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Theorem divsabl 15173
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 15155 . . . . 5  |-  ( G  e.  Abel  ->  (NrmSGrp `  G
)  =  (SubGrp `  G ) )
21eleq2d 2363 . . . 4  |-  ( G  e.  Abel  ->  ( S  e.  (NrmSGrp `  G
)  <->  S  e.  (SubGrp `  G ) ) )
32biimpar 471 . . 3  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  S  e.  (NrmSGrp `  G ) )
4 divsabl.h . . . 4  |-  H  =  ( G  /.s  ( G ~QG  S
) )
54divsgrp 14688 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )
63, 5syl 15 . 2  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  H  e.  Grp )
7 vex 2804 . . . . . . 7  |-  x  e. 
_V
87elqs 6728 . . . . . 6  |-  ( x  e.  ( ( Base `  G ) /. ( G ~QG  S ) )  <->  E. a  e.  ( Base `  G
) x  =  [
a ] ( G ~QG  S ) )
94a1i 10 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  H  =  ( G  /.s  ( G ~QG  S ) ) )
10 eqidd 2297 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( Base `  G )  =  (
Base `  G )
)
11 ovex 5899 . . . . . . . . 9  |-  ( G ~QG  S )  e.  _V
1211a1i 10 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( G ~QG  S
)  e.  _V )
13 simpl 443 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  G  e.  Abel )
149, 10, 12, 13divsbas 13463 . . . . . . 7  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( ( Base `  G ) /. ( G ~QG  S ) )  =  ( Base `  H
) )
1514eleq2d 2363 . . . . . 6  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( x  e.  ( ( Base `  G
) /. ( G ~QG  S ) )  <->  x  e.  ( Base `  H )
) )
168, 15syl5bbr 250 . . . . 5  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( E. a  e.  ( Base `  G ) x  =  [ a ] ( G ~QG  S )  <->  x  e.  ( Base `  H )
) )
17 vex 2804 . . . . . . 7  |-  y  e. 
_V
1817elqs 6728 . . . . . 6  |-  ( y  e.  ( ( Base `  G ) /. ( G ~QG  S ) )  <->  E. b  e.  ( Base `  G
) y  =  [
b ] ( G ~QG  S ) )
1914eleq2d 2363 . . . . . 6  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( y  e.  ( ( Base `  G
) /. ( G ~QG  S ) )  <->  y  e.  ( Base `  H )
) )
2018, 19syl5bbr 250 . . . . 5  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( E. b  e.  ( Base `  G ) y  =  [ b ] ( G ~QG  S )  <->  y  e.  ( Base `  H )
) )
2116, 20anbi12d 691 . . . 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 2720 . . . . 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 2296 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
24 eqid 2296 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
2523, 24ablcom 15122 . . . . . . . . . . 11  |-  ( ( G  e.  Abel  /\  a  e.  ( Base `  G
)  /\  b  e.  ( Base `  G )
)  ->  ( a
( +g  `  G ) b )  =  ( b ( +g  `  G
) a ) )
26253expb 1152 . . . . . . . . . 10  |-  ( ( G  e.  Abel  /\  (
a  e.  ( Base `  G )  /\  b  e.  ( Base `  G
) ) )  -> 
( a ( +g  `  G ) b )  =  ( b ( +g  `  G ) a ) )
2726adantlr 695 . . . . . . . . 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 6712 . . . . . . . . 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 15 . . . . . . . 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 451 . . . . . . . . 9  |-  ( ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) )  /\  ( a  e.  (
Base `  G )  /\  b  e.  ( Base `  G ) ) )  ->  S  e.  (NrmSGrp `  G ) )
31 simprl 732 . . . . . . . . 9  |-  ( ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) )  /\  ( a  e.  (
Base `  G )  /\  b  e.  ( Base `  G ) ) )  ->  a  e.  ( Base `  G )
)
32 simprr 733 . . . . . . . . 9  |-  ( ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) )  /\  ( a  e.  (
Base `  G )  /\  b  e.  ( Base `  G ) ) )  ->  b  e.  ( Base `  G )
)
33 eqid 2296 . . . . . . . . . 10  |-  ( +g  `  H )  =  ( +g  `  H )
344, 23, 24, 33divsadd 14690 . . . . . . . . 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 1182 . . . . . . . 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 14690 . . . . . . . . 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 1182 . . . . . . . 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 2338 . . . . . . 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 5883 . . . . . . . 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 5883 . . . . . . . . 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 439 . . . . . . . 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 2310 . . . . . . 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 213 . . . . . 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 2687 . . . . 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 209 . . . 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 226 . . 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 2647 . 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 2296 . . 3  |-  ( Base `  H )  =  (
Base `  H )
4948, 33isabl2 15113 . 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 645 1  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  H  e.  Abel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801   ` cfv 5271  (class class class)co 5874   [cec 6674   /.cqs 6675   Basecbs 13164   +g cplusg 13224    /.s cqus 13424   Grpcgrp 14378  SubGrpcsubg 14631  NrmSGrpcnsg 14632   ~QG cqg 14633   Abelcabel 15106
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  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-ec 6678  df-qs 6682  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-imas 13427  df-divs 13428  df-mnd 14383  df-grp 14505  df-minusg 14506  df-subg 14634  df-nsg 14635  df-eqg 14636  df-cmn 15107  df-abl 15108
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