![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3980 | . . . . 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 2734 | . . . . . . . . 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 27303 | . . . . . . . 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 2960 | . . . . . 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 27303 | . . . . . . . 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 2960 | . . . . . 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 2777 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)) → (𝑋‘𝑘) = (𝑌‘𝑘)) |
23 | 22 | ralrimiva 3148 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ ((Base‘𝑍) ∖ 𝑈)(𝑋‘𝑘) = (𝑌‘𝑘)) |
24 | 2, 3, 4, 5, 7 | dchrf 27295 | . . . . 5 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ) |
25 | 24 | ffnd 6747 | . . . 4 ⊢ (𝜑 → 𝑋 Fn (Base‘𝑍)) |
26 | 2, 3, 4, 5, 15 | dchrf 27295 | . . . . 5 ⊢ (𝜑 → 𝑌:(Base‘𝑍)⟶ℂ) |
27 | 26 | ffnd 6747 | . . . 4 ⊢ (𝜑 → 𝑌 Fn (Base‘𝑍)) |
28 | eqfnfv 7062 | . . . 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 20397 | . . . . . 6 ⊢ 𝑈 ⊆ (Base‘𝑍) |
31 | undif 4501 | . . . . . 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 3327 | . . . 4 ⊢ (∀𝑘 ∈ (𝑈 ∪ ((Base‘𝑍) ∖ 𝑈))(𝑋‘𝑘) = (𝑌‘𝑘) ↔ ∀𝑘 ∈ (Base‘𝑍)(𝑋‘𝑘) = (𝑌‘𝑘)) |
34 | ralunb 4214 | . . . 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 2103 ≠ wne 2942 ∀wral 3063 ∖ cdif 3967 ∪ cun 3968 ⊆ wss 3970 Fn wfn 6567 ‘cfv 6572 ℂcc 11178 0cc0 11180 Basecbs 17253 Unitcui 20376 ℤ/nℤczn 21531 DChrcdchr 27285 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-addf 11259 ax-mulf 11260 |
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 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-tpos 8263 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-ec 8761 df-qs 8765 df-map 8882 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-sup 9507 df-inf 9508 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-fz 13564 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-0g 17496 df-imas 17563 df-qus 17564 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-mhm 18813 df-grp 18971 df-minusg 18972 df-sbg 18973 df-subg 19158 df-nsg 19159 df-eqg 19160 df-cmn 19819 df-abl 19820 df-mgp 20157 df-rng 20175 df-ur 20204 df-ring 20257 df-cring 20258 df-oppr 20355 df-dvdsr 20378 df-unit 20379 df-invr 20409 df-subrng 20567 df-subrg 20592 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-2idl 21278 df-cnfld 21383 df-zring 21476 df-zn 21535 df-dchr 27286 |
This theorem is referenced by: dchrresb 27312 dchrinv 27314 dchrsum2 27321 |
Copyright terms: Public domain | W3C validator |