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Mirrors > Home > MPE Home > Th. List > dchreq | Structured version Visualization version GIF version |
Description: A Dirichlet character is determined by its values on the unit group. (Contributed by Mario Carneiro, 28-Apr-2016.) |
Ref | Expression |
---|---|
dchrresb.g | ⊢ 𝐺 = (DChr‘𝑁) |
dchrresb.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
dchrresb.b | ⊢ 𝐷 = (Base‘𝐺) |
dchrresb.u | ⊢ 𝑈 = (Unit‘𝑍) |
dchrresb.x | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
dchrresb.Y | ⊢ (𝜑 → 𝑌 ∈ 𝐷) |
Ref | Expression |
---|---|
dchreq | ⊢ (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3986 | . . . . 5 ⊢ (𝑘 ∈ ((Base‘𝑍) ∖ 𝑈) ↔ (𝑘 ∈ (Base‘𝑍) ∧ ¬ 𝑘 ∈ 𝑈)) | |
2 | dchrresb.g | . . . . . . . . 9 ⊢ 𝐺 = (DChr‘𝑁) | |
3 | dchrresb.z | . . . . . . . . 9 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
4 | dchrresb.b | . . . . . . . . 9 ⊢ 𝐷 = (Base‘𝐺) | |
5 | eqid 2740 | . . . . . . . . 9 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
6 | dchrresb.u | . . . . . . . . 9 ⊢ 𝑈 = (Unit‘𝑍) | |
7 | dchrresb.x | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
8 | 7 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → 𝑋 ∈ 𝐷) |
9 | simpr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → 𝑘 ∈ (Base‘𝑍)) | |
10 | 2, 3, 4, 5, 6, 8, 9 | dchrn0 27314 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑋‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈)) |
11 | 10 | biimpd 229 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)) |
12 | 11 | necon1bd 2964 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → (¬ 𝑘 ∈ 𝑈 → (𝑋‘𝑘) = 0)) |
13 | 12 | impr 454 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑍) ∧ ¬ 𝑘 ∈ 𝑈)) → (𝑋‘𝑘) = 0) |
14 | 1, 13 | sylan2b 593 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑋‘𝑘) = 0) |
15 | dchrresb.Y | . . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝐷) | |
16 | 15 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → 𝑌 ∈ 𝐷) |
17 | 2, 3, 4, 5, 6, 16, 9 | dchrn0 27314 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑌‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈)) |
18 | 17 | biimpd 229 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑌‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)) |
19 | 18 | necon1bd 2964 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → (¬ 𝑘 ∈ 𝑈 → (𝑌‘𝑘) = 0)) |
20 | 19 | impr 454 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑍) ∧ ¬ 𝑘 ∈ 𝑈)) → (𝑌‘𝑘) = 0) |
21 | 1, 20 | sylan2b 593 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑌‘𝑘) = 0) |
22 | 14, 21 | eqtr4d 2783 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑋‘𝑘) = (𝑌‘𝑘)) |
23 | 22 | ralrimiva 3152 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘)) |
24 | 2, 3, 4, 5, 7 | dchrf 27306 | . . . . 5 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ) |
25 | 24 | ffnd 6750 | . . . 4 ⊢ (𝜑 → 𝑋 Fn (Base‘𝑍)) |
26 | 2, 3, 4, 5, 15 | dchrf 27306 | . . . . 5 ⊢ (𝜑 → 𝑌:(Base‘𝑍)⟶ℂ) |
27 | 26 | ffnd 6750 | . . . 4 ⊢ (𝜑 → 𝑌 Fn (Base‘𝑍)) |
28 | eqfnfv 7066 | . . . 4 ⊢ ((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘))) | |
29 | 25, 27, 28 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘))) |
30 | 5, 6 | unitss 20404 | . . . . . 6 ⊢ 𝑈 ⊆ (Base‘𝑍) |
31 | undif 4505 | . . . . . 6 ⊢ (𝑈 ⊆ (Base‘𝑍) ↔ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈)) = (Base‘𝑍)) | |
32 | 30, 31 | mpbi 230 | . . . . 5 ⊢ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈)) = (Base‘𝑍) |
33 | 32 | raleqi 3332 | . . . 4 ⊢ (∀𝑘 ∈ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈))(𝑋‘𝑘) = (𝑌‘𝑘) ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘)) |
34 | ralunb 4220 | . . . 4 ⊢ (∀𝑘 ∈ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈))(𝑋‘𝑘) = (𝑌‘𝑘) ↔ (∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘) ∧ ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘))) | |
35 | 33, 34 | bitr3i 277 | . . 3 ⊢ (∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘) ↔ (∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘) ∧ ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘))) |
36 | 29, 35 | bitrdi 287 | . 2 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ (∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘) ∧ ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘)))) |
37 | 23, 36 | mpbiran2d 707 | 1 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∖ cdif 3973 ∪ cun 3974 ⊆ wss 3976 Fn wfn 6570 ‘cfv 6575 ℂcc 11184 0cc0 11186 Basecbs 17260 Unitcui 20383 ℤ/nℤczn 21538 DChrcdchr 27296 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-addf 11265 ax-mulf 11266 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-tpos 8269 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-er 8765 df-ec 8767 df-qs 8771 df-map 8888 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-sup 9513 df-inf 9514 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-7 12363 df-8 12364 df-9 12365 df-n0 12556 df-z 12642 df-dec 12761 df-uz 12906 df-fz 13570 df-struct 17196 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-ress 17290 df-plusg 17326 df-mulr 17327 df-starv 17328 df-sca 17329 df-vsca 17330 df-ip 17331 df-tset 17332 df-ple 17333 df-ds 17335 df-unif 17336 df-0g 17503 df-imas 17570 df-qus 17571 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-mhm 18820 df-grp 18978 df-minusg 18979 df-sbg 18980 df-subg 19165 df-nsg 19166 df-eqg 19167 df-cmn 19826 df-abl 19827 df-mgp 20164 df-rng 20182 df-ur 20211 df-ring 20264 df-cring 20265 df-oppr 20362 df-dvdsr 20385 df-unit 20386 df-invr 20416 df-subrng 20574 df-subrg 20599 df-lmod 20884 df-lss 20955 df-lsp 20995 df-sra 21197 df-rgmod 21198 df-lidl 21243 df-rsp 21244 df-2idl 21285 df-cnfld 21390 df-zring 21483 df-zn 21542 df-dchr 27297 |
This theorem is referenced by: dchrresb 27323 dchrinv 27325 dchrsum2 27332 |
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