  <->  a  e.  ran  H ) )
1614eleq2d 2363 . . . . . 6  |-  ( ph  ->  ( b  e.  Y  <->  b  e.  ran  H ) )
1715, 16anbi12d 691 . . . . 5  |-  ( ph  ->  ( ( a  e.  Y  /\  b  e.  Y )  <->  ( a  e.  ran  H  /\  b  e.  ran  H ) ) )
1817biimpar 471 . . . 4  |-  ( (
ph  /\  ( a  e.  ran  H  /\  b  e.  ran  H ) )  ->  ( a  e.  Y  /\  b  e.  Y ) )
197adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  G  e.  AbelOp )
20 simprl 732 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  x  e.  X )
21 simprr 733 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
y  e.  X )
224ablocom 20968 . . . . . . . . . . . . 13  |-  ( ( G  e.  AbelOp  /\  x  e.  X  /\  y  e.  X )  ->  (
x G y )  =  ( y G x ) )
2319, 20, 21, 22syl3anc 1182 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x G y )  =  ( y G x ) )
2423fveq2d 5545 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( F `
 ( y G x ) ) )
251, 2, 3ghgrplem2 21050 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) H ( F `  y ) ) )
261, 2, 3ghgrplem2 21050 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( y  e.  X  /\  x  e.  X ) )  -> 
( F `  (
y G x ) )  =  ( ( F `  y ) H ( F `  x ) ) )
2726ancom2s 777 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
y G x ) )  =  ( ( F `  y ) H ( F `  x ) ) )
2824, 25, 273eqtr3d 2336 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( ( F `  x ) H ( F `  y ) )  =  ( ( F `  y ) H ( F `  x ) ) )
2928ancom2s 777 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  X  /\  x  e.  X ) )  -> 
( ( F `  x ) H ( F `  y ) )  =  ( ( F `  y ) H ( F `  x ) ) )
3029expr 598 . . . . . . . 8  |-  ( (
ph  /\  y  e.  X )  ->  (
x  e.  X  -> 
( ( F `  x ) H ( F `  y ) )  =  ( ( F `  y ) H ( F `  x ) ) ) )
31 oveq2 5882 . . . . . . . . . 10  |-  ( b  =  ( F `  y )  ->  (
( F `  x
) H b )  =  ( ( F `
 x ) H ( F `  y
) ) )
32 oveq1 5881 . . . . . . . . . 10  |-  ( b  =  ( F `  y )  ->  (
b H ( F `
 x ) )  =  ( ( F `
 y ) H ( F `  x
) ) )
3331, 32eqeq12d 2310 . . . . . . . . 9  |-  ( b  =  ( F `  y )  ->  (
( ( F `  x ) H b )  =  ( b H ( F `  x ) )  <->  ( ( F `  x ) H ( F `  y ) )  =  ( ( F `  y ) H ( F `  x ) ) ) )
3433imbi2d 307 . . . . . . . 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 21049 . . . . . . 7  |-  ( (
ph  /\  b  e.  Y )  ->  (
x  e.  X  -> 
( ( F `  x ) H b )  =  ( b H ( F `  x ) ) ) )
3635impancom 427 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
b  e.  Y  -> 
( ( F `  x ) H b )  =  ( b H ( F `  x ) ) ) )
37 oveq1 5881 . . . . . . . 8  |-  ( a  =  ( F `  x )  ->  (
a H b )  =  ( ( F `
 x ) H b ) )
38 oveq2 5882 . . . . . . . 8  |-  ( a  =  ( F `  x )  ->  (
b H a )  =  ( b H ( F `  x
) ) )
3937, 38eqeq12d 2310 . . . . . . 7  |-  ( a  =  ( F `  x )  ->  (
( a H b )  =  ( b H a )  <->  ( ( F `  x ) H b )  =  ( b H ( F `  x ) ) ) )
4039imbi2d 307 . . . . . 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 21049 . . . . 5  |-  ( (
ph  /\  a  e.  Y )  ->  (
b  e.  Y  -> 
( a H b )  =  ( b H a ) ) )
4241impr 602 . . . 4  |-  ( (
ph  /\  ( a  e.  Y  /\  b  e.  Y ) )  -> 
( a H b )  =  ( b H a ) )
4318, 42syldan 456 . . 3  |-  ( (
ph  /\  ( a  e.  ran  H  /\  b  e.  ran  H ) )  ->  ( a H b )  =  ( b H a ) )
4443ralrimivva 2648 . 2  |-  ( ph  ->  A. a  e.  ran  H A. b  e.  ran  H ( a H b )  =  ( b H a ) )
45 eqid 2296 . . 3  |-  ran  H  =  ran  H
4645isablo 20966 . 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 645 1  |-  ( ph  ->  H  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165    X. cxp 4703   dom cdm 4705   ran crn 4706    |` cres 4707    Fn wfn 5266   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   GrpOpcgr 20869   AbelOpcablo 20964
This theorem is referenced by:  ghsubgolem  21053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965
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