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Theorem divsghm 15034
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 2435 . 2  |-  ( Base `  H )  =  (
Base `  H )
3 eqid 2435 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqid 2435 . 2  |-  ( +g  `  H )  =  ( +g  `  H )
5 nsgsubg 14964 . . 3  |-  ( Y  e.  (NrmSGrp `  G
)  ->  Y  e.  (SubGrp `  G ) )
6 subgrcl 14941 . . 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 14987 . 2  |-  ( Y  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )
108, 1, 2divseccl 14988 . . 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 5885 . 2  |-  ( Y  e.  (NrmSGrp `  G
)  ->  F : X
--> ( Base `  H
) )
138, 1, 3, 4divsadd 14989 . . . 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 6933 . . . . . 6  |-  ( x  =  y  ->  [ x ] ( G ~QG  Y )  =  [ y ] ( G ~QG  Y ) )
16 ovex 6098 . . . . . . 7  |-  ( G ~QG  Y )  e.  _V
17 ecexg 6901 . . . . . . 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 5801 . . . . 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 6933 . . . . . 6  |-  ( x  =  z  ->  [ x ] ( G ~QG  Y )  =  [ z ] ( G ~QG  Y ) )
2221, 11, 18fvmpt3i 5801 . . . . 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 6091 . . 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 14810 . . . . . 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 6933 . . . . 5  |-  ( x  =  ( y ( +g  `  G ) z )  ->  [ x ] ( G ~QG  Y )  =  [ ( y ( +g  `  G
) z ) ] ( G ~QG  Y ) )
2928, 11, 18fvmpt3i 5801 . . . 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 2478 . 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 15007 1  |-  ( Y  e.  (NrmSGrp `  G
)  ->  F  e.  ( G  GrpHom  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   [cec 6895   Basecbs 13461   +g cplusg 13521    /.s cqus 13723   Grpcgrp 14677  SubGrpcsubg 14930  NrmSGrpcnsg 14931   ~QG cqg 14932    GrpHom cghm 14995
This theorem is referenced by:  divsrhm  16300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-ec 6899  df-qs 6903  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-0g 13719  df-imas 13726  df-divs 13727  df-mnd 14682  df-grp 14804  df-minusg 14805  df-subg 14933  df-nsg 14934  df-eqg 14935  df-ghm 14996
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