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

Theorem dchrval 25004
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 25003 . . . 4 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))⟩})
32a1i 11 . . 3 (𝜑 → 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))⟩}))
4 fvexd 6241 . . . 4 ((𝜑𝑛 = 𝑁) → (ℤ/nℤ‘𝑛) ∈ V)
5 ovex 6718 . . . . . . 7 ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∈ V
65rabex 4845 . . . . . 6 {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥} ∈ V
76a1i 11 . . . . 5 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥} ∈ V)
8 dchrval.d . . . . . . . . . . 11 (𝜑𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵𝑈) × {0}) ⊆ 𝑥})
98ad2antrr 762 . . . . . . . . . 10 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵𝑈) × {0}) ⊆ 𝑥})
10 simpr 476 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 = 𝑁) → 𝑛 = 𝑁)
1110fveq2d 6233 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 = 𝑁) → (ℤ/nℤ‘𝑛) = (ℤ/nℤ‘𝑁))
12 dchrval.z . . . . . . . . . . . . . . . 16 𝑍 = (ℤ/nℤ‘𝑁)
1311, 12syl6reqr 2704 . . . . . . . . . . . . . . 15 ((𝜑𝑛 = 𝑁) → 𝑍 = (ℤ/nℤ‘𝑛))
1413eqeq2d 2661 . . . . . . . . . . . . . 14 ((𝜑𝑛 = 𝑁) → (𝑧 = 𝑍𝑧 = (ℤ/nℤ‘𝑛)))
1514biimpar 501 . . . . . . . . . . . . 13 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → 𝑧 = 𝑍)
1615fveq2d 6233 . . . . . . . . . . . 12 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (mulGrp‘𝑧) = (mulGrp‘𝑍))
1716oveq1d 6705 . . . . . . . . . . 11 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) = ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)))
1815fveq2d 6233 . . . . . . . . . . . . . . 15 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Base‘𝑧) = (Base‘𝑍))
19 dchrval.b . . . . . . . . . . . . . . 15 𝐵 = (Base‘𝑍)
2018, 19syl6eqr 2703 . . . . . . . . . . . . . 14 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Base‘𝑧) = 𝐵)
2115fveq2d 6233 . . . . . . . . . . . . . . 15 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Unit‘𝑧) = (Unit‘𝑍))
22 dchrval.u . . . . . . . . . . . . . . 15 𝑈 = (Unit‘𝑍)
2321, 22syl6eqr 2703 . . . . . . . . . . . . . 14 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (Unit‘𝑧) = 𝑈)
2420, 23difeq12d 3762 . . . . . . . . . . . . 13 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → ((Base‘𝑧) ∖ (Unit‘𝑧)) = (𝐵𝑈))
2524xpeq1d 5172 . . . . . . . . . . . 12 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) = ((𝐵𝑈) × {0}))
2625sseq1d 3665 . . . . . . . . . . 11 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → ((((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥 ↔ ((𝐵𝑈) × {0}) ⊆ 𝑥))
2717, 26rabeqbidv 3226 . . . . . . . . . 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})
289, 27eqtr4d 2688 . . . . . . . . 9 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥})
2928eqeq2d 2661 . . . . . . . 8 (((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) → (𝑏 = 𝐷𝑏 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥}))
3029biimpar 501 . . . . . . 7 ((((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) ∧ 𝑏 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥}) → 𝑏 = 𝐷)
3130opeq2d 4440 . . . . . 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⟩)
3230sqxpeqd 5175 . . . . . . . 8 ((((𝜑𝑛 = 𝑁) ∧ 𝑧 = (ℤ/nℤ‘𝑛)) ∧ 𝑏 = {𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥}) → (𝑏 × 𝑏) = (𝐷 × 𝐷))
3332reseq2d 5428 . . . . . . 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)))
3433opeq2d 4440 . . . . . 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))⟩)
3531, 34preq12d 4308 . . . . 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))⟩})
367, 35csbied 3593 . . . 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))⟩})
374, 36csbied 3593 . . 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))⟩})
38 dchrval.n . . 3 (𝜑𝑁 ∈ ℕ)
39 prex 4939 . . . 4 {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘𝑓 · ↾ (𝐷 × 𝐷))⟩} ∈ V
4039a1i 11 . . 3 (𝜑 → {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘𝑓 · ↾ (𝐷 × 𝐷))⟩} ∈ V)
413, 37, 38, 40fvmptd 6327 . 2 (𝜑 → (DChr‘𝑁) = {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘𝑓 · ↾ (𝐷 × 𝐷))⟩})
421, 41syl5eq 2697 1 (𝜑𝐺 = {⟨(Base‘ndx), 𝐷⟩, ⟨(+g‘ndx), ( ∘𝑓 · ↾ (𝐷 × 𝐷))⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  csb 3566  cdif 3604  wss 3607  {csn 4210  {cpr 4212  cop 4216  cmpt 4762   × cxp 5141  cres 5145  cfv 5926  (class class class)co 6690  𝑓 cof 6937  0cc0 9974   · cmul 9979  cn 11058  ndxcnx 15901  Basecbs 15904  +gcplusg 15988   MndHom cmhm 17380  mulGrpcmgp 18535  Unitcui 18685  fldccnfld 19794  ℤ/nczn 19899  DChrcdchr 25002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-dchr 25003
This theorem is referenced by:  dchrbas  25005  dchrplusg  25017
  Copyright terms: Public domain W3C validator