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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 2737 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 4 | eqid 2737 | . . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
| 5 | dchr1.u | . . . 4 ⊢ 𝑈 = (Unit‘𝑍) | |
| 6 | eqid 2737 | . . . 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 27295 | . . 3 ⊢ (𝜑 → (𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0)) ∈ (Base‘𝐺)) |
| 9 | eleq1w 2824 | . . . . . 6 ⊢ (𝑘 = 𝑥 → (𝑘 ∈ 𝑈 ↔ 𝑥 ∈ 𝑈)) | |
| 10 | 9 | ifbid 4549 | . . . . 5 ⊢ (𝑘 = 𝑥 → if(𝑘 ∈ 𝑈, 1, 0) = if(𝑥 ∈ 𝑈, 1, 0)) |
| 11 | 10 | cbvmptv 5255 | . . . 4 ⊢ (𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0)) = (𝑥 ∈ (Base‘𝑍) ↦ if(𝑥 ∈ 𝑈, 1, 0)) |
| 12 | eqid 2737 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 13 | 1, 2, 3, 4, 5, 11, 12, 8 | dchrmullid 27296 | . . 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 27298 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) |
| 15 | ablgrp 19803 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
| 16 | dchr1.o | . . . . 5 ⊢ 1 = (0g‘𝐺) | |
| 17 | 3, 12, 16 | isgrpid2 18994 | . . . 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 712 | . 2 ⊢ (𝜑 → 1 = (𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0))) |
| 20 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝑘 = 𝐴) | |
| 21 | dchr1.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
| 22 | 21 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐴 ∈ 𝑈) |
| 23 | 20, 22 | eqeltrd 2841 | . . 3 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝑘 ∈ 𝑈) |
| 24 | 23 | iftrued 4533 | . 2 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → if(𝑘 ∈ 𝑈, 1, 0) = 1) |
| 25 | 4, 5 | unitss 20376 | . . 3 ⊢ 𝑈 ⊆ (Base‘𝑍) |
| 26 | 25, 21 | sselid 3981 | . 2 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑍)) |
| 27 | 1cnd 11256 | . 2 ⊢ (𝜑 → 1 ∈ ℂ) | |
| 28 | 19, 24, 26, 27 | fvmptd 7023 | 1 ⊢ (𝜑 → ( 1 ‘𝐴) = 1) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ifcif 4525 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 ℕcn 12266 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Grpcgrp 18951 Abelcabl 19799 Unitcui 20355 ℤ/nℤczn 21513 DChrcdchr 27276 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-ec 8747 df-qs 8751 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-imas 17553 df-qus 17554 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-nsg 19142 df-eqg 19143 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-lss 20930 df-lsp 20970 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-rsp 21219 df-2idl 21260 df-cnfld 21365 df-zring 21458 df-zn 21517 df-dchr 27277 |
| This theorem is referenced by: dchrinv 27305 dchr1re 27307 dchrsum2 27312 rpvmasumlem 27531 rpvmasum2 27556 |
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