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Theorem divsghm 14719
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 2283 . 2  |-  ( Base `  H )  =  (
Base `  H )
3 eqid 2283 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqid 2283 . 2  |-  ( +g  `  H )  =  ( +g  `  H )
5 nsgsubg 14649 . . 3  |-  ( Y  e.  (NrmSGrp `  G
)  ->  Y  e.  (SubGrp `  G ) )
6 subgrcl 14626 . . 3  |-  ( Y  e.  (SubGrp `  G
)  ->  G  e.  Grp )
75, 6syl 15 . 2  |-  ( Y  e.  (NrmSGrp `  G
)  ->  G  e.  Grp )
8 divsghm.h . . 3  |-  H  =  ( G  /.s  ( G ~QG  Y
) )
98divsgrp 14672 . 2  |-  ( Y  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )
108, 1, 2divseccl 14673 . . 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 5684 . 2  |-  ( Y  e.  (NrmSGrp `  G
)  ->  F : X
--> ( Base `  H
) )
138, 1, 3, 4divsadd 14674 . . . 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 1152 . . 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 6696 . . . . . 6  |-  ( x  =  y  ->  [ x ] ( G ~QG  Y )  =  [ y ] ( G ~QG  Y ) )
16 ovex 5883 . . . . . . 7  |-  ( G ~QG  Y )  e.  _V
17 ecexg 6664 . . . . . . 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 5605 . . . . 5  |-  ( y  e.  X  ->  ( F `  y )  =  [ y ] ( G ~QG  Y ) )
2019ad2antrl 708 . . . 4  |-  ( ( Y  e.  (NrmSGrp `  G
)  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( F `  y
)  =  [ y ] ( G ~QG  Y ) )
21 eceq1 6696 . . . . . 6  |-  ( x  =  z  ->  [ x ] ( G ~QG  Y )  =  [ z ] ( G ~QG  Y ) )
2221, 11, 18fvmpt3i 5605 . . . . 5  |-  ( z  e.  X  ->  ( F `  z )  =  [ z ] ( G ~QG  Y ) )
2322ad2antll 709 . . . 4  |-  ( ( Y  e.  (NrmSGrp `  G
)  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( F `  z
)  =  [ z ] ( G ~QG  Y ) )
2420, 23oveq12d 5876 . . 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 14495 . . . . . 6  |-  ( ( G  e.  Grp  /\  y  e.  X  /\  z  e.  X )  ->  ( y ( +g  `  G ) z )  e.  X )
26253expb 1152 . . . . 5  |-  ( ( G  e.  Grp  /\  ( y  e.  X  /\  z  e.  X
) )  ->  (
y ( +g  `  G
) z )  e.  X )
277, 26sylan 457 . . . 4  |-  ( ( Y  e.  (NrmSGrp `  G
)  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( y ( +g  `  G ) z )  e.  X )
28 eceq1 6696 . . . . 5  |-  ( x  =  ( y ( +g  `  G ) z )  ->  [ x ] ( G ~QG  Y )  =  [ ( y ( +g  `  G
) z ) ] ( G ~QG  Y ) )
2928, 11, 18fvmpt3i 5605 . . . 4  |-  ( ( y ( +g  `  G
) z )  e.  X  ->  ( F `  ( y ( +g  `  G ) z ) )  =  [ ( y ( +g  `  G
) z ) ] ( G ~QG  Y ) )
3027, 29syl 15 . . 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 2326 . 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 14692 1  |-  ( Y  e.  (NrmSGrp `  G
)  ->  F  e.  ( G  GrpHom  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   [cec 6658   Basecbs 13148   +g cplusg 13208    /.s cqus 13408   Grpcgrp 14362  SubGrpcsubg 14615  NrmSGrpcnsg 14616   ~QG cqg 14617    GrpHom cghm 14680
This theorem is referenced by:  divsrhm  15989
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-ghm 14681
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