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Theorem divsghm 14969
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 2387 . 2  |-  ( Base `  H )  =  (
Base `  H )
3 eqid 2387 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqid 2387 . 2  |-  ( +g  `  H )  =  ( +g  `  H )
5 nsgsubg 14899 . . 3  |-  ( Y  e.  (NrmSGrp `  G
)  ->  Y  e.  (SubGrp `  G ) )
6 subgrcl 14876 . . 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 14922 . 2  |-  ( Y  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )
108, 1, 2divseccl 14923 . . 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 5832 . 2  |-  ( Y  e.  (NrmSGrp `  G
)  ->  F : X
--> ( Base `  H
) )
138, 1, 3, 4divsadd 14924 . . . 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 1154 . . 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 6877 . . . . . 6  |-  ( x  =  y  ->  [ x ] ( G ~QG  Y )  =  [ y ] ( G ~QG  Y ) )
16 ovex 6045 . . . . . . 7  |-  ( G ~QG  Y )  e.  _V
17 ecexg 6845 . . . . . . 7  |-  ( ( G ~QG  Y )  e.  _V  ->  [ x ] ( G ~QG  Y )  e.  _V )
1816, 17ax-mp 8 . . . . . 6  |-  [ x ] ( G ~QG  Y )  e.  _V
1915, 11, 18fvmpt3i 5748 . . . . 5  |-  ( y  e.  X  ->  ( F `  y )  =  [ y ] ( G ~QG  Y ) )
2019ad2antrl 709 . . . 4  |-  ( ( Y  e.  (NrmSGrp `  G
)  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( F `  y
)  =  [ y ] ( G ~QG  Y ) )
21 eceq1 6877 . . . . . 6  |-  ( x  =  z  ->  [ x ] ( G ~QG  Y )  =  [ z ] ( G ~QG  Y ) )
2221, 11, 18fvmpt3i 5748 . . . . 5  |-  ( z  e.  X  ->  ( F `  z )  =  [ z ] ( G ~QG  Y ) )
2322ad2antll 710 . . . 4  |-  ( ( Y  e.  (NrmSGrp `  G
)  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( F `  z
)  =  [ z ] ( G ~QG  Y ) )
2420, 23oveq12d 6038 . . 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 14745 . . . . . 6  |-  ( ( G  e.  Grp  /\  y  e.  X  /\  z  e.  X )  ->  ( y ( +g  `  G ) z )  e.  X )
26253expb 1154 . . . . 5  |-  ( ( G  e.  Grp  /\  ( y  e.  X  /\  z  e.  X
) )  ->  (
y ( +g  `  G
) z )  e.  X )
277, 26sylan 458 . . . 4  |-  ( ( Y  e.  (NrmSGrp `  G
)  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( y ( +g  `  G ) z )  e.  X )
28 eceq1 6877 . . . . 5  |-  ( x  =  ( y ( +g  `  G ) z )  ->  [ x ] ( G ~QG  Y )  =  [ ( y ( +g  `  G
) z ) ] ( G ~QG  Y ) )
2928, 11, 18fvmpt3i 5748 . . . 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 2430 . 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 14942 1  |-  ( Y  e.  (NrmSGrp `  G
)  ->  F  e.  ( G  GrpHom  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899    e. cmpt 4207   ` cfv 5394  (class class class)co 6020   [cec 6839   Basecbs 13396   +g cplusg 13456    /.s cqus 13658   Grpcgrp 14612  SubGrpcsubg 14865  NrmSGrpcnsg 14866   ~QG cqg 14867    GrpHom cghm 14930
This theorem is referenced by:  divsrhm  16235
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-ec 6843  df-qs 6847  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-fz 10976  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-0g 13654  df-imas 13661  df-divs 13662  df-mnd 14617  df-grp 14739  df-minusg 14740  df-subg 14868  df-nsg 14869  df-eqg 14870  df-ghm 14931
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