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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 25743 | . . . . 5 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
11 | 1, 2, 3, 5, 6, 7, 10 | dchr1cl 25754 | . . 3 ⊢ (𝜑 → 1 ∈ 𝐷) |
12 | 1, 2, 3, 4, 11, 8 | dchrmul 25751 | . 2 ⊢ (𝜑 → ( 1 · 𝑋) = ( 1 ∘f · 𝑋)) |
13 | oveq1 7152 | . . . . . 6 ⊢ (1 = if(𝑘 ∈ 𝑈, 1, 0) → (1 · (𝑋‘𝑘)) = (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) | |
14 | 13 | eqeq1d 2820 | . . . . 5 ⊢ (1 = if(𝑘 ∈ 𝑈, 1, 0) → ((1 · (𝑋‘𝑘)) = (𝑋‘𝑘) ↔ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘))) |
15 | oveq1 7152 | . . . . . 6 ⊢ (0 = if(𝑘 ∈ 𝑈, 1, 0) → (0 · (𝑋‘𝑘)) = (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) | |
16 | 15 | eqeq1d 2820 | . . . . 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 25745 | . . . . . . . 8 ⊢ (𝜑 → 𝑋:𝐵⟶ℂ) |
18 | 17 | ffvelrnda 6843 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑋‘𝑘) ∈ ℂ) |
19 | 18 | adantr 481 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ∈ ℂ) |
20 | 19 | mulid2d 10647 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (1 · (𝑋‘𝑘)) = (𝑋‘𝑘)) |
21 | 0cn 10621 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
22 | 21 | mul02i 10817 | . . . . . 6 ⊢ (0 · 0) = 0 |
23 | 1, 2, 5, 6, 10, 3 | dchrelbas2 25740 | . . . . . . . . . . . 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 3205 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)) |
27 | 26 | necon1bd 3031 | . . . . . . . 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 7161 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (0 · (𝑋‘𝑘)) = (0 · 0)) |
30 | 22, 29, 28 | 3eqtr4a 2879 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (0 · (𝑋‘𝑘)) = (𝑋‘𝑘)) |
31 | 14, 16, 20, 30 | ifbothda 4500 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘)) |
32 | 31 | mpteq2dva 5152 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘))) |
33 | 5 | fvexi 6677 | . . . . 5 ⊢ 𝐵 ∈ V |
34 | 33 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
35 | ax-1cn 10583 | . . . . . 6 ⊢ 1 ∈ ℂ | |
36 | 35, 21 | ifcli 4509 | . . . . 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 6726 | . . . 4 ⊢ (𝜑 → 𝑋 = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘))) |
40 | 34, 37, 18, 38, 39 | offval2 7415 | . . 3 ⊢ (𝜑 → ( 1 ∘f · 𝑋) = (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)))) |
41 | 32, 40, 39 | 3eqtr4d 2863 | . 2 ⊢ (𝜑 → ( 1 ∘f · 𝑋) = 𝑋) |
42 | 12, 41 | eqtrd 2853 | 1 ⊢ (𝜑 → ( 1 · 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 Vcvv 3492 ifcif 4463 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 ∘f cof 7396 ℂcc 10523 0cc0 10525 1c1 10526 · cmul 10530 ℕcn 11626 Basecbs 16471 +gcplusg 16553 MndHom cmhm 17942 mulGrpcmgp 19168 Unitcui 19318 ℂfldccnfld 20473 ℤ/nℤczn 20578 DChrcdchr 25735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-ec 8280 df-qs 8284 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-imas 16769 df-qus 16770 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-nsg 18215 df-eqg 18216 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-subrg 19462 df-lmod 19565 df-lss 19633 df-lsp 19673 df-sra 19873 df-rgmod 19874 df-lidl 19875 df-rsp 19876 df-2idl 19933 df-cnfld 20474 df-zring 20546 df-zn 20582 df-dchr 25736 |
This theorem is referenced by: dchrabl 25757 dchr1 25760 |
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