Theorem dchrval 25802

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 25801	. . 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 6678	. . . 4 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → (ℤ/nℤ‘𝑛) ∈ V)
4		ovex 7181	. . . . . . 7 ⊢ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∈ V
5	4	rabex 5226	. . . . . 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 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥})
9		simpr 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → 𝑛 = 𝑁)
10	9	fveq2d 6667	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → (ℤ/nℤ‘𝑛) = (ℤ/nℤ‘𝑁))
11		dchrval.z	. . . . . . . . . . . . . . . 16 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
12	10, 11	syl6reqr 2873	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → 𝑍 = (ℤ/nℤ‘𝑛))
13	12	eqeq2d 2830	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → (𝑧 = 𝑍 ↔ 𝑧 = (ℤ/nℤ‘𝑛)))
14	13	biimpar 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → 𝑧 = 𝑍)
15	14	fveq2d 6667	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (mulGrp‘𝑧) = (mulGrp‘𝑍))
16	15	oveq1d 7163	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) = ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
17	14	fveq2d 6667	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Base‘𝑧) = (Base‘𝑍))
18		dchrval.b	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = (Base‘𝑍)
19	17, 18	syl6eqr 2872	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Base‘𝑧) = 𝐵)
20	14	fveq2d 6667	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Unit‘𝑧) = (Unit‘𝑍))
21		dchrval.u	. . . . . . . . . . . . . . 15 ⊢ 𝑈 = (Unit‘𝑍)
22	20, 21	syl6eqr 2872	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Unit‘𝑧) = 𝑈)
23	19, 22	difeq12d 4098	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → ((Base‘𝑧) ∖ (Unit‘𝑧)) = (𝐵 ∖ 𝑈))
24	23	xpeq1d 5577	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) = ((𝐵 ∖ 𝑈) × {0}))
25	24	sseq1d 3996	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → ((((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥 ↔ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥))
26	16, 25	rabeqbidv 3484	. . . . . . . . . 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 2857	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥})
28	27	eqeq2d 2830	. . . . . . . 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 480	. . . . . . 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 4802	. . . . . 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 5580	. . . . . . . 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 5846	. . . . . . 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 4802	. . . . . 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 4669	. . . . 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 3917	. . . 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 3917	. . 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 5323	. . . 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 6769	. 2 ⊢ (𝜑 → (DChr‘𝑁) = {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))⟩})
41	1, 40	syl5eq 2866	1 ⊢ (𝜑 → 𝐺 = {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))⟩})

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1531   ∈ wcel 2108  {crab 3140  Vcvv 3493  ⦋csb 3881   ∖ cdif 3931   ⊆ wss 3934  {csn 4559  {cpr 4561  ⟨cop 4565   × cxp 5546   ↾ cres 5550  ‘cfv 6348  (class class class)co 7148   ∘_f cof 7399  0cc0 10529   · cmul 10534  ℕcn 11630  ndxcnx 16472  Basecbs 16475  +_gcplusg 16557   MndHom cmhm 17946  mulGrpcmgp 19231  Unitcui 19381  ℂ_fldccnfld 20537  ℤ/nℤczn 20642  DChrcdchr 25800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-dchr 25801

This theorem is referenced by:  dchrbas  25803  dchrplusg  25815