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Mirrors > Home > MPE Home > Th. List > dchrmulid2 | Structured version Visualization version GIF version |
Description: Left identity for the principal Dirichlet character. (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) |
dchrn0.b | ⊢ 𝐵 = (Base‘𝑍) |
dchrn0.u | ⊢ 𝑈 = (Unit‘𝑍) |
dchr1cl.o | ⊢ 1 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0)) |
dchrmulid2.t | ⊢ · = (+g‘𝐺) |
dchrmulid2.x | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
Ref | Expression |
---|---|
dchrmulid2 | ⊢ (𝜑 → ( 1 · 𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchrmhm.g | . . 3 ⊢ 𝐺 = (DChr‘𝑁) | |
2 | dchrmhm.z | . . 3 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
3 | dchrmhm.b | . . 3 ⊢ 𝐷 = (Base‘𝐺) | |
4 | dchrmulid2.t | . . 3 ⊢ · = (+g‘𝐺) | |
5 | dchrn0.b | . . . 4 ⊢ 𝐵 = (Base‘𝑍) | |
6 | dchrn0.u | . . . 4 ⊢ 𝑈 = (Unit‘𝑍) | |
7 | dchr1cl.o | . . . 4 ⊢ 1 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0)) | |
8 | dchrmulid2.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
9 | 1, 3 | dchrrcl 25503 | . . . . 5 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
11 | 1, 2, 3, 5, 6, 7, 10 | dchr1cl 25514 | . . 3 ⊢ (𝜑 → 1 ∈ 𝐷) |
12 | 1, 2, 3, 4, 11, 8 | dchrmul 25511 | . 2 ⊢ (𝜑 → ( 1 · 𝑋) = ( 1 ∘𝑓 · 𝑋)) |
13 | oveq1 7028 | . . . . . 6 ⊢ (1 = if(𝑘 ∈ 𝑈, 1, 0) → (1 · (𝑋‘𝑘)) = (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) | |
14 | 13 | eqeq1d 2797 | . . . . 5 ⊢ (1 = if(𝑘 ∈ 𝑈, 1, 0) → ((1 · (𝑋‘𝑘)) = (𝑋‘𝑘) ↔ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘))) |
15 | oveq1 7028 | . . . . . 6 ⊢ (0 = if(𝑘 ∈ 𝑈, 1, 0) → (0 · (𝑋‘𝑘)) = (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) | |
16 | 15 | eqeq1d 2797 | . . . . 5 ⊢ (0 = if(𝑘 ∈ 𝑈, 1, 0) → ((0 · (𝑋‘𝑘)) = (𝑋‘𝑘) ↔ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘))) |
17 | 1, 2, 3, 5, 8 | dchrf 25505 | . . . . . . . 8 ⊢ (𝜑 → 𝑋:𝐵⟶ℂ) |
18 | 17 | ffvelrnda 6721 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑋‘𝑘) ∈ ℂ) |
19 | 18 | adantr 481 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ∈ ℂ) |
20 | 19 | mulid2d 10510 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (1 · (𝑋‘𝑘)) = (𝑋‘𝑘)) |
21 | 0cn 10484 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
22 | 21 | mul02i 10681 | . . . . . 6 ⊢ (0 · 0) = 0 |
23 | 1, 2, 5, 6, 10, 3 | dchrelbas2 25500 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)))) |
24 | 8, 23 | mpbid 233 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈))) |
25 | 24 | simprd 496 | . . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)) |
26 | 25 | r19.21bi 3175 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)) |
27 | 26 | necon1bd 3002 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝑈 → (𝑋‘𝑘) = 0)) |
28 | 27 | imp 407 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) = 0) |
29 | 28 | oveq2d 7037 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (0 · (𝑋‘𝑘)) = (0 · 0)) |
30 | 22, 29, 28 | 3eqtr4a 2857 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (0 · (𝑋‘𝑘)) = (𝑋‘𝑘)) |
31 | 14, 16, 20, 30 | ifbothda 4422 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘)) |
32 | 31 | mpteq2dva 5060 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘))) |
33 | 5 | fvexi 6557 | . . . . 5 ⊢ 𝐵 ∈ V |
34 | 33 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
35 | ax-1cn 10446 | . . . . . 6 ⊢ 1 ∈ ℂ | |
36 | 35, 21 | ifcli 4431 | . . . . 5 ⊢ if(𝑘 ∈ 𝑈, 1, 0) ∈ ℂ |
37 | 36 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝑈, 1, 0) ∈ ℂ) |
38 | 7 | a1i 11 | . . . 4 ⊢ (𝜑 → 1 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0))) |
39 | 17 | feqmptd 6606 | . . . 4 ⊢ (𝜑 → 𝑋 = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘))) |
40 | 34, 37, 18, 38, 39 | offval2 7289 | . . 3 ⊢ (𝜑 → ( 1 ∘𝑓 · 𝑋) = (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)))) |
41 | 32, 40, 39 | 3eqtr4d 2841 | . 2 ⊢ (𝜑 → ( 1 ∘𝑓 · 𝑋) = 𝑋) |
42 | 12, 41 | eqtrd 2831 | 1 ⊢ (𝜑 → ( 1 · 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ∀wral 3105 Vcvv 3437 ifcif 4385 ↦ cmpt 5045 ‘cfv 6230 (class class class)co 7021 ∘𝑓 cof 7270 ℂcc 10386 0cc0 10388 1c1 10389 · cmul 10393 ℕcn 11491 Basecbs 16317 +gcplusg 16399 MndHom cmhm 17777 mulGrpcmgp 18934 Unitcui 19084 ℂfldccnfld 20232 ℤ/nℤczn 20337 DChrcdchr 25495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 ax-addf 10467 ax-mulf 10468 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-of 7272 df-om 7442 df-1st 7550 df-2nd 7551 df-tpos 7748 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-oadd 7962 df-er 8144 df-ec 8146 df-qs 8150 df-map 8263 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-sup 8757 df-inf 8758 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-7 11558 df-8 11559 df-9 11560 df-n0 11751 df-z 11835 df-dec 11953 df-uz 12099 df-fz 12748 df-struct 16319 df-ndx 16320 df-slot 16321 df-base 16323 df-sets 16324 df-ress 16325 df-plusg 16412 df-mulr 16413 df-starv 16414 df-sca 16415 df-vsca 16416 df-ip 16417 df-tset 16418 df-ple 16419 df-ds 16421 df-unif 16422 df-0g 16549 df-imas 16615 df-qus 16616 df-mgm 17686 df-sgrp 17728 df-mnd 17739 df-mhm 17779 df-grp 17869 df-minusg 17870 df-sbg 17871 df-subg 18035 df-nsg 18036 df-eqg 18037 df-cmn 18640 df-abl 18641 df-mgp 18935 df-ur 18947 df-ring 18994 df-cring 18995 df-oppr 19068 df-dvdsr 19086 df-unit 19087 df-subrg 19228 df-lmod 19331 df-lss 19399 df-lsp 19439 df-sra 19639 df-rgmod 19640 df-lidl 19641 df-rsp 19642 df-2idl 19699 df-cnfld 20233 df-zring 20305 df-zn 20341 df-dchr 25496 |
This theorem is referenced by: dchrabl 25517 dchr1 25520 |
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