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Theorem dchrrcl 25743
Description: Reverse closure for a Dirichlet character. (Contributed by Mario Carneiro, 12-May-2016.)
Hypotheses
Ref Expression
dchrrcl.g 𝐺 = (DChr‘𝑁)
dchrrcl.b 𝐷 = (Base‘𝐺)
Assertion
Ref Expression
dchrrcl (𝑋𝐷𝑁 ∈ ℕ)

Proof of Theorem dchrrcl
Dummy variables 𝑛 𝑏 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dchr 25736 . . 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))⟩})
21dmmptss 6088 . 2 dom DChr ⊆ ℕ
3 n0i 4296 . . 3 (𝑋𝐷 → ¬ 𝐷 = ∅)
4 dchrrcl.g . . . . 5 𝐺 = (DChr‘𝑁)
5 ndmfv 6693 . . . . 5 𝑁 ∈ dom DChr → (DChr‘𝑁) = ∅)
64, 5syl5eq 2865 . . . 4 𝑁 ∈ dom DChr → 𝐺 = ∅)
7 fveq2 6663 . . . . 5 (𝐺 = ∅ → (Base‘𝐺) = (Base‘∅))
8 dchrrcl.b . . . . 5 𝐷 = (Base‘𝐺)
9 base0 16524 . . . . 5 ∅ = (Base‘∅)
107, 8, 93eqtr4g 2878 . . . 4 (𝐺 = ∅ → 𝐷 = ∅)
116, 10syl 17 . . 3 𝑁 ∈ dom DChr → 𝐷 = ∅)
123, 11nsyl2 143 . 2 (𝑋𝐷𝑁 ∈ dom DChr)
132, 12sseldi 3962 1 (𝑋𝐷𝑁 ∈ ℕ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1528  wcel 2105  {crab 3139  csb 3880  cdif 3930  wss 3933  c0 4288  {csn 4557  {cpr 4559  cop 4563   × cxp 5546  dom cdm 5548  cres 5550  cfv 6348  (class class class)co 7145  f cof 7396  0cc0 10525   · cmul 10530  cn 11626  ndxcnx 16468  Basecbs 16471  +gcplusg 16553   MndHom cmhm 17942  mulGrpcmgp 19168  Unitcui 19318  fldccnfld 20473  ℤ/nczn 20578  DChrcdchr 25735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  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-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-slot 16475  df-base 16477  df-dchr 25736
This theorem is referenced by:  dchrmhm  25744  dchrf  25745  dchrelbas4  25746  dchrzrh1  25747  dchrzrhcl  25748  dchrzrhmul  25749  dchrmul  25751  dchrmulcl  25752  dchrn0  25753  dchrmulid2  25755  dchrinvcl  25756  dchrghm  25759  dchrabs  25763  dchrinv  25764  dchrsum2  25771  dchrsum  25772  dchr2sum  25776
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