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Mirrors > Home > MPE Home > Th. List > dchrrcl | Structured version Visualization version GIF version |
Description: Reverse closure for a Dirichlet character. (Contributed by Mario Carneiro, 12-May-2016.) |
Ref | Expression |
---|---|
dchrrcl.g | ⊢ 𝐺 = (DChr‘𝑁) |
dchrrcl.b | ⊢ 𝐷 = (Base‘𝐺) |
Ref | Expression |
---|---|
dchrrcl | ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
Step | Hyp | Ref | 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))〉}) | |
2 | 1 | dmmptss 6088 | . 2 ⊢ dom DChr ⊆ ℕ |
3 | n0i 4296 | . . 3 ⊢ (𝑋 ∈ 𝐷 → ¬ 𝐷 = ∅) | |
4 | dchrrcl.g | . . . . 5 ⊢ 𝐺 = (DChr‘𝑁) | |
5 | ndmfv 6693 | . . . . 5 ⊢ (¬ 𝑁 ∈ dom DChr → (DChr‘𝑁) = ∅) | |
6 | 4, 5 | syl5eq 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‘∅) | |
10 | 7, 8, 9 | 3eqtr4g 2878 | . . . 4 ⊢ (𝐺 = ∅ → 𝐷 = ∅) |
11 | 6, 10 | syl 17 | . . 3 ⊢ (¬ 𝑁 ∈ dom DChr → 𝐷 = ∅) |
12 | 3, 11 | nsyl2 143 | . 2 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ dom DChr) |
13 | 2, 12 | sseldi 3962 | 1 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ 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 ℕcn 11626 ndxcnx 16468 Basecbs 16471 +gcplusg 16553 MndHom cmhm 17942 mulGrpcmgp 19168 Unitcui 19318 ℂfldccnfld 20473 ℤ/nℤczn 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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