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Mirrors > Home > MPE Home > Th. List > dchrf | Structured version Visualization version GIF version |
Description: A Dirichlet character is a function. (Contributed by Mario Carneiro, 18-Apr-2016.) |
Ref | Expression |
---|---|
dchrmhm.g | ⊢ 𝐺 = (DChr‘𝑁) |
dchrmhm.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
dchrmhm.b | ⊢ 𝐷 = (Base‘𝐺) |
dchrf.b | ⊢ 𝐵 = (Base‘𝑍) |
dchrf.x | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
Ref | Expression |
---|---|
dchrf | ⊢ (𝜑 → 𝑋:𝐵⟶ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchrf.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
2 | dchrmhm.g | . . . 4 ⊢ 𝐺 = (DChr‘𝑁) | |
3 | dchrmhm.z | . . . 4 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
4 | dchrf.b | . . . 4 ⊢ 𝐵 = (Base‘𝑍) | |
5 | eqid 2726 | . . . 4 ⊢ (Unit‘𝑍) = (Unit‘𝑍) | |
6 | dchrmhm.b | . . . . . 6 ⊢ 𝐷 = (Base‘𝐺) | |
7 | 2, 6 | dchrrcl 27266 | . . . . 5 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
9 | 2, 3, 4, 5, 8, 6 | dchrelbas3 27264 | . . 3 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋:𝐵⟶ℂ ∧ (∀𝑥 ∈ (Unit‘𝑍)∀𝑦 ∈ (Unit‘𝑍)(𝑋‘(𝑥(.r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)) ∧ (𝑋‘(1r‘𝑍)) = 1 ∧ ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍)))))) |
10 | 1, 9 | mpbid 231 | . 2 ⊢ (𝜑 → (𝑋:𝐵⟶ℂ ∧ (∀𝑥 ∈ (Unit‘𝑍)∀𝑦 ∈ (Unit‘𝑍)(𝑋‘(𝑥(.r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)) ∧ (𝑋‘(1r‘𝑍)) = 1 ∧ ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ (Unit‘𝑍))))) |
11 | 10 | simpld 493 | 1 ⊢ (𝜑 → 𝑋:𝐵⟶ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ⟶wf 6542 ‘cfv 6546 (class class class)co 7416 ℂcc 11147 0cc0 11149 1c1 11150 · cmul 11154 ℕcn 12258 Basecbs 17208 .rcmulr 17262 1rcur 20160 Unitcui 20333 ℤ/nℤczn 21488 DChrcdchr 27258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-addf 11228 ax-mulf 11229 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-tpos 8233 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8726 df-ec 8728 df-qs 8732 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9478 df-inf 9479 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-uz 12869 df-fz 13533 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-sca 17277 df-vsca 17278 df-ip 17279 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-0g 17451 df-imas 17518 df-qus 17519 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-mhm 18768 df-grp 18926 df-minusg 18927 df-sbg 18928 df-subg 19113 df-nsg 19114 df-eqg 19115 df-cmn 19776 df-abl 19777 df-mgp 20114 df-rng 20132 df-ur 20161 df-ring 20214 df-cring 20215 df-oppr 20312 df-dvdsr 20335 df-unit 20336 df-subrng 20524 df-subrg 20549 df-lmod 20834 df-lss 20905 df-lsp 20945 df-sra 21147 df-rgmod 21148 df-lidl 21193 df-rsp 21194 df-2idl 21235 df-cnfld 21340 df-zring 21433 df-zn 21492 df-dchr 27259 |
This theorem is referenced by: dchrzrhcl 27271 dchrmulcl 27275 dchrmullid 27278 dchrinvcl 27279 dchrabl 27280 dchrfi 27281 dchrghm 27282 dchreq 27284 dchrresb 27285 dchrabs 27286 dchrinv 27287 dchr1re 27289 dchrsum2 27294 dchrsum 27295 sumdchr2 27296 dchrhash 27297 dchr2sum 27299 sum2dchr 27300 dchrisumlem1 27515 rpvmasum2 27538 dchrisum0re 27539 |
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