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Theorem divsabl 15157
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 15139 . . . . 5  |-  ( G  e.  Abel  ->  (NrmSGrp `  G
)  =  (SubGrp `  G ) )
21eleq2d 2350 . . . 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 14672 . . 3  |-  ( S  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )
63, 5syl 15 . 2  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  H  e.  Grp )
7 vex 2791 . . . . . . 7  |-  x  e. 
_V
87elqs 6712 . . . . . 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 2284 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( Base `  G )  =  (
Base `  G )
)
11 ovex 5883 . . . . . . . . 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 13447 . . . . . . 7  |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G )
)  ->  ( ( Base `  G ) /. ( G ~QG  S ) )  =  ( Base `  H
) )
1514eleq2d 2350 . . . . . 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 2791 . . . . . . 7  |-  y  e. 
_V
1817elqs 6712 . . . . . 6  |-  ( y  e.  ( ( Base `  G ) /. ( G ~QG  S ) )  <->  E. b  e.  ( Base `  G
) y  =  [
b ] ( G ~QG  S ) )
1914eleq2d 2350 . . . . . 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 2707 . . . . 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 2283 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
24 eqid 2283 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
2523, 24ablcom 15106 . . . . . . . . . . 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 6696 . . . . . . . . 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 2283 . . . . . . . . . 10  |-  ( +g  `  H )  =  ( +g  `  H )
344, 23, 24, 33divsadd 14674 . . . . . . . . 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 14674 . . . . . . . . 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 2325 . . . . . . 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 5867 . . . . . . . 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 5867 . . . . . . . . 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 2297 . . . . . . 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 2674 . . . . 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 2634 . 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 2283 . . 3  |-  ( Base `  H )  =  (
Base `  H )
4948, 33isabl2 15097 . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788   ` cfv 5255  (class class class)co 5858   [cec 6658   /.cqs 6659   Basecbs 13148   +g cplusg 13208    /.s cqus 13408   Grpcgrp 14362  SubGrpcsubg 14615  NrmSGrpcnsg 14616   ~QG cqg 14617   Abelcabel 15090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-imas 13411  df-divs 13412  df-mnd 14367  df-grp 14489  df-minusg 14490  df-subg 14618  df-nsg 14619  df-eqg 14620  df-cmn 15091  df-abl 15092
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