Theorem dchrval 27199

Description: Value of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)

Hypotheses

Ref	Expression
dchrval.g	⊢ 𝐺 = (DChr‘𝑁)
dchrval.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
dchrval.b	⊢ 𝐵 = (Base‘𝑍)
dchrval.u	⊢ 𝑈 = (Unit‘𝑍)
dchrval.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
dchrval.d	⊢ (𝜑 → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥})

Assertion

Ref	Expression
dchrval	⊢ (𝜑 → 𝐺 = {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))⟩})

Distinct variable groups:   𝑥,𝐵   𝑥,𝑁   𝑥,𝑈   𝜑,𝑥   𝑥,𝑍

Allowed substitution hints:   𝐷(𝑥)   𝐺(𝑥)

Proof of Theorem dchrval

Dummy variables 𝑧 𝑛 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dchrval.g	. 2 ⊢ 𝐺 = (DChr‘𝑁)
2		df-dchr 27198	. . 3 ⊢ DChr = (𝑛 ∈ ℕ ↦ ⦋(ℤ/nℤ‘𝑛) / 𝑧⦌⦋{𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥} / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝑏 × 𝑏))⟩})
3		fvexd 6847	. . . 4 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → (ℤ/nℤ‘𝑛) ∈ V)
4		ovex 7389	. . . . . . 7 ⊢ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∈ V
5	4	rabex 5282	. . . . . 6 ⊢ {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥} ∈ V
6	5	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥} ∈ V)
7		dchrval.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥})
8	7	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥})
9		dchrval.z	. . . . . . . . . . . . . . . 16 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
10		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → 𝑛 = 𝑁)
11	10	fveq2d 6836	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → (ℤ/nℤ‘𝑛) = (ℤ/nℤ‘𝑁))
12	9, 11	eqtr4id 2788	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → 𝑍 = (ℤ/nℤ‘𝑛))
13	12	eqeq2d 2745	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → (𝑧 = 𝑍 ↔ 𝑧 = (ℤ/nℤ‘𝑛)))
14	13	biimpar 477	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → 𝑧 = 𝑍)
15	14	fveq2d 6836	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (mulGrp‘𝑧) = (mulGrp‘𝑍))
16	15	oveq1d 7371	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) = ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
17	14	fveq2d 6836	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Base‘𝑧) = (Base‘𝑍))
18		dchrval.b	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = (Base‘𝑍)
19	17, 18	eqtr4di 2787	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Base‘𝑧) = 𝐵)
20	14	fveq2d 6836	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Unit‘𝑧) = (Unit‘𝑍))
21		dchrval.u	. . . . . . . . . . . . . . 15 ⊢ 𝑈 = (Unit‘𝑍)
22	20, 21	eqtr4di 2787	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Unit‘𝑧) = 𝑈)
23	19, 22	difeq12d 4077	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → ((Base‘𝑧) ∖ (Unit‘𝑧)) = (𝐵 ∖ 𝑈))
24	23	xpeq1d 5651	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) = ((𝐵 ∖ 𝑈) × {0}))
25	24	sseq1d 3963	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → ((((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥 ↔ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥))
26	16, 25	rabeqbidv 3415	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥} = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥})
27	8, 26	eqtr4d 2772	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥})
28	27	eqeq2d 2745	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (𝑏 = 𝐷 ↔ 𝑏 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥}))
29	28	biimpar 477	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) ∧ 𝑏 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥}) → 𝑏 = 𝐷)
30	29	opeq2d 4834	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) ∧ 𝑏 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥}) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐷⟩)
31	29	sqxpeqd 5654	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) ∧ 𝑏 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥}) → (𝑏 × 𝑏) = (𝐷 × 𝐷))
32	31	reseq2d 5936	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) ∧ 𝑏 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥}) → ( ∘f · ↾ (𝑏 × 𝑏)) = ( ∘f · ↾ (𝐷 × 𝐷)))
33	32	opeq2d 4834	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) ∧ 𝑏 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥}) → ⟨(+g‘ndx), ( ∘f · ↾ (𝑏 × 𝑏))⟩ = ⟨(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))⟩)
34	30, 33	preq12d 4696	. . . . 5 ⊢ ((((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) ∧ 𝑏 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥}) → {⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝑏 × 𝑏))⟩} = {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))⟩})
35	6, 34	csbied 3883	. . . 4 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → ⦋{𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥} / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝑏 × 𝑏))⟩} = {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))⟩})
36	3, 35	csbied 3883	. . 3 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → ⦋(ℤ/nℤ‘𝑛) / 𝑧⦌⦋{𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥} / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝑏 × 𝑏))⟩} = {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))⟩})
37		dchrval.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
38		prex 5380	. . . 4 ⊢ {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))⟩} ∈ V
39	38	a1i 11	. . 3 ⊢ (𝜑 → {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))⟩} ∈ V)
40	2, 36, 37, 39	fvmptd2 6947	. 2 ⊢ (𝜑 → (DChr‘𝑁) = {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))⟩})
41	1, 40	eqtrid 2781	1 ⊢ (𝜑 → 𝐺 = {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))⟩})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3397  Vcvv 3438  ⦋csb 3847   ∖ cdif 3896   ⊆ wss 3899  {csn 4578  {cpr 4580  ⟨cop 4584   × cxp 5620   ↾ cres 5624  ‘cfv 6490  (class class class)co 7356   ∘_f cof 7618  0cc0 11024   · cmul 11029  ℕcn 12143  ndxcnx 17118  Basecbs 17134  +_gcplusg 17175   MndHom cmhm 18704  mulGrpcmgp 20073  Unitcui 20289  ℂ_fldccnfld 21307  ℤ/nℤczn 21455  DChrcdchr 27197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-res 5634  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7359  df-dchr 27198

This theorem is referenced by:  dchrbas  27200  dchrplusg  27212