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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 | dchrresb.g | . . . . . 6 ⊢ 𝐺 = (DChr‘𝑁) | |
2 | dchrresb.z | . . . . . 6 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
3 | dchrresb.b | . . . . . 6 ⊢ 𝐷 = (Base‘𝐺) | |
4 | eqid 2799 | . . . . . 6 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
5 | dchrresb.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
6 | 1, 2, 3, 4, 5 | dchrf 25319 | . . . . 5 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ) |
7 | 6 | ffnd 6257 | . . . 4 ⊢ (𝜑 → 𝑋 Fn (Base‘𝑍)) |
8 | dchrresb.Y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐷) | |
9 | 1, 2, 3, 4, 8 | dchrf 25319 | . . . . 5 ⊢ (𝜑 → 𝑌:(Base‘𝑍)⟶ℂ) |
10 | 9 | ffnd 6257 | . . . 4 ⊢ (𝜑 → 𝑌 Fn (Base‘𝑍)) |
11 | eqfnfv 6537 | . . . 4 ⊢ ((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘))) | |
12 | 7, 10, 11 | syl2anc 580 | . . 3 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘))) |
13 | dchrresb.u | . . . . . . 7 ⊢ 𝑈 = (Unit‘𝑍) | |
14 | 4, 13 | unitss 18976 | . . . . . 6 ⊢ 𝑈 ⊆ (Base‘𝑍) |
15 | undif 4243 | . . . . . 6 ⊢ (𝑈 ⊆ (Base‘𝑍) ↔ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈)) = (Base‘𝑍)) | |
16 | 14, 15 | mpbi 222 | . . . . 5 ⊢ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈)) = (Base‘𝑍) |
17 | 16 | raleqi 3325 | . . . 4 ⊢ (∀𝑘 ∈ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈))(𝑋‘𝑘) = (𝑌‘𝑘) ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘)) |
18 | ralunb 3992 | . . . 4 ⊢ (∀𝑘 ∈ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈))(𝑋‘𝑘) = (𝑌‘𝑘) ↔ (∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘) ∧ ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘))) | |
19 | 17, 18 | bitr3i 269 | . . 3 ⊢ (∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘) ↔ (∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘) ∧ ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘))) |
20 | 12, 19 | syl6bb 279 | . 2 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ (∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘) ∧ ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘)))) |
21 | eldif 3779 | . . . . . 6 ⊢ (𝑘 ∈ ((Base‘𝑍) ∖ 𝑈) ↔ (𝑘 ∈ (Base‘𝑍) ∧ ¬ 𝑘 ∈ 𝑈)) | |
22 | 5 | adantr 473 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → 𝑋 ∈ 𝐷) |
23 | simpr 478 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → 𝑘 ∈ (Base‘𝑍)) | |
24 | 1, 2, 3, 4, 13, 22, 23 | dchrn0 25327 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑋‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈)) |
25 | 24 | biimpd 221 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)) |
26 | 25 | necon1bd 2989 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → (¬ 𝑘 ∈ 𝑈 → (𝑋‘𝑘) = 0)) |
27 | 26 | impr 447 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑍) ∧ ¬ 𝑘 ∈ 𝑈)) → (𝑋‘𝑘) = 0) |
28 | 21, 27 | sylan2b 588 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑋‘𝑘) = 0) |
29 | 8 | adantr 473 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → 𝑌 ∈ 𝐷) |
30 | 1, 2, 3, 4, 13, 29, 23 | dchrn0 25327 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑌‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈)) |
31 | 30 | biimpd 221 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑌‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)) |
32 | 31 | necon1bd 2989 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → (¬ 𝑘 ∈ 𝑈 → (𝑌‘𝑘) = 0)) |
33 | 32 | impr 447 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑍) ∧ ¬ 𝑘 ∈ 𝑈)) → (𝑌‘𝑘) = 0) |
34 | 21, 33 | sylan2b 588 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑌‘𝑘) = 0) |
35 | 28, 34 | eqtr4d 2836 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑋‘𝑘) = (𝑌‘𝑘)) |
36 | 35 | ralrimiva 3147 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘)) |
37 | 36 | biantrud 528 | . 2 ⊢ (𝜑 → (∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘) ↔ (∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘) ∧ ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘)))) |
38 | 20, 37 | bitr4d 274 | 1 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 ∀wral 3089 ∖ cdif 3766 ∪ cun 3767 ⊆ wss 3769 Fn wfn 6096 ‘cfv 6101 ℂcc 10222 0cc0 10224 Basecbs 16184 Unitcui 18955 ℤ/nℤczn 20173 DChrcdchr 25309 |
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 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-addf 10303 ax-mulf 10304 |
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 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-tpos 7590 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-ec 7984 df-qs 7988 df-map 8097 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-sup 8590 df-inf 8591 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-fz 12581 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-starv 16282 df-sca 16283 df-vsca 16284 df-ip 16285 df-tset 16286 df-ple 16287 df-ds 16289 df-unif 16290 df-0g 16417 df-imas 16483 df-qus 16484 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-mhm 17650 df-grp 17741 df-minusg 17742 df-sbg 17743 df-subg 17904 df-nsg 17905 df-eqg 17906 df-cmn 18510 df-abl 18511 df-mgp 18806 df-ur 18818 df-ring 18865 df-cring 18866 df-oppr 18939 df-dvdsr 18957 df-unit 18958 df-invr 18988 df-subrg 19096 df-lmod 19183 df-lss 19251 df-lsp 19293 df-sra 19495 df-rgmod 19496 df-lidl 19497 df-rsp 19498 df-2idl 19555 df-cnfld 20069 df-zring 20141 df-zn 20177 df-dchr 25310 |
This theorem is referenced by: dchrresb 25336 dchrinv 25338 dchrsum2 25345 |
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