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Theorem ghablo 21957
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 21872 . . . 4  |-  ( G  e.  AbelOp  ->  G  e.  GrpOp )
97, 8syl 16 . . 3  |-  ( ph  ->  G  e.  GrpOp )
101, 2, 3, 4, 5, 6, 9ghgrp 21956 . 2  |-  ( ph  ->  H  e.  GrpOp )
11 fndm 5544 . . . . . . . . 9  |-  ( O  Fn  ( A  X.  A )  ->  dom  O  =  ( A  X.  A ) )
126, 11syl 16 . . . . . . . 8  |-  ( ph  ->  dom  O  =  ( A  X.  A ) )
133resgrprn 21868 . . . . . . . 8  |-  ( ( dom  O  =  ( A  X.  A )  /\  H  e.  GrpOp  /\  Y  C_  A )  ->  Y  =  ran  H
)
1412, 10, 5, 13syl3anc 1184 . . . . . . 7  |-  ( ph  ->  Y  =  ran  H
)
1514eleq2d 2503 . . . . . 6  |-  ( ph  ->  ( a  e.  Y  <->  a  e.  ran  H ) )
1614eleq2d 2503 . . . . . 6  |-  ( ph  ->  ( b  e.  Y  <->  b  e.  ran  H ) )
1715, 16anbi12d 692 . . . . 5  |-  ( ph  ->  ( ( a  e.  Y  /\  b  e.  Y )  <->  ( a  e.  ran  H  /\  b  e.  ran  H ) ) )
1817biimpar 472 . . . 4  |-  ( (
ph  /\  ( a  e.  ran  H  /\  b  e.  ran  H ) )  ->  ( a  e.  Y  /\  b  e.  Y ) )
197adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  G  e.  AbelOp )
20 simprl 733 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  x  e.  X )
21 simprr 734 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
y  e.  X )
224ablocom 21873 . . . . . . . . . . . . 13  |-  ( ( G  e.  AbelOp  /\  x  e.  X  /\  y  e.  X )  ->  (
x G y )  =  ( y G x ) )
2319, 20, 21, 22syl3anc 1184 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x G y )  =  ( y G x ) )
2423fveq2d 5732 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( F `
 ( y G x ) ) )
251, 2, 3ghgrplem2 21955 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
x G y ) )  =  ( ( F `  x ) H ( F `  y ) ) )
261, 2, 3ghgrplem2 21955 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( y  e.  X  /\  x  e.  X ) )  -> 
( F `  (
y G x ) )  =  ( ( F `  y ) H ( F `  x ) ) )
2726ancom2s 778 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( F `  (
y G x ) )  =  ( ( F `  y ) H ( F `  x ) ) )
2824, 25, 273eqtr3d 2476 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( ( F `  x ) H ( F `  y ) )  =  ( ( F `  y ) H ( F `  x ) ) )
2928ancom2s 778 . . . . . . . . 9  |-  ( (
ph  /\  ( y  e.  X  /\  x  e.  X ) )  -> 
( ( F `  x ) H ( F `  y ) )  =  ( ( F `  y ) H ( F `  x ) ) )
3029expr 599 . . . . . . . 8  |-  ( (
ph  /\  y  e.  X )  ->  (
x  e.  X  -> 
( ( F `  x ) H ( F `  y ) )  =  ( ( F `  y ) H ( F `  x ) ) ) )
31 oveq2 6089 . . . . . . . . . 10  |-  ( b  =  ( F `  y )  ->  (
( F `  x
) H b )  =  ( ( F `
 x ) H ( F `  y
) ) )
32 oveq1 6088 . . . . . . . . . 10  |-  ( b  =  ( F `  y )  ->  (
b H ( F `
 x ) )  =  ( ( F `
 y ) H ( F `  x
) ) )
3331, 32eqeq12d 2450 . . . . . . . . 9  |-  ( b  =  ( F `  y )  ->  (
( ( F `  x ) H b )  =  ( b H ( F `  x ) )  <->  ( ( F `  x ) H ( F `  y ) )  =  ( ( F `  y ) H ( F `  x ) ) ) )
3433imbi2d 308 . . . . . . . 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 21954 . . . . . . 7  |-  ( (
ph  /\  b  e.  Y )  ->  (
x  e.  X  -> 
( ( F `  x ) H b )  =  ( b H ( F `  x ) ) ) )
3635impancom 428 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
b  e.  Y  -> 
( ( F `  x ) H b )  =  ( b H ( F `  x ) ) ) )
37 oveq1 6088 . . . . . . . 8  |-  ( a  =  ( F `  x )  ->  (
a H b )  =  ( ( F `
 x ) H b ) )
38 oveq2 6089 . . . . . . . 8  |-  ( a  =  ( F `  x )  ->  (
b H a )  =  ( b H ( F `  x
) ) )
3937, 38eqeq12d 2450 . . . . . . 7  |-  ( a  =  ( F `  x )  ->  (
( a H b )  =  ( b H a )  <->  ( ( F `  x ) H b )  =  ( b H ( F `  x ) ) ) )
4039imbi2d 308 . . . . . 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 21954 . . . . 5  |-  ( (
ph  /\  a  e.  Y )  ->  (
b  e.  Y  -> 
( a H b )  =  ( b H a ) ) )
4241impr 603 . . . 4  |-  ( (
ph  /\  ( a  e.  Y  /\  b  e.  Y ) )  -> 
( a H b )  =  ( b H a ) )
4318, 42syldan 457 . . 3  |-  ( (
ph  /\  ( a  e.  ran  H  /\  b  e.  ran  H ) )  ->  ( a H b )  =  ( b H a ) )
4443ralrimivva 2798 . 2  |-  ( ph  ->  A. a  e.  ran  H A. b  e.  ran  H ( a H b )  =  ( b H a ) )
45 eqid 2436 . . 3  |-  ran  H  =  ran  H
4645isablo 21871 . 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 646 1  |-  ( ph  ->  H  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320    X. cxp 4876   dom cdm 4878   ran crn 4879    |` cres 4880    Fn wfn 5449   -onto->wfo 5452   ` cfv 5454  (class class class)co 6081   GrpOpcgr 21774   AbelOpcablo 21869
This theorem is referenced by:  ghsubgolem  21958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-grpo 21779  df-gid 21780  df-ginv 21781  df-gdiv 21782  df-ablo 21870
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