Theorem qusghm 13412

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 2196	. 2 |- ( Base ` H ) = ( Base ` H )
3		eqid 2196	. 2 |- ( +g ` G ) = ( +g ` G )
4		eqid 2196	. 2 |- ( +g ` H ) = ( +g ` H )
5		nsgsubg 13335	. . 3 |- ( Y e. (NrmSGrp ` G ) -> Y e. (SubGrp ` G ) )
6		subgrcl 13309	. . 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 13362	. 2 |- ( Y e. (NrmSGrp ` G ) -> H e. Grp )
10	8, 1, 2	quseccl 13363	. . 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 5716	. 2 |- ( Y e. (NrmSGrp ` G ) -> F : X --> ( Base ` H ) )
13	8, 1, 3, 4	qusadd 13364	. . . 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 1206	. . 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 6627	. . . . 5 |- ( x = y -> [ x ] ( G ~QG Y ) = [ y ] ( G ~QG Y ) )
16		simprl 529	. . . . 5 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> y e. X )
17		eqgex 13351	. . . . . . . 8 |- ( ( G e. Grp /\ Y e. (NrmSGrp ` G ) ) -> ( G ~QG Y ) e. _V )
18	7, 17	mpancom 422	. . . . . . 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 6596	. . . . . 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 5655	. . . 4 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> ( F ` y ) = [ y ] ( G ~QG Y ) )
23		eceq1 6627	. . . . 5 |- ( x = z -> [ x ] ( G ~QG Y ) = [ z ] ( G ~QG Y ) )
24		simprr 531	. . . . 5 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> z e. X )
25		ecexg 6596	. . . . . 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 5655	. . . 4 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> ( F ` z ) = [ z ] ( G ~QG Y ) )
28	22, 27	oveq12d 5940	. . 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 6627	. . . 4 |- ( x = ( y ( +g ` G ) z ) -> [ x ] ( G ~QG Y ) = [ ( y ( +g ` G ) z ) ] ( G ~QG Y ) )
30	1, 3	grpcl 13140	. . . . . 6 |- ( ( G e. Grp /\ y e. X /\ z e. X ) -> ( y ( +g ` G ) z ) e. X )
31	30	3expb 1206	. . . . 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 6596	. . . . 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 5655	. . 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 2240	. 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 13382	1 |- ( Y e. (NrmSGrp ` G ) -> F e. ( G GrpHom H ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   _Vcvv 2763   |-> cmpt 4094   ` cfv 5258  (class class class)co 5922   [cec 6590   Basecbs 12678   +g cplusg 12755   /._s cqus 12943   Grpcgrp 13132  SubGrpcsubg 13297  NrmSGrpcnsg 13298   ~_QG cqg 13299   GrpHom cghm 13370

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-tp 3630  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-er 6592  df-ec 6594  df-qs 6598  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-0g 12929  df-iimas 12945  df-qus 12946  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-subg 13300  df-nsg 13301  df-eqg 13302  df-ghm 13371

This theorem is referenced by:  qusrhm  14084