| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > dchrn0 | Structured version Visualization version GIF version | ||
| Description: A Dirichlet character is nonzero on the units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.) |
| Ref | Expression |
|---|---|
| dchrmhm.g | ⊢ 𝐺 = (DChr‘𝑁) |
| dchrmhm.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
| dchrmhm.b | ⊢ 𝐷 = (Base‘𝐺) |
| dchrn0.b | ⊢ 𝐵 = (Base‘𝑍) |
| dchrn0.u | ⊢ 𝑈 = (Unit‘𝑍) |
| dchrn0.x | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
| dchrn0.a | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
| Ref | Expression |
|---|---|
| dchrn0 | ⊢ (𝜑 → ((𝑋‘𝐴) ≠ 0 ↔ 𝐴 ∈ 𝑈)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fveq2 6906 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑋‘𝑥) = (𝑋‘𝐴)) | |
| 2 | 1 | neeq1d 3000 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑋‘𝑥) ≠ 0 ↔ (𝑋‘𝐴) ≠ 0)) |
| 3 | eleq1 2829 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈)) | |
| 4 | 2, 3 | imbi12d 344 | . . . 4 ⊢ (𝑥 = 𝐴 → (((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈) ↔ ((𝑋‘𝐴) ≠ 0 → 𝐴 ∈ 𝑈))) |
| 5 | dchrn0.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
| 6 | dchrmhm.g | . . . . . . 7 ⊢ 𝐺 = (DChr‘𝑁) | |
| 7 | dchrmhm.z | . . . . . . 7 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
| 8 | dchrn0.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑍) | |
| 9 | dchrn0.u | . . . . . . 7 ⊢ 𝑈 = (Unit‘𝑍) | |
| 10 | dchrmhm.b | . . . . . . . . 9 ⊢ 𝐷 = (Base‘𝐺) | |
| 11 | 6, 10 | dchrrcl 27284 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
| 12 | 5, 11 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 13 | 6, 7, 8, 9, 12, 10 | dchrelbas2 27281 | . . . . . 6 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈)))) |
| 14 | 5, 13 | mpbid 232 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈))) |
| 15 | 14 | simprd 495 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈)) |
| 16 | dchrn0.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
| 17 | 4, 15, 16 | rspcdva 3623 | . . 3 ⊢ (𝜑 → ((𝑋‘𝐴) ≠ 0 → 𝐴 ∈ 𝑈)) |
| 18 | 17 | imp 406 | . 2 ⊢ ((𝜑 ∧ (𝑋‘𝐴) ≠ 0) → 𝐴 ∈ 𝑈) |
| 19 | ax-1ne0 11224 | . . . . 5 ⊢ 1 ≠ 0 | |
| 20 | 19 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 1 ≠ 0) |
| 21 | 12 | nnnn0d 12587 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
| 22 | 7 | zncrng 21563 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing) |
| 23 | crngring 20242 | . . . . . . . 8 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring) | |
| 24 | 21, 22, 23 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ Ring) |
| 25 | eqid 2737 | . . . . . . . 8 ⊢ (invr‘𝑍) = (invr‘𝑍) | |
| 26 | eqid 2737 | . . . . . . . 8 ⊢ (.r‘𝑍) = (.r‘𝑍) | |
| 27 | eqid 2737 | . . . . . . . 8 ⊢ (1r‘𝑍) = (1r‘𝑍) | |
| 28 | 9, 25, 26, 27 | unitrinv 20394 | . . . . . . 7 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ 𝑈) → (𝐴(.r‘𝑍)((invr‘𝑍)‘𝐴)) = (1r‘𝑍)) |
| 29 | 24, 28 | sylan 580 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝐴(.r‘𝑍)((invr‘𝑍)‘𝐴)) = (1r‘𝑍)) |
| 30 | 29 | fveq2d 6910 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘(𝐴(.r‘𝑍)((invr‘𝑍)‘𝐴))) = (𝑋‘(1r‘𝑍))) |
| 31 | 14 | simpld 494 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) |
| 32 | 31 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) |
| 33 | 16 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝐴 ∈ 𝐵) |
| 34 | 9, 25, 8 | ringinvcl 20392 | . . . . . . 7 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ 𝑈) → ((invr‘𝑍)‘𝐴) ∈ 𝐵) |
| 35 | 24, 34 | sylan 580 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((invr‘𝑍)‘𝐴) ∈ 𝐵) |
| 36 | eqid 2737 | . . . . . . . 8 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍) | |
| 37 | 36, 8 | mgpbas 20142 | . . . . . . 7 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍)) |
| 38 | 36, 26 | mgpplusg 20141 | . . . . . . 7 ⊢ (.r‘𝑍) = (+g‘(mulGrp‘𝑍)) |
| 39 | eqid 2737 | . . . . . . . 8 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
| 40 | cnfldmul 21372 | . . . . . . . 8 ⊢ · = (.r‘ℂfld) | |
| 41 | 39, 40 | mgpplusg 20141 | . . . . . . 7 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
| 42 | 37, 38, 41 | mhmlin 18806 | . . . . . 6 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝐴 ∈ 𝐵 ∧ ((invr‘𝑍)‘𝐴) ∈ 𝐵) → (𝑋‘(𝐴(.r‘𝑍)((invr‘𝑍)‘𝐴))) = ((𝑋‘𝐴) · (𝑋‘((invr‘𝑍)‘𝐴)))) |
| 43 | 32, 33, 35, 42 | syl3anc 1373 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘(𝐴(.r‘𝑍)((invr‘𝑍)‘𝐴))) = ((𝑋‘𝐴) · (𝑋‘((invr‘𝑍)‘𝐴)))) |
| 44 | 36, 27 | ringidval 20180 | . . . . . . 7 ⊢ (1r‘𝑍) = (0g‘(mulGrp‘𝑍)) |
| 45 | cnfld1 21406 | . . . . . . . 8 ⊢ 1 = (1r‘ℂfld) | |
| 46 | 39, 45 | ringidval 20180 | . . . . . . 7 ⊢ 1 = (0g‘(mulGrp‘ℂfld)) |
| 47 | 44, 46 | mhm0 18807 | . . . . . 6 ⊢ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) → (𝑋‘(1r‘𝑍)) = 1) |
| 48 | 32, 47 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘(1r‘𝑍)) = 1) |
| 49 | 30, 43, 48 | 3eqtr3d 2785 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((𝑋‘𝐴) · (𝑋‘((invr‘𝑍)‘𝐴))) = 1) |
| 50 | cnfldbas 21368 | . . . . . . . . 9 ⊢ ℂ = (Base‘ℂfld) | |
| 51 | 39, 50 | mgpbas 20142 | . . . . . . . 8 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
| 52 | 37, 51 | mhmf 18802 | . . . . . . 7 ⊢ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) → 𝑋:𝐵⟶ℂ) |
| 53 | 32, 52 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝑋:𝐵⟶ℂ) |
| 54 | 53, 35 | ffvelcdmd 7105 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘((invr‘𝑍)‘𝐴)) ∈ ℂ) |
| 55 | 54 | mul02d 11459 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (0 · (𝑋‘((invr‘𝑍)‘𝐴))) = 0) |
| 56 | 20, 49, 55 | 3netr4d 3018 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((𝑋‘𝐴) · (𝑋‘((invr‘𝑍)‘𝐴))) ≠ (0 · (𝑋‘((invr‘𝑍)‘𝐴)))) |
| 57 | oveq1 7438 | . . . 4 ⊢ ((𝑋‘𝐴) = 0 → ((𝑋‘𝐴) · (𝑋‘((invr‘𝑍)‘𝐴))) = (0 · (𝑋‘((invr‘𝑍)‘𝐴)))) | |
| 58 | 57 | necon3i 2973 | . . 3 ⊢ (((𝑋‘𝐴) · (𝑋‘((invr‘𝑍)‘𝐴))) ≠ (0 · (𝑋‘((invr‘𝑍)‘𝐴))) → (𝑋‘𝐴) ≠ 0) |
| 59 | 56, 58 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘𝐴) ≠ 0) |
| 60 | 18, 59 | impbida 801 | 1 ⊢ (𝜑 → ((𝑋‘𝐴) ≠ 0 ↔ 𝐴 ∈ 𝑈)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 · cmul 11160 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 .rcmulr 17298 MndHom cmhm 18794 mulGrpcmgp 20137 1rcur 20178 Ringcrg 20230 CRingccrg 20231 Unitcui 20355 invrcinvr 20387 ℂfldccnfld 21364 ℤ/nℤczn 21513 DChrcdchr 27276 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-ec 8747 df-qs 8751 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-imas 17553 df-qus 17554 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-nsg 19142 df-eqg 19143 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-lss 20930 df-lsp 20970 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-rsp 21219 df-2idl 21260 df-cnfld 21365 df-zring 21458 df-zn 21517 df-dchr 27277 |
| This theorem is referenced by: dchrinvcl 27297 dchrfi 27299 dchrghm 27300 dchreq 27302 dchrabs 27304 dchrabs2 27306 dchr1re 27307 dchrpt 27311 dchrsum 27313 sum2dchr 27318 rpvmasumlem 27531 dchrisum0flblem1 27552 |
| Copyright terms: Public domain | W3C validator |