Theorem qusghm 14062

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
qusghm.x	|- X = ( Base ` G )
qusghm.h	|- H = ( G /.s ( G ~QG Y ) )
qusghm.f	|- F = ( x e. X |-> [ x ] ( G ~QG Y ) )

Assertion

Ref	Expression
qusghm	|- ( 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 qusghm

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qusghm.x	. 2 |- X = ( Base ` G )
2		eqid 2238	. 2 |- ( Base ` H ) = ( Base ` H )
3		eqid 2238	. 2 |- ( +g ` G ) = ( +g ` G )
4		eqid 2238	. 2 |- ( +g ` H ) = ( +g ` H )
5		nsgsubg 13985	. . 3 |- ( Y e. (NrmSGrp ` G ) -> Y e. (SubGrp ` G ) )
6		subgrcl 13959	. . 3 |- ( Y e. (SubGrp ` G ) -> G e. Grp )
7	5, 6	syl 14	. 2 |- ( Y e. (NrmSGrp ` G ) -> G e. Grp )
8		qusghm.h	. . 3 |- H = ( G /.s ( G ~QG Y ) )
9	8	qusgrp 14012	. 2 |- ( Y e. (NrmSGrp ` G ) -> H e. Grp )
10	8, 1, 2	quseccl 14013	. . 3 |- ( ( Y e. (NrmSGrp ` G ) /\ x e. X ) -> [ x ] ( G ~QG Y ) e. ( Base ` H ) )
11		qusghm.f	. . 3 |- F = ( x e. X |-> [ x ] ( G ~QG Y ) )
12	10, 11	fmptd 5853	. 2 |- ( Y e. (NrmSGrp ` G ) -> F : X --> ( Base ` H ) )
13	8, 1, 3, 4	qusadd 14014	. . . 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 ) )
14	13	3expb 1235	. . 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 6832	. . . . 5 |- ( x = y -> [ x ] ( G ~QG Y ) = [ y ] ( G ~QG Y ) )
16		simprl 535	. . . . 5 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> y e. X )
17		eqgex 14001	. . . . . . . 8 |- ( ( G e. Grp /\ Y e. (NrmSGrp ` G ) ) -> ( G ~QG Y ) e. _V )
18	7, 17	mpancom 426	. . . . . . 7 |- ( Y e. (NrmSGrp ` G ) -> ( G ~QG Y ) e. _V )
19	18	adantr 276	. . . . . 6 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> ( G ~QG Y ) e. _V )
20		ecexg 6801	. . . . . 6 |- ( ( G ~QG Y ) e. _V -> [ y ] ( G ~QG Y ) e. _V )
21	19, 20	syl 14	. . . . 5 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> [ y ] ( G ~QG Y ) e. _V )
22	11, 15, 16, 21	fvmptd3 5793	. . . 4 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> ( F ` y ) = [ y ] ( G ~QG Y ) )
23		eceq1 6832	. . . . 5 |- ( x = z -> [ x ] ( G ~QG Y ) = [ z ] ( G ~QG Y ) )
24		simprr 537	. . . . 5 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> z e. X )
25		ecexg 6801	. . . . . 6 |- ( ( G ~QG Y ) e. _V -> [ z ] ( G ~QG Y ) e. _V )
26	19, 25	syl 14	. . . . 5 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> [ z ] ( G ~QG Y ) e. _V )
27	11, 23, 24, 26	fvmptd3 5793	. . . 4 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> ( F ` z ) = [ z ] ( G ~QG Y ) )
28	22, 27	oveq12d 6093	. . 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 ) ) )
29		eceq1 6832	. . . 4 |- ( x = ( y ( +g ` G ) z ) -> [ x ] ( G ~QG Y ) = [ ( y ( +g ` G ) z ) ] ( G ~QG Y ) )
30	1, 3	grpcl 13790	. . . . . 6 |- ( ( G e. Grp /\ y e. X /\ z e. X ) -> ( y ( +g ` G ) z ) e. X )
31	30	3expb 1235	. . . . 5 |- ( ( G e. Grp /\ ( y e. X /\ z e. X ) ) -> ( y ( +g ` G ) z ) e. X )
32	7, 31	sylan 283	. . . 4 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> ( y ( +g ` G ) z ) e. X )
33		ecexg 6801	. . . . 5 |- ( ( G ~QG Y ) e. _V -> [ ( y ( +g ` G ) z ) ] ( G ~QG Y ) e. _V )
34	19, 33	syl 14	. . . 4 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> [ ( y ( +g ` G ) z ) ] ( G ~QG Y ) e. _V )
35	11, 29, 32, 34	fvmptd3 5793	. . 3 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> ( F ` ( y ( +g ` G ) z ) ) = [ ( y ( +g ` G ) z ) ] ( G ~QG Y ) )
36	14, 28, 35	3eqtr4rd 2282	. 2 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> ( F ` ( y ( +g ` G ) z ) ) = ( ( F ` y ) ( +g ` H ) ( F ` z ) ) )
37	1, 2, 3, 4, 7, 9, 12, 36	isghmd 14032	1 |- ( Y e. (NrmSGrp ` G ) -> F e. ( G GrpHom H ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   |-> cmpt 4187   ` cfv 5372  (class class class)co 6075   [cec 6795   Basecbs 13330   +g cplusg 13408   /._s cqus 13600   Grpcgrp 13782  SubGrpcsubg 13947  NrmSGrpcnsg 13948   ~_QG cqg 13949   GrpHom cghm 14020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-tp 3713  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-er 6797  df-ec 6799  df-qs 6803  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-mulr 13422  df-0g 13589  df-iimas 13601  df-qus 13602  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-subg 13950  df-nsg 13951  df-eqg 13952  df-ghm 14021

This theorem is referenced by:  qusrhm  14837