Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dchrval Structured version   Visualization version   GIF version

Theorem dchrval 25931
 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 (𝜑𝑁 ∈ ℕ)
dchrval.d (𝜑𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵𝑈) × {0}) ⊆ 𝑥})
Assertion
Ref Expression
dchrval (𝜑𝐺 = {⟨(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.
StepHypRef Expression
1 dchrval.g . 2 𝐺 = (DChr‘𝑁)
2 df-dchr 25930 . . 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 ((𝜑𝑛 = 𝑁) → (ℤ/nℤ‘𝑛) ∈ V)
4 ovex 7189 . . . . . . 7 ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∈ V
54rabex 5206 . . . . . 6 {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥} ∈ V
65a1i 11 . . . . 5 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥} ∈ V)
7 dchrval.d . . . . . . . . . . 11 (𝜑𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵𝑈) × {0}) ⊆ 𝑥})
87ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵𝑈) × {0}) ⊆ 𝑥})
9 dchrval.z . . . . . . . . . . . . . . . 16 𝑍 = (ℤ/nℤ‘𝑁)
10 simpr 488 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 = 𝑁) → 𝑛 = 𝑁)
1110fveq2d 6667 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 = 𝑁) → (ℤ/nℤ‘𝑛) = (ℤ/nℤ‘𝑁))
129, 11eqtr4id 2812 . . . . . . . . . . . . . . 15 ((𝜑𝑛 = 𝑁) → 𝑍 = (ℤ/nℤ‘𝑛))
1312eqeq2d 2769 . . . . . . . . . . . . . 14 ((𝜑𝑛 = 𝑁) → (𝑧 = 𝑍𝑧 = (ℤ/nℤ‘𝑛)))
1413biimpar 481 . . . . . . . . . . . . 13 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → 𝑧 = 𝑍)
1514fveq2d 6667 . . . . . . . . . . . 12 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (mulGrp‘𝑧) = (mulGrp‘𝑍))
1615oveq1d 7171 . . . . . . . . . . 11 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) = ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
1714fveq2d 6667 . . . . . . . . . . . . . . 15 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Base‘𝑧) = (Base‘𝑍))
18 dchrval.b . . . . . . . . . . . . . . 15 𝐵 = (Base‘𝑍)
1917, 18eqtr4di 2811 . . . . . . . . . . . . . 14 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Base‘𝑧) = 𝐵)
2014fveq2d 6667 . . . . . . . . . . . . . . 15 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Unit‘𝑧) = (Unit‘𝑍))
21 dchrval.u . . . . . . . . . . . . . . 15 𝑈 = (Unit‘𝑍)
2220, 21eqtr4di 2811 . . . . . . . . . . . . . 14 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Unit‘𝑧) = 𝑈)
2319, 22difeq12d 4031 . . . . . . . . . . . . 13 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → ((Base‘𝑧) ∖ (Unit‘𝑧)) = (𝐵𝑈))
2423xpeq1d 5557 . . . . . . . . . . . 12 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) = ((𝐵𝑈) × {0}))
2524sseq1d 3925 . . . . . . . . . . 11 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → ((((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥 ↔ ((𝐵𝑈) × {0}) ⊆ 𝑥))
2616, 25rabeqbidv 3398 . . . . . . . . . 10 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥} = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵𝑈) × {0}) ⊆ 𝑥})
278, 26eqtr4d 2796 . . . . . . . . 9 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥})
2827eqeq2d 2769 . . . . . . . 8 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (𝑏 = 𝐷𝑏 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥}))
2928biimpar 481 . . . . . . 7 ((((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) ∧ 𝑏 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥}) → 𝑏 = 𝐷)
3029opeq2d 4773 . . . . . 6 ((((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) ∧ 𝑏 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥}) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐷⟩)
3129sqxpeqd 5560 . . . . . . . 8 ((((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) ∧ 𝑏 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥}) → (𝑏 × 𝑏) = (𝐷 × 𝐷))
3231reseq2d 5828 . . . . . . 7 ((((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) ∧ 𝑏 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥}) → ( ∘f · ↾ (𝑏 × 𝑏)) = ( ∘f · ↾ (𝐷 × 𝐷)))
3332opeq2d 4773 . . . . . 6 ((((𝜑𝑛 = 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))⟩)
3430, 33preq12d 4637 . . . . 5 ((((𝜑𝑛 = 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))⟩})
356, 34csbied 3843 . . . 4 (((𝜑𝑛 = 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))⟩})
363, 35csbied 3843 . . 3 ((𝜑𝑛 = 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 (𝜑𝑁 ∈ ℕ)
38 prex 5305 . . . 4 {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))⟩} ∈ V
3938a1i 11 . . 3 (𝜑 → {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))⟩} ∈ V)
402, 36, 37, 39fvmptd2 6772 . 2 (𝜑 → (DChr‘𝑁) = {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))⟩})
411, 40syl5eq 2805 1 (𝜑𝐺 = {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3074  Vcvv 3409  ⦋csb 3807   ∖ cdif 3857   ⊆ wss 3860  {csn 4525  {cpr 4527  ⟨cop 4531   × cxp 5526   ↾ cres 5530  ‘cfv 6340  (class class class)co 7156   ∘f cof 7409  0cc0 10588   · cmul 10593  ℕcn 11687  ndxcnx 16552  Basecbs 16555  +gcplusg 16637   MndHom cmhm 18034  mulGrpcmgp 19321  Unitcui 19474  ℂfldccnfld 20180  ℤ/nℤczn 20286  DChrcdchr 25929 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-res 5540  df-iota 6299  df-fun 6342  df-fv 6348  df-ov 7159  df-dchr 25930 This theorem is referenced by:  dchrbas  25932  dchrplusg  25944
 Copyright terms: Public domain W3C validator