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Theorem dchrrcl 24946
 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 24939 . . 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 5619 . 2 dom DChr ⊆ ℕ
3 n0i 3912 . . 3 (𝑋𝐷 → ¬ 𝐷 = ∅)
4 dchrrcl.g . . . . 5 𝐺 = (DChr‘𝑁)
5 ndmfv 6205 . . . . 5 𝑁 ∈ dom DChr → (DChr‘𝑁) = ∅)
64, 5syl5eq 2666 . . . 4 𝑁 ∈ dom DChr → 𝐺 = ∅)
7 fveq2 6178 . . . . 5 (𝐺 = ∅ → (Base‘𝐺) = (Base‘∅))
8 dchrrcl.b . . . . 5 𝐷 = (Base‘𝐺)
9 base0 15893 . . . . 5 ∅ = (Base‘∅)
107, 8, 93eqtr4g 2679 . . . 4 (𝐺 = ∅ → 𝐷 = ∅)
116, 10syl 17 . . 3 𝑁 ∈ dom DChr → 𝐷 = ∅)
123, 11nsyl2 142 . 2 (𝑋𝐷𝑁 ∈ dom DChr)
132, 12sseldi 3593 1 (𝑋𝐷𝑁 ∈ ℕ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1481   ∈ wcel 1988  {crab 2913  ⦋csb 3526   ∖ cdif 3564   ⊆ wss 3567  ∅c0 3907  {csn 4168  {cpr 4170  ⟨cop 4174   × cxp 5102  dom cdm 5104   ↾ cres 5106  ‘cfv 5876  (class class class)co 6635   ∘𝑓 cof 6880  0cc0 9921   · cmul 9926  ℕcn 11005  ndxcnx 15835  Basecbs 15838  +gcplusg 15922   MndHom cmhm 17314  mulGrpcmgp 18470  Unitcui 18620  ℂfldccnfld 19727  ℤ/nℤczn 19832  DChrcdchr 24938 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fv 5884  df-slot 15842  df-base 15844  df-dchr 24939 This theorem is referenced by:  dchrmhm  24947  dchrf  24948  dchrelbas4  24949  dchrzrh1  24950  dchrzrhcl  24951  dchrzrhmul  24952  dchrmul  24954  dchrmulcl  24955  dchrn0  24956  dchrmulid2  24958  dchrinvcl  24959  dchrghm  24962  dchrabs  24966  dchrinv  24967  dchrsum2  24974  dchrsum  24975  dchr2sum  24979
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