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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 25529 | . . 3 ⊢ (𝜑 → (𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0)) ∈ (Base‘𝐺)) |
9 | eleq1w 2848 | . . . . . 6 ⊢ (𝑘 = 𝑥 → (𝑘 ∈ 𝑈 ↔ 𝑥 ∈ 𝑈)) | |
10 | 9 | ifbid 4372 | . . . . 5 ⊢ (𝑘 = 𝑥 → if(𝑘 ∈ 𝑈, 1, 0) = if(𝑥 ∈ 𝑈, 1, 0)) |
11 | 10 | cbvmptv 5028 | . . . 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 25530 | . . 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 25532 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) |
15 | ablgrp 18671 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
16 | dchr1.o | . . . . 5 ⊢ 1 = (0g‘𝐺) | |
17 | 3, 12, 16 | isgrpid2 17927 | . . . 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 699 | . 2 ⊢ (𝜑 → 1 = (𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0))) |
20 | simpr 477 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝑘 = 𝐴) | |
21 | dchr1.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
22 | 21 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐴 ∈ 𝑈) |
23 | 20, 22 | eqeltrd 2866 | . . 3 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝑘 ∈ 𝑈) |
24 | 23 | iftrued 4358 | . 2 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → if(𝑘 ∈ 𝑈, 1, 0) = 1) |
25 | 4, 5 | unitss 19133 | . . 3 ⊢ 𝑈 ⊆ (Base‘𝑍) |
26 | 25, 21 | sseldi 3856 | . 2 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑍)) |
27 | 1cnd 10434 | . 2 ⊢ (𝜑 → 1 ∈ ℂ) | |
28 | 19, 24, 26, 27 | fvmptd 6601 | 1 ⊢ (𝜑 → ( 1 ‘𝐴) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ifcif 4350 ↦ cmpt 5008 ‘cfv 6188 (class class class)co 6976 ℂcc 10333 0cc0 10335 1c1 10336 ℕcn 11439 Basecbs 16339 +gcplusg 16421 0gc0g 16569 Grpcgrp 17891 Abelcabl 18667 Unitcui 19112 ℤ/nℤczn 20352 DChrcdchr 25510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-addf 10414 ax-mulf 10415 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-of 7227 df-om 7397 df-1st 7501 df-2nd 7502 df-tpos 7695 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-ec 8091 df-qs 8095 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-sup 8701 df-inf 8702 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-uz 12059 df-fz 12709 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-starv 16436 df-sca 16437 df-vsca 16438 df-ip 16439 df-tset 16440 df-ple 16441 df-ds 16443 df-unif 16444 df-0g 16571 df-imas 16637 df-qus 16638 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-mhm 17803 df-grp 17894 df-minusg 17895 df-sbg 17896 df-subg 18060 df-nsg 18061 df-eqg 18062 df-cmn 18668 df-abl 18669 df-mgp 18963 df-ur 18975 df-ring 19022 df-cring 19023 df-oppr 19096 df-dvdsr 19114 df-unit 19115 df-invr 19145 df-subrg 19256 df-lmod 19358 df-lss 19426 df-lsp 19466 df-sra 19666 df-rgmod 19667 df-lidl 19668 df-rsp 19669 df-2idl 19726 df-cnfld 20248 df-zring 20320 df-zn 20356 df-dchr 25511 |
This theorem is referenced by: dchrinv 25539 dchr1re 25541 dchrsum2 25546 rpvmasumlem 25765 rpvmasum2 25790 |
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