Theorem qusghm 13651

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 2205	. 2 |- ( Base ` H ) = ( Base ` H )
3		eqid 2205	. 2 |- ( +g ` G ) = ( +g ` G )
4		eqid 2205	. 2 |- ( +g ` H ) = ( +g ` H )
5		nsgsubg 13574	. . 3 |- ( Y e. (NrmSGrp ` G ) -> Y e. (SubGrp ` G ) )
6		subgrcl 13548	. . 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 13601	. 2 |- ( Y e. (NrmSGrp ` G ) -> H e. Grp )
10	8, 1, 2	quseccl 13602	. . 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 5736	. 2 |- ( Y e. (NrmSGrp ` G ) -> F : X --> ( Base ` H ) )
13	8, 1, 3, 4	qusadd 13603	. . . 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 1207	. . 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 6657	. . . . 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 13590	. . . . . . . 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 6626	. . . . . 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 5675	. . . 4 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> ( F ` y ) = [ y ] ( G ~QG Y ) )
23		eceq1 6657	. . . . 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 6626	. . . . . 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 5675	. . . 4 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> ( F ` z ) = [ z ] ( G ~QG Y ) )
28	22, 27	oveq12d 5964	. . 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 6657	. . . 4 |- ( x = ( y ( +g ` G ) z ) -> [ x ] ( G ~QG Y ) = [ ( y ( +g ` G ) z ) ] ( G ~QG Y ) )
30	1, 3	grpcl 13373	. . . . . 6 |- ( ( G e. Grp /\ y e. X /\ z e. X ) -> ( y ( +g ` G ) z ) e. X )
31	30	3expb 1207	. . . . 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 6626	. . . . 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 5675	. . 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 2249	. 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 13621	1 |- ( Y e. (NrmSGrp ` G ) -> F e. ( G GrpHom H ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   _Vcvv 2772   |-> cmpt 4106   ` cfv 5272  (class class class)co 5946   [cec 6620   Basecbs 12865   +g cplusg 12942   /._s cqus 13165   Grpcgrp 13365  SubGrpcsubg 13536  NrmSGrpcnsg 13537   ~_QG cqg 13538   GrpHom cghm 13609

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-pre-ltirr 8039  ax-pre-lttrn 8041  ax-pre-ltadd 8043

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-tp 3641  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-er 6622  df-ec 6624  df-qs 6628  df-pnf 8111  df-mnf 8112  df-ltxr 8114  df-inn 9039  df-2 9097  df-3 9098  df-ndx 12868  df-slot 12869  df-base 12871  df-sets 12872  df-iress 12873  df-plusg 12955  df-mulr 12956  df-0g 13123  df-iimas 13167  df-qus 13168  df-mgm 13221  df-sgrp 13267  df-mnd 13282  df-grp 13368  df-minusg 13369  df-subg 13539  df-nsg 13540  df-eqg 13541  df-ghm 13610

This theorem is referenced by:  qusrhm  14323