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Theorem divsabl 16327
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 16309 . . . . 5  |-  ( G  e.  Abel  ->  (NrmSGrp `  G
)  =  (SubGrp `  G ) )
21eleq2d 2500 . . . 4  |-  ( G  e.  Abel  ->  ( S  e.  (NrmSGrp `  G
)  <->  S  e.  (SubGrp `  G ) ) )
32biimpar 482 . . 3  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  S  e.  (NrmSGrp `  G ) )
4 divsabl.h . . . 4  |-  H  =  ( G  /.s  ( G ~QG  S
) )
54divsgrp 15716 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )
63, 5syl 16 . 2  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  H  e.  Grp )
7 vex 2965 . . . . . . 7  |-  x  e. 
_V
87elqs 7141 . . . . . 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 2434 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( Base `  G )  =  (
Base `  G )
)
11 ovex 6105 . . . . . . . . 9  |-  ( G ~QG  S )  e.  _V
1211a1i 11 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( G ~QG  S
)  e.  _V )
13 simpl 454 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  G  e.  Abel )
149, 10, 12, 13divsbas 14466 . . . . . . 7  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( ( Base `  G ) /. ( G ~QG  S ) )  =  ( Base `  H
) )
1514eleq2d 2500 . . . . . 6  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( x  e.  ( ( Base `  G
) /. ( G ~QG  S ) )  <->  x  e.  ( Base `  H )
) )
168, 15syl5bbr 259 . . . . 5  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( E. a  e.  ( Base `  G ) x  =  [ a ] ( G ~QG  S )  <->  x  e.  ( Base `  H )
) )
17 vex 2965 . . . . . . 7  |-  y  e. 
_V
1817elqs 7141 . . . . . 6  |-  ( y  e.  ( ( Base `  G ) /. ( G ~QG  S ) )  <->  E. b  e.  ( Base `  G
) y  =  [
b ] ( G ~QG  S ) )
1914eleq2d 2500 . . . . . 6  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( y  e.  ( ( Base `  G
) /. ( G ~QG  S ) )  <->  y  e.  ( Base `  H )
) )
2018, 19syl5bbr 259 . . . . 5  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( E. b  e.  ( Base `  G ) y  =  [ b ] ( G ~QG  S )  <->  y  e.  ( Base `  H )
) )
2116, 20anbi12d 703 . . . 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 2878 . . . . 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 2433 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
24 eqid 2433 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
2523, 24ablcom 16274 . . . . . . . . . . 11  |-  ( ( G  e.  Abel  /\  a  e.  ( Base `  G
)  /\  b  e.  ( Base `  G )
)  ->  ( a
( +g  `  G ) b )  =  ( b ( +g  `  G
) a ) )
26253expb 1181 . . . . . . . . . 10  |-  ( ( G  e.  Abel  /\  (
a  e.  ( Base `  G )  /\  b  e.  ( Base `  G
) ) )  -> 
( a ( +g  `  G ) b )  =  ( b ( +g  `  G ) a ) )
2726adantlr 707 . . . . . . . . 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 7125 . . . . . . . . 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 462 . . . . . . . . 9  |-  ( ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) )  /\  ( a  e.  (
Base `  G )  /\  b  e.  ( Base `  G ) ) )  ->  S  e.  (NrmSGrp `  G ) )
31 simprl 748 . . . . . . . . 9  |-  ( ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) )  /\  ( a  e.  (
Base `  G )  /\  b  e.  ( Base `  G ) ) )  ->  a  e.  ( Base `  G )
)
32 simprr 749 . . . . . . . . 9  |-  ( ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) )  /\  ( a  e.  (
Base `  G )  /\  b  e.  ( Base `  G ) ) )  ->  b  e.  ( Base `  G )
)
33 eqid 2433 . . . . . . . . . 10  |-  ( +g  `  H )  =  ( +g  `  H )
344, 23, 24, 33divsadd 15718 . . . . . . . . 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 1211 . . . . . . . 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 15718 . . . . . . . . 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 1211 . . . . . . . 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 2475 . . . . . . 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 6089 . . . . . . . 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 6089 . . . . . . . . 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 450 . . . . . . . 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 2447 . . . . . . 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 222 . . . . . 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 2838 . . . . 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 218 . . . 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 235 . . 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 2797 . 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 2433 . . 3  |-  ( Base `  H )  =  (
Base `  H )
4948, 33isabl2 16265 . 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 657 1  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  H  e.  Abel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   A.wral 2705   E.wrex 2706   _Vcvv 2962   ` cfv 5406  (class class class)co 6080   [cec 7087   /.cqs 7088   Basecbs 14157   +g cplusg 14221    /.s cqus 14426   Grpcgrp 15393  SubGrpcsubg 15655  NrmSGrpcnsg 15656   ~QG cqg 15657   Abelcabel 16258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-ec 7091  df-qs 7095  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-fz 11425  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-0g 14363  df-imas 14429  df-divs 14430  df-mnd 15398  df-grp 15525  df-minusg 15526  df-subg 15658  df-nsg 15659  df-eqg 15660  df-cmn 16259  df-abl 16260
This theorem is referenced by: (None)
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