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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 3945 | . . . . 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 2821 | . . . . . . . . 9 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
6 | dchrresb.u | . . . . . . . . 9 ⊢ 𝑈 = (Unit‘𝑍) | |
7 | dchrresb.x | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
8 | 7 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → 𝑋 ∈ 𝐷) |
9 | simpr 487 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → 𝑘 ∈ (Base‘𝑍)) | |
10 | 2, 3, 4, 5, 6, 8, 9 | dchrn0 25820 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑋‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈)) |
11 | 10 | biimpd 231 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)) |
12 | 11 | necon1bd 3034 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → (¬ 𝑘 ∈ 𝑈 → (𝑋‘𝑘) = 0)) |
13 | 12 | impr 457 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑍) ∧ ¬ 𝑘 ∈ 𝑈)) → (𝑋‘𝑘) = 0) |
14 | 1, 13 | sylan2b 595 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑋‘𝑘) = 0) |
15 | dchrresb.Y | . . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝐷) | |
16 | 15 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → 𝑌 ∈ 𝐷) |
17 | 2, 3, 4, 5, 6, 16, 9 | dchrn0 25820 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑌‘𝑘) ≠ 0 ↔ 𝑘 ∈ 𝑈)) |
18 | 17 | biimpd 231 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → ((𝑌‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)) |
19 | 18 | necon1bd 3034 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (Base‘𝑍)) → (¬ 𝑘 ∈ 𝑈 → (𝑌‘𝑘) = 0)) |
20 | 19 | impr 457 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘𝑍) ∧ ¬ 𝑘 ∈ 𝑈)) → (𝑌‘𝑘) = 0) |
21 | 1, 20 | sylan2b 595 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑌‘𝑘) = 0) |
22 | 14, 21 | eqtr4d 2859 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑋‘𝑘) = (𝑌‘𝑘)) |
23 | 22 | ralrimiva 3182 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘)) |
24 | 2, 3, 4, 5, 7 | dchrf 25812 | . . . . 5 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ) |
25 | 24 | ffnd 6509 | . . . 4 ⊢ (𝜑 → 𝑋 Fn (Base‘𝑍)) |
26 | 2, 3, 4, 5, 15 | dchrf 25812 | . . . . 5 ⊢ (𝜑 → 𝑌:(Base‘𝑍)⟶ℂ) |
27 | 26 | ffnd 6509 | . . . 4 ⊢ (𝜑 → 𝑌 Fn (Base‘𝑍)) |
28 | eqfnfv 6796 | . . . 4 ⊢ ((𝑋 Fn (Base‘𝑍) ∧ 𝑌 Fn (Base‘𝑍)) → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘))) | |
29 | 25, 27, 28 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘))) |
30 | 5, 6 | unitss 19404 | . . . . . 6 ⊢ 𝑈 ⊆ (Base‘𝑍) |
31 | undif 4429 | . . . . . 6 ⊢ (𝑈 ⊆ (Base‘𝑍) ↔ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈)) = (Base‘𝑍)) | |
32 | 30, 31 | mpbi 232 | . . . . 5 ⊢ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈)) = (Base‘𝑍) |
33 | 32 | raleqi 3413 | . . . 4 ⊢ (∀𝑘 ∈ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈))(𝑋‘𝑘) = (𝑌‘𝑘) ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘)) |
34 | ralunb 4166 | . . . 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 279 | . . 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 | syl6bb 289 | . 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 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∖ cdif 3932 ∪ cun 3933 ⊆ wss 3935 Fn wfn 6344 ‘cfv 6349 ℂcc 10529 0cc0 10531 Basecbs 16477 Unitcui 19383 ℤ/nℤczn 20644 DChrcdchr 25802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-ec 8285 df-qs 8289 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-imas 16775 df-qus 16776 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-nsg 18271 df-eqg 18272 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-subrg 19527 df-lmod 19630 df-lss 19698 df-lsp 19738 df-sra 19938 df-rgmod 19939 df-lidl 19940 df-rsp 19941 df-2idl 19999 df-cnfld 20540 df-zring 20612 df-zn 20648 df-dchr 25803 |
This theorem is referenced by: dchrresb 25829 dchrinv 25831 dchrsum2 25838 |
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