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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 2821 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | eqid 2821 | . . . 4 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
5 | dchr1.u | . . . 4 ⊢ 𝑈 = (Unit‘𝑍) | |
6 | eqid 2821 | . . . 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 25827 | . . 3 ⊢ (𝜑 → (𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0)) ∈ (Base‘𝐺)) |
9 | eleq1w 2895 | . . . . . 6 ⊢ (𝑘 = 𝑥 → (𝑘 ∈ 𝑈 ↔ 𝑥 ∈ 𝑈)) | |
10 | 9 | ifbid 4489 | . . . . 5 ⊢ (𝑘 = 𝑥 → if(𝑘 ∈ 𝑈, 1, 0) = if(𝑥 ∈ 𝑈, 1, 0)) |
11 | 10 | cbvmptv 5169 | . . . 4 ⊢ (𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0)) = (𝑥 ∈ (Base‘𝑍) ↦ if(𝑥 ∈ 𝑈, 1, 0)) |
12 | eqid 2821 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
13 | 1, 2, 3, 4, 5, 11, 12, 8 | dchrmulid2 25828 | . . 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 25830 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) |
15 | ablgrp 18911 | . . . 4 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
16 | dchr1.o | . . . . 5 ⊢ 1 = (0g‘𝐺) | |
17 | 3, 12, 16 | isgrpid2 18140 | . . . 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 710 | . 2 ⊢ (𝜑 → 1 = (𝑘 ∈ (Base‘𝑍) ↦ if(𝑘 ∈ 𝑈, 1, 0))) |
20 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝑘 = 𝐴) | |
21 | dchr1.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
22 | 21 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐴 ∈ 𝑈) |
23 | 20, 22 | eqeltrd 2913 | . . 3 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝑘 ∈ 𝑈) |
24 | 23 | iftrued 4475 | . 2 ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → if(𝑘 ∈ 𝑈, 1, 0) = 1) |
25 | 4, 5 | unitss 19410 | . . 3 ⊢ 𝑈 ⊆ (Base‘𝑍) |
26 | 25, 21 | sseldi 3965 | . 2 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝑍)) |
27 | 1cnd 10636 | . 2 ⊢ (𝜑 → 1 ∈ ℂ) | |
28 | 19, 24, 26, 27 | fvmptd 6775 | 1 ⊢ (𝜑 → ( 1 ‘𝐴) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ifcif 4467 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 0cc0 10537 1c1 10538 ℕcn 11638 Basecbs 16483 +gcplusg 16565 0gc0g 16713 Grpcgrp 18103 Abelcabl 18907 Unitcui 19389 ℤ/nℤczn 20650 DChrcdchr 25808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-ec 8291 df-qs 8295 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-0g 16715 df-imas 16781 df-qus 16782 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-nsg 18277 df-eqg 18278 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-subrg 19533 df-lmod 19636 df-lss 19704 df-lsp 19744 df-sra 19944 df-rgmod 19945 df-lidl 19946 df-rsp 19947 df-2idl 20005 df-cnfld 20546 df-zring 20618 df-zn 20654 df-dchr 25809 |
This theorem is referenced by: dchrinv 25837 dchr1re 25839 dchrsum2 25844 rpvmasumlem 26063 rpvmasum2 26088 |
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