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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 27292 | . . 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 6263 | . 2 ⊢ dom DChr ⊆ ℕ |
3 | n0i 4346 | . . 3 ⊢ (𝑋 ∈ 𝐷 → ¬ 𝐷 = ∅) | |
4 | dchrrcl.g | . . . . 5 ⊢ 𝐺 = (DChr‘𝑁) | |
5 | ndmfv 6942 | . . . . 5 ⊢ (¬ 𝑁 ∈ dom DChr → (DChr‘𝑁) = ∅) | |
6 | 4, 5 | eqtrid 2787 | . . . 4 ⊢ (¬ 𝑁 ∈ dom DChr → 𝐺 = ∅) |
7 | fveq2 6907 | . . . . 5 ⊢ (𝐺 = ∅ → (Base‘𝐺) = (Base‘∅)) | |
8 | dchrrcl.b | . . . . 5 ⊢ 𝐷 = (Base‘𝐺) | |
9 | base0 17250 | . . . . 5 ⊢ ∅ = (Base‘∅) | |
10 | 7, 8, 9 | 3eqtr4g 2800 | . . . 4 ⊢ (𝐺 = ∅ → 𝐷 = ∅) |
11 | 6, 10 | syl 17 | . . 3 ⊢ (¬ 𝑁 ∈ dom DChr → 𝐷 = ∅) |
12 | 3, 11 | nsyl2 141 | . 2 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ dom DChr) |
13 | 2, 12 | sselid 3993 | 1 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2106 {crab 3433 ⦋csb 3908 ∖ cdif 3960 ⊆ wss 3963 ∅c0 4339 {csn 4631 {cpr 4633 〈cop 4637 × cxp 5687 dom cdm 5689 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 0cc0 11153 · cmul 11158 ℕcn 12264 ndxcnx 17227 Basecbs 17245 +gcplusg 17298 MndHom cmhm 18807 mulGrpcmgp 20152 Unitcui 20372 ℂfldccnfld 21382 ℤ/nℤczn 21531 DChrcdchr 27291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-slot 17216 df-ndx 17228 df-base 17246 df-dchr 27292 |
This theorem is referenced by: dchrmhm 27300 dchrf 27301 dchrelbas4 27302 dchrzrh1 27303 dchrzrhcl 27304 dchrzrhmul 27305 dchrmul 27307 dchrmulcl 27308 dchrn0 27309 dchrmullid 27311 dchrinvcl 27312 dchrghm 27315 dchrabs 27319 dchrinv 27320 dchrsum2 27327 dchrsum 27328 dchr2sum 27332 |
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