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Theorem divsghm 15763
Description: If  Y is a normal subgroup of  G, then the "natural map" from elements to their cosets is a group homomorphism from  G to  G  /  Y. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 18-Sep-2015.)
Hypotheses
Ref Expression
divsghm.x  |-  X  =  ( Base `  G
)
divsghm.h  |-  H  =  ( G  /.s  ( G ~QG  Y
) )
divsghm.f  |-  F  =  ( x  e.  X  |->  [ x ] ( G ~QG  Y ) )
Assertion
Ref Expression
divsghm  |-  ( Y  e.  (NrmSGrp `  G
)  ->  F  e.  ( G  GrpHom  H ) )
Distinct variable groups:    x, G    x, H    x, X    x, Y
Allowed substitution hint:    F( x)

Proof of Theorem divsghm
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 divsghm.x . 2  |-  X  =  ( Base `  G
)
2 eqid 2433 . 2  |-  ( Base `  H )  =  (
Base `  H )
3 eqid 2433 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqid 2433 . 2  |-  ( +g  `  H )  =  ( +g  `  H )
5 nsgsubg 15693 . . 3  |-  ( Y  e.  (NrmSGrp `  G
)  ->  Y  e.  (SubGrp `  G ) )
6 subgrcl 15666 . . 3  |-  ( Y  e.  (SubGrp `  G
)  ->  G  e.  Grp )
75, 6syl 16 . 2  |-  ( Y  e.  (NrmSGrp `  G
)  ->  G  e.  Grp )
8 divsghm.h . . 3  |-  H  =  ( G  /.s  ( G ~QG  Y
) )
98divsgrp 15716 . 2  |-  ( Y  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )
108, 1, 2divseccl 15717 . . 3  |-  ( ( Y  e.  (NrmSGrp `  G
)  /\  x  e.  X )  ->  [ x ] ( G ~QG  Y )  e.  ( Base `  H
) )
11 divsghm.f . . 3  |-  F  =  ( x  e.  X  |->  [ x ] ( G ~QG  Y ) )
1210, 11fmptd 5855 . 2  |-  ( Y  e.  (NrmSGrp `  G
)  ->  F : X
--> ( Base `  H
) )
138, 1, 3, 4divsadd 15718 . . . 4  |-  ( ( Y  e.  (NrmSGrp `  G
)  /\  y  e.  X  /\  z  e.  X
)  ->  ( [
y ] ( G ~QG  Y ) ( +g  `  H
) [ z ] ( G ~QG  Y ) )  =  [ ( y ( +g  `  G ) z ) ] ( G ~QG  Y ) )
14133expb 1181 . . 3  |-  ( ( Y  e.  (NrmSGrp `  G
)  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( [ y ] ( G ~QG  Y ) ( +g  `  H ) [ z ] ( G ~QG  Y ) )  =  [ ( y ( +g  `  G
) z ) ] ( G ~QG  Y ) )
15 eceq1 7125 . . . . . 6  |-  ( x  =  y  ->  [ x ] ( G ~QG  Y )  =  [ y ] ( G ~QG  Y ) )
16 ovex 6105 . . . . . . 7  |-  ( G ~QG  Y )  e.  _V
17 ecexg 7093 . . . . . . 7  |-  ( ( G ~QG  Y )  e.  _V  ->  [ x ] ( G ~QG  Y )  e.  _V )
1816, 17ax-mp 5 . . . . . 6  |-  [ x ] ( G ~QG  Y )  e.  _V
1915, 11, 18fvmpt3i 5766 . . . . 5  |-  ( y  e.  X  ->  ( F `  y )  =  [ y ] ( G ~QG  Y ) )
2019ad2antrl 720 . . . 4  |-  ( ( Y  e.  (NrmSGrp `  G
)  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( F `  y
)  =  [ y ] ( G ~QG  Y ) )
21 eceq1 7125 . . . . . 6  |-  ( x  =  z  ->  [ x ] ( G ~QG  Y )  =  [ z ] ( G ~QG  Y ) )
2221, 11, 18fvmpt3i 5766 . . . . 5  |-  ( z  e.  X  ->  ( F `  z )  =  [ z ] ( G ~QG  Y ) )
2322ad2antll 721 . . . 4  |-  ( ( Y  e.  (NrmSGrp `  G
)  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( F `  z
)  =  [ z ] ( G ~QG  Y ) )
2420, 23oveq12d 6098 . . 3  |-  ( ( Y  e.  (NrmSGrp `  G
)  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( ( F `  y ) ( +g  `  H ) ( F `
 z ) )  =  ( [ y ] ( G ~QG  Y ) ( +g  `  H
) [ z ] ( G ~QG  Y ) ) )
251, 3grpcl 15531 . . . . . 6  |-  ( ( G  e.  Grp  /\  y  e.  X  /\  z  e.  X )  ->  ( y ( +g  `  G ) z )  e.  X )
26253expb 1181 . . . . 5  |-  ( ( G  e.  Grp  /\  ( y  e.  X  /\  z  e.  X
) )  ->  (
y ( +g  `  G
) z )  e.  X )
277, 26sylan 468 . . . 4  |-  ( ( Y  e.  (NrmSGrp `  G
)  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( y ( +g  `  G ) z )  e.  X )
28 eceq1 7125 . . . . 5  |-  ( x  =  ( y ( +g  `  G ) z )  ->  [ x ] ( G ~QG  Y )  =  [ ( y ( +g  `  G
) z ) ] ( G ~QG  Y ) )
2928, 11, 18fvmpt3i 5766 . . . 4  |-  ( ( y ( +g  `  G
) z )  e.  X  ->  ( F `  ( y ( +g  `  G ) z ) )  =  [ ( y ( +g  `  G
) z ) ] ( G ~QG  Y ) )
3027, 29syl 16 . . 3  |-  ( ( Y  e.  (NrmSGrp `  G
)  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( F `  (
y ( +g  `  G
) z ) )  =  [ ( y ( +g  `  G
) z ) ] ( G ~QG  Y ) )
3114, 24, 303eqtr4rd 2476 . 2  |-  ( ( Y  e.  (NrmSGrp `  G
)  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( F `  (
y ( +g  `  G
) z ) )  =  ( ( F `
 y ) ( +g  `  H ) ( F `  z
) ) )
321, 2, 3, 4, 7, 9, 12, 31isghmd 15736 1  |-  ( Y  e.  (NrmSGrp `  G
)  ->  F  e.  ( G  GrpHom  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   _Vcvv 2962    e. cmpt 4338   ` cfv 5406  (class class class)co 6080   [cec 7087   Basecbs 14157   +g cplusg 14221    /.s cqus 14426   Grpcgrp 15393  SubGrpcsubg 15655  NrmSGrpcnsg 15656   ~QG cqg 15657    GrpHom cghm 15724
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-ghm 15725
This theorem is referenced by:  divsrhm  17241
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