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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 25313 | . . . . 5 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
11 | 1, 2, 3, 5, 6, 7, 10 | dchr1cl 25324 | . . 3 ⊢ (𝜑 → 1 ∈ 𝐷) |
12 | 1, 2, 3, 4, 11, 8 | dchrmul 25321 | . 2 ⊢ (𝜑 → ( 1 · 𝑋) = ( 1 ∘𝑓 · 𝑋)) |
13 | oveq1 6883 | . . . . . 6 ⊢ (1 = if(𝑘 ∈ 𝑈, 1, 0) → (1 · (𝑋‘𝑘)) = (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) | |
14 | 13 | eqeq1d 2799 | . . . . 5 ⊢ (1 = if(𝑘 ∈ 𝑈, 1, 0) → ((1 · (𝑋‘𝑘)) = (𝑋‘𝑘) ↔ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘))) |
15 | oveq1 6883 | . . . . . 6 ⊢ (0 = if(𝑘 ∈ 𝑈, 1, 0) → (0 · (𝑋‘𝑘)) = (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) | |
16 | 15 | eqeq1d 2799 | . . . . 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 25315 | . . . . . . . 8 ⊢ (𝜑 → 𝑋:𝐵⟶ℂ) |
18 | 17 | ffvelrnda 6583 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑋‘𝑘) ∈ ℂ) |
19 | 18 | adantr 473 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ∈ ℂ) |
20 | 19 | mulid2d 10345 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (1 · (𝑋‘𝑘)) = (𝑋‘𝑘)) |
21 | 0cn 10318 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
22 | 21 | mul02i 10513 | . . . . . 6 ⊢ (0 · 0) = 0 |
23 | 1, 2, 5, 6, 10, 3 | dchrelbas2 25310 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)))) |
24 | 8, 23 | mpbid 224 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈))) |
25 | 24 | simprd 490 | . . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)) |
26 | 25 | r19.21bi 3111 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)) |
27 | 26 | necon1bd 2987 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝑈 → (𝑋‘𝑘) = 0)) |
28 | 27 | imp 396 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) = 0) |
29 | 28 | oveq2d 6892 | . . . . . 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 4312 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘)) |
32 | 31 | mpteq2dva 4935 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘))) |
33 | 5 | fvexi 6423 | . . . . 5 ⊢ 𝐵 ∈ V |
34 | 33 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
35 | ax-1cn 10280 | . . . . . 6 ⊢ 1 ∈ ℂ | |
36 | 35, 21 | ifcli 4321 | . . . . 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 6472 | . . . 4 ⊢ (𝜑 → 𝑋 = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘))) |
40 | 34, 37, 18, 38, 39 | offval2 7146 | . . 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 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2969 ∀wral 3087 Vcvv 3383 ifcif 4275 ↦ cmpt 4920 ‘cfv 6099 (class class class)co 6876 ∘𝑓 cof 7127 ℂcc 10220 0cc0 10222 1c1 10223 · cmul 10227 ℕcn 11310 Basecbs 16180 +gcplusg 16263 MndHom cmhm 17644 mulGrpcmgp 18801 Unitcui 18951 ℂfldccnfld 20064 ℤ/nℤczn 20169 DChrcdchr 25305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-addf 10301 ax-mulf 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-int 4666 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-of 7129 df-om 7298 df-1st 7399 df-2nd 7400 df-tpos 7588 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-oadd 7801 df-er 7980 df-ec 7982 df-qs 7986 df-map 8095 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-sup 8588 df-inf 8589 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-9 11379 df-n0 11577 df-z 11663 df-dec 11780 df-uz 11927 df-fz 12577 df-struct 16182 df-ndx 16183 df-slot 16184 df-base 16186 df-sets 16187 df-ress 16188 df-plusg 16276 df-mulr 16277 df-starv 16278 df-sca 16279 df-vsca 16280 df-ip 16281 df-tset 16282 df-ple 16283 df-ds 16285 df-unif 16286 df-0g 16413 df-imas 16479 df-qus 16480 df-mgm 17553 df-sgrp 17595 df-mnd 17606 df-mhm 17646 df-grp 17737 df-minusg 17738 df-sbg 17739 df-subg 17900 df-nsg 17901 df-eqg 17902 df-cmn 18506 df-abl 18507 df-mgp 18802 df-ur 18814 df-ring 18861 df-cring 18862 df-oppr 18935 df-dvdsr 18953 df-unit 18954 df-subrg 19092 df-lmod 19179 df-lss 19247 df-lsp 19289 df-sra 19491 df-rgmod 19492 df-lidl 19493 df-rsp 19494 df-2idl 19551 df-cnfld 20065 df-zring 20137 df-zn 20173 df-dchr 25306 |
This theorem is referenced by: dchrabl 25327 dchr1 25330 |
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