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Theorem divsabl 15435
Description: If  Y is a subgroup of the abelian group  G, then  H  =  G  /  Y is an abelian group. (Contributed by Mario Carneiro, 26-Apr-2016.)
Hypothesis
Ref Expression
divsabl.h  |-  H  =  ( G  /.s  ( G ~QG  S
) )
Assertion
Ref Expression
divsabl  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  H  e.  Abel )

Proof of Theorem divsabl
Dummy variables  a 
b  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ablnsg 15417 . . . . 5  |-  ( G  e.  Abel  ->  (NrmSGrp `  G
)  =  (SubGrp `  G ) )
21eleq2d 2471 . . . 4  |-  ( G  e.  Abel  ->  ( S  e.  (NrmSGrp `  G
)  <->  S  e.  (SubGrp `  G ) ) )
32biimpar 472 . . 3  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  S  e.  (NrmSGrp `  G ) )
4 divsabl.h . . . 4  |-  H  =  ( G  /.s  ( G ~QG  S
) )
54divsgrp 14950 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )
63, 5syl 16 . 2  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  H  e.  Grp )
7 vex 2919 . . . . . . 7  |-  x  e. 
_V
87elqs 6916 . . . . . 6  |-  ( x  e.  ( ( Base `  G ) /. ( G ~QG  S ) )  <->  E. a  e.  ( Base `  G
) x  =  [
a ] ( G ~QG  S ) )
94a1i 11 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  H  =  ( G  /.s  ( G ~QG  S ) ) )
10 eqidd 2405 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( Base `  G )  =  (
Base `  G )
)
11 ovex 6065 . . . . . . . . 9  |-  ( G ~QG  S )  e.  _V
1211a1i 11 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( G ~QG  S
)  e.  _V )
13 simpl 444 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  G  e.  Abel )
149, 10, 12, 13divsbas 13725 . . . . . . 7  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( ( Base `  G ) /. ( G ~QG  S ) )  =  ( Base `  H
) )
1514eleq2d 2471 . . . . . 6  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( x  e.  ( ( Base `  G
) /. ( G ~QG  S ) )  <->  x  e.  ( Base `  H )
) )
168, 15syl5bbr 251 . . . . 5  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( E. a  e.  ( Base `  G ) x  =  [ a ] ( G ~QG  S )  <->  x  e.  ( Base `  H )
) )
17 vex 2919 . . . . . . 7  |-  y  e. 
_V
1817elqs 6916 . . . . . 6  |-  ( y  e.  ( ( Base `  G ) /. ( G ~QG  S ) )  <->  E. b  e.  ( Base `  G
) y  =  [
b ] ( G ~QG  S ) )
1914eleq2d 2471 . . . . . 6  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( y  e.  ( ( Base `  G
) /. ( G ~QG  S ) )  <->  y  e.  ( Base `  H )
) )
2018, 19syl5bbr 251 . . . . 5  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( E. b  e.  ( Base `  G ) y  =  [ b ] ( G ~QG  S )  <->  y  e.  ( Base `  H )
) )
2116, 20anbi12d 692 . . . 4  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( ( E. a  e.  ( Base `  G ) x  =  [ a ] ( G ~QG  S )  /\  E. b  e.  ( Base `  G ) y  =  [ b ] ( G ~QG  S ) )  <->  ( x  e.  ( Base `  H
)  /\  y  e.  ( Base `  H )
) ) )
22 reeanv 2835 . . . . 5  |-  ( E. a  e.  ( Base `  G ) E. b  e.  ( Base `  G
) ( x  =  [ a ] ( G ~QG  S )  /\  y  =  [ b ] ( G ~QG  S ) )  <->  ( E. a  e.  ( Base `  G ) x  =  [ a ] ( G ~QG  S )  /\  E. b  e.  ( Base `  G ) y  =  [ b ] ( G ~QG  S ) ) )
23 eqid 2404 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
24 eqid 2404 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
2523, 24ablcom 15384 . . . . . . . . . . 11  |-  ( ( G  e.  Abel  /\  a  e.  ( Base `  G
)  /\  b  e.  ( Base `  G )
)  ->  ( a
( +g  `  G ) b )  =  ( b ( +g  `  G
) a ) )
26253expb 1154 . . . . . . . . . 10  |-  ( ( G  e.  Abel  /\  (
a  e.  ( Base `  G )  /\  b  e.  ( Base `  G
) ) )  -> 
( a ( +g  `  G ) b )  =  ( b ( +g  `  G ) a ) )
2726adantlr 696 . . . . . . . . 9  |-  ( ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) )  /\  ( a  e.  (
Base `  G )  /\  b  e.  ( Base `  G ) ) )  ->  ( a
( +g  `  G ) b )  =  ( b ( +g  `  G
) a ) )
28 eceq1 6900 . . . . . . . . 9  |-  ( ( a ( +g  `  G
) b )  =  ( b ( +g  `  G ) a )  ->  [ ( a ( +g  `  G
) b ) ] ( G ~QG  S )  =  [
( b ( +g  `  G ) a ) ] ( G ~QG  S ) )
2927, 28syl 16 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) )  /\  ( a  e.  (
Base `  G )  /\  b  e.  ( Base `  G ) ) )  ->  [ (
a ( +g  `  G
) b ) ] ( G ~QG  S )  =  [
( b ( +g  `  G ) a ) ] ( G ~QG  S ) )
303adantr 452 . . . . . . . . 9  |-  ( ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) )  /\  ( a  e.  (
Base `  G )  /\  b  e.  ( Base `  G ) ) )  ->  S  e.  (NrmSGrp `  G ) )
31 simprl 733 . . . . . . . . 9  |-  ( ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) )  /\  ( a  e.  (
Base `  G )  /\  b  e.  ( Base `  G ) ) )  ->  a  e.  ( Base `  G )
)
32 simprr 734 . . . . . . . . 9  |-  ( ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) )  /\  ( a  e.  (
Base `  G )  /\  b  e.  ( Base `  G ) ) )  ->  b  e.  ( Base `  G )
)
33 eqid 2404 . . . . . . . . . 10  |-  ( +g  `  H )  =  ( +g  `  H )
344, 23, 24, 33divsadd 14952 . . . . . . . . 9  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( [ a ] ( G ~QG  S ) ( +g  `  H
) [ b ] ( G ~QG  S ) )  =  [ ( a ( +g  `  G ) b ) ] ( G ~QG  S ) )
3530, 31, 32, 34syl3anc 1184 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) )  /\  ( a  e.  (
Base `  G )  /\  b  e.  ( Base `  G ) ) )  ->  ( [
a ] ( G ~QG  S ) ( +g  `  H
) [ b ] ( G ~QG  S ) )  =  [ ( a ( +g  `  G ) b ) ] ( G ~QG  S ) )
364, 23, 24, 33divsadd 14952 . . . . . . . . 9  |-  ( ( S  e.  (NrmSGrp `  G
)  /\  b  e.  ( Base `  G )  /\  a  e.  ( Base `  G ) )  ->  ( [ b ] ( G ~QG  S ) ( +g  `  H
) [ a ] ( G ~QG  S ) )  =  [ ( b ( +g  `  G ) a ) ] ( G ~QG  S ) )
3730, 32, 31, 36syl3anc 1184 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) )  /\  ( a  e.  (
Base `  G )  /\  b  e.  ( Base `  G ) ) )  ->  ( [
b ] ( G ~QG  S ) ( +g  `  H
) [ a ] ( G ~QG  S ) )  =  [ ( b ( +g  `  G ) a ) ] ( G ~QG  S ) )
3829, 35, 373eqtr4d 2446 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) )  /\  ( a  e.  (
Base `  G )  /\  b  e.  ( Base `  G ) ) )  ->  ( [
a ] ( G ~QG  S ) ( +g  `  H
) [ b ] ( G ~QG  S ) )  =  ( [ b ] ( G ~QG  S ) ( +g  `  H ) [ a ] ( G ~QG  S ) ) )
39 oveq12 6049 . . . . . . . 8  |-  ( ( x  =  [ a ] ( G ~QG  S )  /\  y  =  [
b ] ( G ~QG  S ) )  ->  (
x ( +g  `  H
) y )  =  ( [ a ] ( G ~QG  S ) ( +g  `  H ) [ b ] ( G ~QG  S ) ) )
40 oveq12 6049 . . . . . . . . 9  |-  ( ( y  =  [ b ] ( G ~QG  S )  /\  x  =  [
a ] ( G ~QG  S ) )  ->  (
y ( +g  `  H
) x )  =  ( [ b ] ( G ~QG  S ) ( +g  `  H ) [ a ] ( G ~QG  S ) ) )
4140ancoms 440 . . . . . . . 8  |-  ( ( x  =  [ a ] ( G ~QG  S )  /\  y  =  [
b ] ( G ~QG  S ) )  ->  (
y ( +g  `  H
) x )  =  ( [ b ] ( G ~QG  S ) ( +g  `  H ) [ a ] ( G ~QG  S ) ) )
4239, 41eqeq12d 2418 . . . . . . 7  |-  ( ( x  =  [ a ] ( G ~QG  S )  /\  y  =  [
b ] ( G ~QG  S ) )  ->  (
( x ( +g  `  H ) y )  =  ( y ( +g  `  H ) x )  <->  ( [
a ] ( G ~QG  S ) ( +g  `  H
) [ b ] ( G ~QG  S ) )  =  ( [ b ] ( G ~QG  S ) ( +g  `  H ) [ a ] ( G ~QG  S ) ) ) )
4338, 42syl5ibrcom 214 . . . . . 6  |-  ( ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) )  /\  ( a  e.  (
Base `  G )  /\  b  e.  ( Base `  G ) ) )  ->  ( (
x  =  [ a ] ( G ~QG  S )  /\  y  =  [
b ] ( G ~QG  S ) )  ->  (
x ( +g  `  H
) y )  =  ( y ( +g  `  H ) x ) ) )
4443rexlimdvva 2797 . . . . 5  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( E. a  e.  ( Base `  G ) E. b  e.  ( Base `  G
) ( x  =  [ a ] ( G ~QG  S )  /\  y  =  [ b ] ( G ~QG  S ) )  -> 
( x ( +g  `  H ) y )  =  ( y ( +g  `  H ) x ) ) )
4522, 44syl5bir 210 . . . 4  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( ( E. a  e.  ( Base `  G ) x  =  [ a ] ( G ~QG  S )  /\  E. b  e.  ( Base `  G ) y  =  [ b ] ( G ~QG  S ) )  -> 
( x ( +g  `  H ) y )  =  ( y ( +g  `  H ) x ) ) )
4621, 45sylbird 227 . . 3  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( (
x  e.  ( Base `  H )  /\  y  e.  ( Base `  H
) )  ->  (
x ( +g  `  H
) y )  =  ( y ( +g  `  H ) x ) ) )
4746ralrimivv 2757 . 2  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  A. x  e.  ( Base `  H
) A. y  e.  ( Base `  H
) ( x ( +g  `  H ) y )  =  ( y ( +g  `  H
) x ) )
48 eqid 2404 . . 3  |-  ( Base `  H )  =  (
Base `  H )
4948, 33isabl2 15375 . 2  |-  ( H  e.  Abel  <->  ( H  e. 
Grp  /\  A. x  e.  ( Base `  H
) A. y  e.  ( Base `  H
) ( x ( +g  `  H ) y )  =  ( y ( +g  `  H
) x ) ) )
506, 47, 49sylanbrc 646 1  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  H  e.  Abel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   _Vcvv 2916   ` cfv 5413  (class class class)co 6040   [cec 6862   /.cqs 6863   Basecbs 13424   +g cplusg 13484    /.s cqus 13686   Grpcgrp 14640  SubGrpcsubg 14893  NrmSGrpcnsg 14894   ~QG cqg 14895   Abelcabel 15368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-ec 6866  df-qs 6870  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-0g 13682  df-imas 13689  df-divs 13690  df-mnd 14645  df-grp 14767  df-minusg 14768  df-subg 14896  df-nsg 14897  df-eqg 14898  df-cmn 15369  df-abl 15370
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