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Mirrors > Home > MPE Home > Th. List > dchr1 | Structured version Visualization version GIF version |
Description: Value of the principal Dirichlet character. (Contributed by Mario Carneiro, 28-Apr-2016.) |
Ref | Expression |
---|---|
dchr1.g | ⊢ 𝐺 = (DChr‘𝑁) |
dchr1.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
dchr1.o | ⊢ 1 = (0g‘𝐺) |
dchr1.u | ⊢ 𝑈 = (Unit‘𝑍) |
dchr1.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
dchr1.a | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Ref | Expression |
---|---|
dchr1 | ⊢ (𝜑 → ( 1 ‘𝐴) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchr1.g | . . . 4 ⊢ 𝐺 = (DChr‘𝑁) | |
2 | dchr1.z | . . . 4 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
3 | eqid 2778 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | eqid 2778 | . . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
5 | dchr1.u | . . . 4 ⊢ 𝑈 = (Unit‘𝑍) | |
6 | eqid 2778 | . . . 4 ⊢ (𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0)) = (𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0)) | |
7 | dchr1.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | dchr1cl 25428 | . . 3 ⊢ (𝜑 → (𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0)) ∈ (Base‘𝐺)) |
9 | eleq1w 2842 | . . . . . 6 ⊢ (𝑘 = 𝑥 → (𝑘 ∈ 𝑈 ↔ 𝑥 ∈ 𝑈)) | |
10 | 9 | ifbid 4329 | . . . . 5 ⊢ (𝑘 = 𝑥 → if(𝑘 ∈ 𝑈, 1, 0) = if(𝑥 ∈ 𝑈, 1, 0)) |
11 | 10 | cbvmptv 4985 | . . . 4 ⊢ (𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0)) = (𝑥 ∈ (Base‘𝑍) ↦ if(𝑥 ∈ 𝑈, 1, 0)) |
12 | eqid 2778 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
13 | 1, 2, 3, 4, 5, 11, 12, 8 | dchrmulid2 25429 | . . 3 ⊢ (𝜑 → ((𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0))(+g‘𝐺)(𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0))) = (𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0))) |
14 | 1 | dchrabl 25431 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) |
15 | ablgrp 18584 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
16 | dchr1.o | . . . . 5 ⊢ 1 = (0g‘𝐺) | |
17 | 3, 12, 16 | isgrpid2 17845 | . . . 4 ⊢ (𝐺 ∈ Grp → (((𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0)) ∈ (Base‘𝐺) ∧ ((𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0))(+g‘𝐺)(𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0))) = (𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0))) ↔ 1 = (𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0)))) |
18 | 7, 14, 15, 17 | 4syl 19 | . . 3 ⊢ (𝜑 → (((𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0)) ∈ (Base‘𝐺) ∧ ((𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0))(+g‘𝐺)(𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0))) = (𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0))) ↔ 1 = (𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0)))) |
19 | 8, 13, 18 | mpbi2and 702 | . 2 ⊢ (𝜑 → 1 = (𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0))) |
20 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝑘 = 𝐴) | |
21 | dchr1.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
22 | 21 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐴 ∈ 𝑈) |
23 | 20, 22 | eqeltrd 2859 | . . 3 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝑘 ∈ 𝑈) |
24 | 23 | iftrued 4315 | . 2 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → if(𝑘 ∈ 𝑈, 1, 0) = 1) |
25 | 4, 5 | unitss 19047 | . . 3 ⊢ 𝑈 ⊆ (Base‘𝑍) |
26 | 25, 21 | sseldi 3819 | . 2 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑍)) |
27 | 1cnd 10371 | . 2 ⊢ (𝜑 → 1 ∈ ℂ) | |
28 | 19, 24, 26, 27 | fvmptd 6548 | 1 ⊢ (𝜑 → ( 1 ‘𝐴) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ifcif 4307 ↦ cmpt 4965 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 0cc0 10272 1c1 10273 ℕcn 11374 Basecbs 16255 +gcplusg 16338 0gc0g 16486 Grpcgrp 17809 Abelcabl 18580 Unitcui 19026 ℤ/nℤczn 20247 DChrcdchr 25409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-ec 8028 df-qs 8032 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-0g 16488 df-imas 16554 df-qus 16555 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-nsg 17976 df-eqg 17977 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-subrg 19170 df-lmod 19257 df-lss 19325 df-lsp 19367 df-sra 19569 df-rgmod 19570 df-lidl 19571 df-rsp 19572 df-2idl 19629 df-cnfld 20143 df-zring 20215 df-zn 20251 df-dchr 25410 |
This theorem is referenced by: dchrinv 25438 dchr1re 25440 dchrsum2 25445 rpvmasumlem 25628 rpvmasum2 25653 |
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