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Theorem dchrrcl 25868
 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 25861 . . 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 6067 . 2 dom DChr ⊆ ℕ
3 n0i 4252 . . 3 (𝑋𝐷 → ¬ 𝐷 = ∅)
4 dchrrcl.g . . . . 5 𝐺 = (DChr‘𝑁)
5 ndmfv 6685 . . . . 5 𝑁 ∈ dom DChr → (DChr‘𝑁) = ∅)
64, 5syl5eq 2845 . . . 4 𝑁 ∈ dom DChr → 𝐺 = ∅)
7 fveq2 6655 . . . . 5 (𝐺 = ∅ → (Base‘𝐺) = (Base‘∅))
8 dchrrcl.b . . . . 5 𝐷 = (Base‘𝐺)
9 base0 16548 . . . . 5 ∅ = (Base‘∅)
107, 8, 93eqtr4g 2858 . . . 4 (𝐺 = ∅ → 𝐷 = ∅)
116, 10syl 17 . . 3 𝑁 ∈ dom DChr → 𝐷 = ∅)
123, 11nsyl2 143 . 2 (𝑋𝐷𝑁 ∈ dom DChr)
132, 12sseldi 3915 1 (𝑋𝐷𝑁 ∈ ℕ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2111  {crab 3110  ⦋csb 3830   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4246  {csn 4528  {cpr 4530  ⟨cop 4534   × cxp 5521  dom cdm 5523   ↾ cres 5525  ‘cfv 6332  (class class class)co 7145   ∘f cof 7398  0cc0 10544   · cmul 10549  ℕcn 11643  ndxcnx 16492  Basecbs 16495  +gcplusg 16577   MndHom cmhm 17966  mulGrpcmgp 19253  Unitcui 19406  ℂfldccnfld 20112  ℤ/nℤczn 20218  DChrcdchr 25860 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 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fv 6340  df-slot 16499  df-base 16501  df-dchr 25861 This theorem is referenced by:  dchrmhm  25869  dchrf  25870  dchrelbas4  25871  dchrzrh1  25872  dchrzrhcl  25873  dchrzrhmul  25874  dchrmul  25876  dchrmulcl  25877  dchrn0  25878  dchrmulid2  25880  dchrinvcl  25881  dchrghm  25884  dchrabs  25888  dchrinv  25889  dchrsum2  25896  dchrsum  25897  dchr2sum  25901
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