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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 26388 | . . . . 5 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
11 | 1, 2, 3, 5, 6, 7, 10 | dchr1cl 26399 | . . 3 ⊢ (𝜑 → 1 ∈ 𝐷) |
12 | 1, 2, 3, 4, 11, 8 | dchrmul 26396 | . 2 ⊢ (𝜑 → ( 1 · 𝑋) = ( 1 ∘f · 𝑋)) |
13 | oveq1 7282 | . . . . . 6 ⊢ (1 = if(𝑘 ∈ 𝑈, 1, 0) → (1 · (𝑋‘𝑘)) = (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) | |
14 | 13 | eqeq1d 2740 | . . . . 5 ⊢ (1 = if(𝑘 ∈ 𝑈, 1, 0) → ((1 · (𝑋‘𝑘)) = (𝑋‘𝑘) ↔ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘))) |
15 | oveq1 7282 | . . . . . 6 ⊢ (0 = if(𝑘 ∈ 𝑈, 1, 0) → (0 · (𝑋‘𝑘)) = (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) | |
16 | 15 | eqeq1d 2740 | . . . . 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 26390 | . . . . . . . 8 ⊢ (𝜑 → 𝑋:𝐵⟶ℂ) |
18 | 17 | ffvelrnda 6961 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑋‘𝑘) ∈ ℂ) |
19 | 18 | adantr 481 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ∈ ℂ) |
20 | 19 | mulid2d 10993 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (1 · (𝑋‘𝑘)) = (𝑋‘𝑘)) |
21 | 0cn 10967 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
22 | 21 | mul02i 11164 | . . . . . 6 ⊢ (0 · 0) = 0 |
23 | 1, 2, 5, 6, 10, 3 | dchrelbas2 26385 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)))) |
24 | 8, 23 | mpbid 231 | . . . . . . . . . . 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 3134 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)) |
27 | 26 | necon1bd 2961 | . . . . . . . 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 7291 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (0 · (𝑋‘𝑘)) = (0 · 0)) |
30 | 22, 29, 28 | 3eqtr4a 2804 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (0 · (𝑋‘𝑘)) = (𝑋‘𝑘)) |
31 | 14, 16, 20, 30 | ifbothda 4497 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘)) |
32 | 31 | mpteq2dva 5174 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘))) |
33 | 5 | fvexi 6788 | . . . . 5 ⊢ 𝐵 ∈ V |
34 | 33 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
35 | ax-1cn 10929 | . . . . . 6 ⊢ 1 ∈ ℂ | |
36 | 35, 21 | ifcli 4506 | . . . . 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 6837 | . . . 4 ⊢ (𝜑 → 𝑋 = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘))) |
40 | 34, 37, 18, 38, 39 | offval2 7553 | . . 3 ⊢ (𝜑 → ( 1 ∘f · 𝑋) = (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)))) |
41 | 32, 40, 39 | 3eqtr4d 2788 | . 2 ⊢ (𝜑 → ( 1 ∘f · 𝑋) = 𝑋) |
42 | 12, 41 | eqtrd 2778 | 1 ⊢ (𝜑 → ( 1 · 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 Vcvv 3432 ifcif 4459 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 ∘f cof 7531 ℂcc 10869 0cc0 10871 1c1 10872 · cmul 10876 ℕcn 11973 Basecbs 16912 +gcplusg 16962 MndHom cmhm 18428 mulGrpcmgp 19720 Unitcui 19881 ℂfldccnfld 20597 ℤ/nℤczn 20704 DChrcdchr 26380 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-addf 10950 ax-mulf 10951 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-ec 8500 df-qs 8504 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-0g 17152 df-imas 17219 df-qus 17220 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-nsg 18753 df-eqg 18754 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-subrg 20022 df-lmod 20125 df-lss 20194 df-lsp 20234 df-sra 20434 df-rgmod 20435 df-lidl 20436 df-rsp 20437 df-2idl 20503 df-cnfld 20598 df-zring 20671 df-zn 20708 df-dchr 26381 |
This theorem is referenced by: dchrabl 26402 dchr1 26405 |
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