Theorem qusghm 14085

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 2238	. 2 ⊢ (Base‘𝐻) = (Base‘𝐻)
3		eqid 2238	. 2 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		eqid 2238	. 2 ⊢ (+g‘𝐻) = (+g‘𝐻)
5		nsgsubg 14008	. . 3 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝑌 ∈ (SubGrp‘𝐺))
6		subgrcl 13982	. . 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 14035	. 2 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝐻 ∈ Grp)
10	8, 1, 2	quseccl 14036	. . 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 5862	. 2 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝐹:𝑋⟶(Base‘𝐻))
13	8, 1, 3, 4	qusadd 14037	. . . 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 1235	. . 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 6842	. . . . 5 ⊢ (𝑥 = 𝑦 → [𝑥](𝐺 ~QG 𝑌) = [𝑦](𝐺 ~QG 𝑌))
16		simprl 535	. . . . 5 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
17		eqgex 14024	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ (NrmSGrp‘𝐺)) → (𝐺 ~QG 𝑌) ∈ V)
18	7, 17	mpancom 426	. . . . . . 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 6811	. . . . . 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 5799	. . . 4 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑦) = [𝑦](𝐺 ~QG 𝑌))
23		eceq1 6842	. . . . 5 ⊢ (𝑥 = 𝑧 → [𝑥](𝐺 ~QG 𝑌) = [𝑧](𝐺 ~QG 𝑌))
24		simprr 537	. . . . 5 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
25		ecexg 6811	. . . . . 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 5799	. . . 4 ⊢ ((𝑌 ∈ (NrmSGrp‘𝐺) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑧) = [𝑧](𝐺 ~QG 𝑌))
28	22, 27	oveq12d 6103	. . 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 6842	. . . 4 ⊢ (𝑥 = (𝑦(+g‘𝐺)𝑧) → [𝑥](𝐺 ~QG 𝑌) = [(𝑦(+g‘𝐺)𝑧)](𝐺 ~QG 𝑌))
30	1, 3	grpcl 13813	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(+g‘𝐺)𝑧) ∈ 𝑋)
31	30	3expb 1235	. . . . 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 6811	. . . . 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 5799	. . 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 2282	. 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 14055	1 ⊢ (𝑌 ∈ (NrmSGrp‘𝐺) → 𝐹 ∈ (𝐺 GrpHom 𝐻))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ↦ cmpt 4192  ‘cfv 5377  (class class class)co 6085  [cec 6805  Basecbs 13352  +_gcplusg 13431   /_s cqus 13623  Grpcgrp 13805  SubGrpcsubg 13970  NrmSGrpcnsg 13971   ~_QG cqg 13972   GrpHom cghm 14043

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-er 6807  df-ec 6809  df-qs 6813  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-mulr 13445  df-0g 13612  df-iimas 13624  df-qus 13625  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-subg 13973  df-nsg 13974  df-eqg 13975  df-ghm 14044

This theorem is used by:  qusrhm  14865