Theorem dchrval 27216

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 27215	. . 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 6843	. . . 4 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → (ℤ/nℤ‘𝑛) ∈ V)
4		ovex 7390	. . . . . . 7 ⊢ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∈ V
5	4	rabex 5268	. . . . . 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 732	. . . . . . . . . 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 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → 𝑛 = 𝑁)
11	10	fveq2d 6832	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → (ℤ/nℤ‘𝑛) = (ℤ/nℤ‘𝑁))
12	9, 11	eqtr4id 2793	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → 𝑍 = (ℤ/nℤ‘𝑛))
13	12	eqeq2d 2750	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → (𝑧 = 𝑍 ↔ 𝑧 = (ℤ/nℤ‘𝑛)))
14	13	biimpar 478	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → 𝑧 = 𝑍)
15	14	fveq2d 6832	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (mulGrp‘𝑧) = (mulGrp‘𝑍))
16	15	oveq1d 7372	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) = ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
17	14	fveq2d 6832	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Base‘𝑧) = (Base‘𝑍))
18		dchrval.b	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = (Base‘𝑍)
19	17, 18	eqtr4di 2792	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Base‘𝑧) = 𝐵)
20	14	fveq2d 6832	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Unit‘𝑧) = (Unit‘𝑍))
21		dchrval.u	. . . . . . . . . . . . . . 15 ⊢ 𝑈 = (Unit‘𝑍)
22	20, 21	eqtr4di 2792	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Unit‘𝑧) = 𝑈)
23	19, 22	difeq12d 4059	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → ((Base‘𝑧) ∖ (Unit‘𝑧)) = (𝐵 ∖ 𝑈))
24	23	xpeq1d 5648	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) = ((𝐵 ∖ 𝑈) × {0}))
25	24	sseq1d 3946	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → ((((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥 ↔ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥))
26	16, 25	rabeqbidv 3409	. . . . . . . . . 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 2777	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥})
28	27	eqeq2d 2750	. . . . . . . 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 478	. . . . . . 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 4812	. . . . . 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 5651	. . . . . . . 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 5932	. . . . . . 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 4812	. . . . . 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 4674	. . . . 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 3867	. . . 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 3867	. . 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 5368	. . . 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 6945	. 2 ⊢ (𝜑 → (DChr‘𝑁) = {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))⟩})
41	1, 40	eqtrid 2786	1 ⊢ (𝜑 → 𝐺 = {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))⟩})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {crab 3391  Vcvv 3431  ⦋csb 3831   ∖ cdif 3880   ⊆ wss 3883  {csn 4556  {cpr 4558  ⟨cop 4562   × cxp 5617   ↾ cres 5621  ‘cfv 6486  (class class class)co 7357   ∘_f cof 7619  0cc0 11030   · cmul 11035  ℕcn 12166  ndxcnx 17155  Basecbs 17171  +_gcplusg 17212   MndHom cmhm 18741  mulGrpcmgp 20113  Unitcui 20327  ℂ_fldccnfld 21348  ℤ/nℤczn 21478  DChrcdchr 27214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-res 5631  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7360  df-dchr 27215

This theorem is referenced by:  dchrbas  27217  dchrplusg  27229