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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 3954 | . . . . 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 2725 | . . . . . . . . 9 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
6 | dchrresb.u | . . . . . . . . 9 ⊢ 𝑈 = (Unit‘𝑍) | |
7 | dchrresb.x | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
8 | 7 | adantr 479 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → 𝑋 ∈ 𝐷) |
9 | simpr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → 𝑘 ∈ (Base‘𝑍)) | |
10 | 2, 3, 4, 5, 6, 8, 9 | dchrn0 27233 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑋‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈)) |
11 | 10 | biimpd 228 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)) |
12 | 11 | necon1bd 2947 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → (¬ 𝑘 ∈ 𝑈 → (𝑋‘𝑘) = 0)) |
13 | 12 | impr 453 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑍) ∧ ¬ 𝑘 ∈ 𝑈)) → (𝑋‘𝑘) = 0) |
14 | 1, 13 | sylan2b 592 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑋‘𝑘) = 0) |
15 | dchrresb.Y | . . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝐷) | |
16 | 15 | adantr 479 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → 𝑌 ∈ 𝐷) |
17 | 2, 3, 4, 5, 6, 16, 9 | dchrn0 27233 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑌‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈)) |
18 | 17 | biimpd 228 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑌‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)) |
19 | 18 | necon1bd 2947 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → (¬ 𝑘 ∈ 𝑈 → (𝑌‘𝑘) = 0)) |
20 | 19 | impr 453 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑍) ∧ ¬ 𝑘 ∈ 𝑈)) → (𝑌‘𝑘) = 0) |
21 | 1, 20 | sylan2b 592 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑌‘𝑘) = 0) |
22 | 14, 21 | eqtr4d 2768 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑋‘𝑘) = (𝑌‘𝑘)) |
23 | 22 | ralrimiva 3135 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘)) |
24 | 2, 3, 4, 5, 7 | dchrf 27225 | . . . . 5 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ) |
25 | 24 | ffnd 6724 | . . . 4 ⊢ (𝜑 → 𝑋 Fn (Base‘𝑍)) |
26 | 2, 3, 4, 5, 15 | dchrf 27225 | . . . . 5 ⊢ (𝜑 → 𝑌:(Base‘𝑍)⟶ℂ) |
27 | 26 | ffnd 6724 | . . . 4 ⊢ (𝜑 → 𝑌 Fn (Base‘𝑍)) |
28 | eqfnfv 7039 | . . . 4 ⊢ ((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘))) | |
29 | 25, 27, 28 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘))) |
30 | 5, 6 | unitss 20332 | . . . . . 6 ⊢ 𝑈 ⊆ (Base‘𝑍) |
31 | undif 4483 | . . . . . 6 ⊢ (𝑈 ⊆ (Base‘𝑍) ↔ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈)) = (Base‘𝑍)) | |
32 | 30, 31 | mpbi 229 | . . . . 5 ⊢ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈)) = (Base‘𝑍) |
33 | 32 | raleqi 3312 | . . . 4 ⊢ (∀𝑘 ∈ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈))(𝑋‘𝑘) = (𝑌‘𝑘) ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘)) |
34 | ralunb 4189 | . . . 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 276 | . . 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 286 | . 2 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ (∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘) ∧ ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘)))) |
37 | 23, 36 | mpbiran2d 706 | 1 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∀wral 3050 ∖ cdif 3941 ∪ cun 3942 ⊆ wss 3944 Fn wfn 6544 ‘cfv 6549 ℂcc 11143 0cc0 11145 Basecbs 17188 Unitcui 20311 ℤ/nℤczn 21450 DChrcdchr 27215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-addf 11224 ax-mulf 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-ec 8727 df-qs 8731 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9472 df-inf 9473 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17189 df-ress 17218 df-plusg 17254 df-mulr 17255 df-starv 17256 df-sca 17257 df-vsca 17258 df-ip 17259 df-tset 17260 df-ple 17261 df-ds 17263 df-unif 17264 df-0g 17431 df-imas 17498 df-qus 17499 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18748 df-grp 18906 df-minusg 18907 df-sbg 18908 df-subg 19091 df-nsg 19092 df-eqg 19093 df-cmn 19754 df-abl 19755 df-mgp 20092 df-rng 20110 df-ur 20139 df-ring 20192 df-cring 20193 df-oppr 20290 df-dvdsr 20313 df-unit 20314 df-invr 20344 df-subrng 20500 df-subrg 20525 df-lmod 20762 df-lss 20833 df-lsp 20873 df-sra 21075 df-rgmod 21076 df-lidl 21121 df-rsp 21122 df-2idl 21162 df-cnfld 21302 df-zring 21395 df-zn 21454 df-dchr 27216 |
This theorem is referenced by: dchrresb 27242 dchrinv 27244 dchrsum2 27251 |
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