Theorem qusghm 14034

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 ∈ 𝑋 ↦ [𝑥](𝐺 ~QG 𝑌))

Assertion

Ref	Expression
qusghm	⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝐹 ∈ (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 2234	. 2 ⊢ (Base‘𝐻) = (Base‘𝐻)
3		eqid 2234	. 2 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2234	. 2 ⊢ (+g‘𝐻) = (+g‘𝐻)
5		nsgsubg 13957	. . 3 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
6		subgrcl 13931	. . 3 ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
7	5, 6	syl 14	. 2 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝐺 ∈ Grp)
8		qusghm.h	. . 3 ⊢ 𝐻 = (𝐺 /s (𝐺 ~QG 𝑌))
9	8	qusgrp 13984	. 2 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp)
10	8, 1, 2	quseccl 13985	. . 3 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ 𝑋) → [𝑥](𝐺 ~QG 𝑌) ∈ (Base‘𝐻))
11		qusghm.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝐺 ~QG 𝑌))
12	10, 11	fmptd 5838	. 2 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝐹:𝑋⟶(Base‘𝐻))
13	8, 1, 3, 4	qusadd 13986	. . . 4 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ([𝑦](𝐺 ~QG 𝑌)(+g‘𝐻)[𝑧](𝐺 ~QG 𝑌)) = [(𝑦(+g‘𝐺)𝑧)](𝐺 ~QG 𝑌))
14	13	3expb 1231	. . 3 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ([𝑦](𝐺 ~QG 𝑌)(+g‘𝐻)[𝑧](𝐺 ~QG 𝑌)) = [(𝑦(+g‘𝐺)𝑧)](𝐺 ~QG 𝑌))
15		eceq1 6817	. . . . 5 ⊢ (𝑥 = 𝑦 → [𝑥](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌))
16		simprl 531	. . . . 5 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
17		eqgex 13973	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ (NrmSGrp‘𝐺)) → (𝐺 ~QG 𝑌) ∈ V)
18	7, 17	mpancom 422	. . . . . . 7 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → (𝐺 ~QG 𝑌) ∈ V)
19	18	adantr 276	. . . . . 6 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐺 ~QG 𝑌) ∈ V)
20		ecexg 6786	. . . . . 6 ⊢ ((𝐺 ~QG 𝑌) ∈ V → [𝑦](𝐺 ~QG 𝑌) ∈ V)
21	19, 20	syl 14	. . . . 5 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → [𝑦](𝐺 ~QG 𝑌) ∈ V)
22	11, 15, 16, 21	fvmptd3 5778	. . . 4 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑦) = [𝑦](𝐺 ~QG 𝑌))
23		eceq1 6817	. . . . 5 ⊢ (𝑥 = 𝑧 → [𝑥](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌))
24		simprr 533	. . . . 5 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
25		ecexg 6786	. . . . . 6 ⊢ ((𝐺 ~QG 𝑌) ∈ V → [𝑧](𝐺 ~QG 𝑌) ∈ V)
26	19, 25	syl 14	. . . . 5 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → [𝑧](𝐺 ~QG 𝑌) ∈ V)
27	11, 23, 24, 26	fvmptd3 5778	. . . 4 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑧) = [𝑧](𝐺 ~QG 𝑌))
28	22, 27	oveq12d 6078	. . 3 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐹‘𝑦)(+g‘𝐻)(𝐹‘𝑧)) = ([𝑦](𝐺 ~QG 𝑌)(+g‘𝐻)[𝑧](𝐺 ~QG 𝑌)))
29		eceq1 6817	. . . 4 ⊢ (𝑥 = (𝑦(+g‘𝐺)𝑧) → [𝑥](𝐺 ~QG 𝑌) = [(𝑦(+g‘𝐺)𝑧)](𝐺 ~QG 𝑌))
30	1, 3	grpcl 13762	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋)
31	30	3expb 1231	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋)
32	7, 31	sylan 283	. . . 4 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋)
33		ecexg 6786	. . . . 5 ⊢ ((𝐺 ~QG 𝑌) ∈ V → [(𝑦(+g‘𝐺)𝑧)](𝐺 ~QG 𝑌) ∈ V)
34	19, 33	syl 14	. . . 4 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → [(𝑦(+g‘𝐺)𝑧)](𝐺 ~QG 𝑌) ∈ V)
35	11, 29, 32, 34	fvmptd3 5778	. . 3 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦(+g‘𝐺)𝑧)) = [(𝑦(+g‘𝐺)𝑧)](𝐺 ~QG 𝑌))
36	14, 28, 35	3eqtr4rd 2278	. 2 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦(+g‘𝐺)𝑧)) = ((𝐹‘𝑦)(+g‘𝐻)(𝐹‘𝑧)))
37	1, 2, 3, 4, 7, 9, 12, 36	isghmd 14004	1 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝐹 ∈ (𝐺 GrpHom 𝐻))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ↦ cmpt 4177  ‘cfv 5359  (class class class)co 6060  [cec 6780  Basecbs 13302  +_gcplusg 13380   /_s cqus 13572  Grpcgrp 13754  SubGrpcsubg 13919  NrmSGrpcnsg 13920   ~_QG cqg 13921   GrpHom cghm 13992

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-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4666  ax-cnex 8236  ax-resscn 8237  ax-1cn 8238  ax-1re 8239  ax-icn 8240  ax-addcl 8241  ax-addrcl 8242  ax-mulcl 8243  ax-addcom 8245  ax-addass 8247  ax-i2m1 8250  ax-0lt1 8251  ax-0id 8253  ax-rnegex 8254  ax-pre-ltirr 8257  ax-pre-lttrn 8259  ax-pre-ltadd 8261

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3677  df-sn 3701  df-pr 3702  df-tp 3703  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-id 4420  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-res 4768  df-ima 4769  df-iota 5319  df-fun 5361  df-fn 5362  df-f 5363  df-f1 5364  df-fo 5365  df-f1o 5366  df-fv 5367  df-riota 6013  df-ov 6063  df-oprab 6064  df-mpo 6065  df-er 6782  df-ec 6784  df-qs 6788  df-pnf 8328  df-mnf 8329  df-ltxr 8331  df-inn 9260  df-2 9318  df-3 9319  df-ndx 13305  df-slot 13306  df-base 13308  df-sets 13309  df-iress 13310  df-plusg 13393  df-mulr 13394  df-0g 13561  df-iimas 13573  df-qus 13574  df-mgm 13625  df-sgrp 13666  df-mnd 13679  df-grp 13757  df-minusg 13758  df-subg 13922  df-nsg 13923  df-eqg 13924  df-ghm 13993

This theorem is referenced by:  qusrhm  14809