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Theorem divsghm 15906
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 2454 . 2  |-  ( Base `  H )  =  (
Base `  H )
3 eqid 2454 . 2  |-  ( +g  `  G )  =  ( +g  `  G )
4 eqid 2454 . 2  |-  ( +g  `  H )  =  ( +g  `  H )
5 nsgsubg 15836 . . 3  |-  ( Y  e.  (NrmSGrp `  G
)  ->  Y  e.  (SubGrp `  G ) )
6 subgrcl 15809 . . 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 15859 . 2  |-  ( Y  e.  (NrmSGrp `  G
)  ->  H  e.  Grp )
108, 1, 2divseccl 15860 . . 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 5979 . 2  |-  ( Y  e.  (NrmSGrp `  G
)  ->  F : X
--> ( Base `  H
) )
138, 1, 3, 4divsadd 15861 . . . 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 1189 . . 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 7250 . . . . . 6  |-  ( x  =  y  ->  [ x ] ( G ~QG  Y )  =  [ y ] ( G ~QG  Y ) )
16 ovex 6228 . . . . . . 7  |-  ( G ~QG  Y )  e.  _V
17 ecexg 7218 . . . . . . 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 5890 . . . . 5  |-  ( y  e.  X  ->  ( F `  y )  =  [ y ] ( G ~QG  Y ) )
2019ad2antrl 727 . . . 4  |-  ( ( Y  e.  (NrmSGrp `  G
)  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( F `  y
)  =  [ y ] ( G ~QG  Y ) )
21 eceq1 7250 . . . . . 6  |-  ( x  =  z  ->  [ x ] ( G ~QG  Y )  =  [ z ] ( G ~QG  Y ) )
2221, 11, 18fvmpt3i 5890 . . . . 5  |-  ( z  e.  X  ->  ( F `  z )  =  [ z ] ( G ~QG  Y ) )
2322ad2antll 728 . . . 4  |-  ( ( Y  e.  (NrmSGrp `  G
)  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( F `  z
)  =  [ z ] ( G ~QG  Y ) )
2420, 23oveq12d 6221 . . 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 15674 . . . . . 6  |-  ( ( G  e.  Grp  /\  y  e.  X  /\  z  e.  X )  ->  ( y ( +g  `  G ) z )  e.  X )
26253expb 1189 . . . . 5  |-  ( ( G  e.  Grp  /\  ( y  e.  X  /\  z  e.  X
) )  ->  (
y ( +g  `  G
) z )  e.  X )
277, 26sylan 471 . . . 4  |-  ( ( Y  e.  (NrmSGrp `  G
)  /\  ( y  e.  X  /\  z  e.  X ) )  -> 
( y ( +g  `  G ) z )  e.  X )
28 eceq1 7250 . . . . 5  |-  ( x  =  ( y ( +g  `  G ) z )  ->  [ x ] ( G ~QG  Y )  =  [ ( y ( +g  `  G
) z ) ] ( G ~QG  Y ) )
2928, 11, 18fvmpt3i 5890 . . . 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 2506 . 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 15879 1  |-  ( Y  e.  (NrmSGrp `  G
)  ->  F  e.  ( G  GrpHom  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078    |-> cmpt 4461   ` cfv 5529  (class class class)co 6203   [cec 7212   Basecbs 14296   +g cplusg 14361    /.s cqus 14566   Grpcgrp 15533  SubGrpcsubg 15798  NrmSGrpcnsg 15799   ~QG cqg 15800    GrpHom cghm 15867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-ec 7216  df-qs 7220  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7806  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-10 10503  df-n0 10695  df-z 10762  df-dec 10871  df-uz 10977  df-fz 11559  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-ress 14303  df-plusg 14374  df-mulr 14375  df-sca 14377  df-vsca 14378  df-ip 14379  df-tset 14380  df-ple 14381  df-ds 14383  df-0g 14503  df-imas 14569  df-divs 14570  df-mnd 15538  df-grp 15668  df-minusg 15669  df-subg 15801  df-nsg 15802  df-eqg 15803  df-ghm 15868
This theorem is referenced by:  divsrhm  17452
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