Theorem qusghm 13733

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 2207	. 2 |- ( Base ` H ) = ( Base ` H )
3		eqid 2207	. 2 |- ( +g ` G ) = ( +g ` G )
4		eqid 2207	. 2 |- ( +g ` H ) = ( +g ` H )
5		nsgsubg 13656	. . 3 |- ( Y e. (NrmSGrp ` G ) -> Y e. (SubGrp ` G ) )
6		subgrcl 13630	. . 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 13683	. 2 |- ( Y e. (NrmSGrp ` G ) -> H e. Grp )
10	8, 1, 2	quseccl 13684	. . 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 5757	. 2 |- ( Y e. (NrmSGrp ` G ) -> F : X --> ( Base ` H ) )
13	8, 1, 3, 4	qusadd 13685	. . . 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 6678	. . . . 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 13672	. . . . . . . 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 6647	. . . . . 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 5696	. . . 4 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> ( F ` y ) = [ y ] ( G ~QG Y ) )
23		eceq1 6678	. . . . 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 6647	. . . . . 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 5696	. . . 4 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> ( F ` z ) = [ z ] ( G ~QG Y ) )
28	22, 27	oveq12d 5985	. . 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 6678	. . . 4 |- ( x = ( y ( +g ` G ) z ) -> [ x ] ( G ~QG Y ) = [ ( y ( +g ` G ) z ) ] ( G ~QG Y ) )
30	1, 3	grpcl 13455	. . . . . 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 6647	. . . . 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 5696	. . 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 2251	. 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 13703	1 |- ( Y e. (NrmSGrp ` G ) -> F e. ( G GrpHom H ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   _Vcvv 2776   |-> cmpt 4121   ` cfv 5290  (class class class)co 5967   [cec 6641   Basecbs 12947   +g cplusg 13024   /._s cqus 13247   Grpcgrp 13447  SubGrpcsubg 13618  NrmSGrpcnsg 13619   ~_QG cqg 13620   GrpHom cghm 13691

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-er 6643  df-ec 6645  df-qs 6649  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955  df-plusg 13037  df-mulr 13038  df-0g 13205  df-iimas 13249  df-qus 13250  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-subg 13621  df-nsg 13622  df-eqg 13623  df-ghm 13692

This theorem is referenced by:  qusrhm  14405