Theorem qusghm 13999

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 2232	. 2 |- ( Base ` H ) = ( Base ` H )
3		eqid 2232	. 2 |- ( +g ` G ) = ( +g ` G )
4		eqid 2232	. 2 |- ( +g ` H ) = ( +g ` H )
5		nsgsubg 13922	. . 3 |- ( Y e. (NrmSGrp ` G ) -> Y e. (SubGrp ` G ) )
6		subgrcl 13896	. . 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 13949	. 2 |- ( Y e. (NrmSGrp ` G ) -> H e. Grp )
10	8, 1, 2	quseccl 13950	. . 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 5831	. 2 |- ( Y e. (NrmSGrp ` G ) -> F : X --> ( Base ` H ) )
13	8, 1, 3, 4	qusadd 13951	. . . 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 1231	. . 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 6802	. . . . 5 |- ( x = y -> [ x ] ( G ~QG Y ) = [ y ] ( G ~QG Y ) )
16		simprl 531	. . . . 5 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> y e. X )
17		eqgex 13938	. . . . . . . 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 6771	. . . . . 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 5771	. . . 4 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> ( F ` y ) = [ y ] ( G ~QG Y ) )
23		eceq1 6802	. . . . 5 |- ( x = z -> [ x ] ( G ~QG Y ) = [ z ] ( G ~QG Y ) )
24		simprr 533	. . . . 5 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> z e. X )
25		ecexg 6771	. . . . . 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 5771	. . . 4 |- ( ( Y e. (NrmSGrp ` G ) /\ ( y e. X /\ z e. X ) ) -> ( F ` z ) = [ z ] ( G ~QG Y ) )
28	22, 27	oveq12d 6068	. . 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 6802	. . . 4 |- ( x = ( y ( +g ` G ) z ) -> [ x ] ( G ~QG Y ) = [ ( y ( +g ` G ) z ) ] ( G ~QG Y ) )
30	1, 3	grpcl 13721	. . . . . 6 |- ( ( G e. Grp /\ y e. X /\ z e. X ) -> ( y ( +g ` G ) z ) e. X )
31	30	3expb 1231	. . . . 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 6771	. . . . 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 5771	. . 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 2276	. 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 13969	1 |- ( Y e. (NrmSGrp ` G ) -> F e. ( G GrpHom H ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   _Vcvv 2813   |-> cmpt 4171   ` cfv 5352  (class class class)co 6050   [cec 6765   Basecbs 13212   +g cplusg 13290   /._s cqus 13513   Grpcgrp 13713  SubGrpcsubg 13884  NrmSGrpcnsg 13885   ~_QG cqg 13886   GrpHom cghm 13957

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-tp 3697  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-er 6767  df-ec 6769  df-qs 6773  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220  df-plusg 13303  df-mulr 13304  df-0g 13471  df-iimas 13515  df-qus 13516  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717  df-subg 13887  df-nsg 13888  df-eqg 13889  df-ghm 13958

This theorem is referenced by:  qusrhm  14676