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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 6672 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑋‘𝑥) = (𝑋‘𝐴)) | |
2 | 1 | neeq1d 3077 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑋‘𝑥) ≠ 0 ↔ (𝑋‘𝐴) ≠ 0)) |
3 | eleq1 2902 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈)) | |
4 | 2, 3 | imbi12d 347 | . . . 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 25818 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
12 | 5, 11 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
13 | 6, 7, 8, 9, 12, 10 | dchrelbas2 25815 | . . . . . 6 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈)))) |
14 | 5, 13 | mpbid 234 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈))) |
15 | 14 | simprd 498 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈)) |
16 | dchrn0.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
17 | 4, 15, 16 | rspcdva 3627 | . . 3 ⊢ (𝜑 → ((𝑋‘𝐴) ≠ 0 → 𝐴 ∈ 𝑈)) |
18 | 17 | imp 409 | . 2 ⊢ ((𝜑 ∧ (𝑋‘𝐴) ≠ 0) → 𝐴 ∈ 𝑈) |
19 | ax-1ne0 10608 | . . . . 5 ⊢ 1 ≠ 0 | |
20 | 19 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 1 ≠ 0) |
21 | 12 | nnnn0d 11958 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
22 | 7 | zncrng 20693 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing) |
23 | crngring 19310 | . . . . . . . 8 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring) | |
24 | 21, 22, 23 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ Ring) |
25 | eqid 2823 | . . . . . . . 8 ⊢ (invr‘𝑍) = (invr‘𝑍) | |
26 | eqid 2823 | . . . . . . . 8 ⊢ (.r‘𝑍) = (.r‘𝑍) | |
27 | eqid 2823 | . . . . . . . 8 ⊢ (1r‘𝑍) = (1r‘𝑍) | |
28 | 9, 25, 26, 27 | unitrinv 19430 | . . . . . . 7 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ 𝑈) → (𝐴(.r‘𝑍)((invr‘𝑍)‘𝐴)) = (1r‘𝑍)) |
29 | 24, 28 | sylan 582 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝐴(.r‘𝑍)((invr‘𝑍)‘𝐴)) = (1r‘𝑍)) |
30 | 29 | fveq2d 6676 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘(𝐴(.r‘𝑍)((invr‘𝑍)‘𝐴))) = (𝑋‘(1r‘𝑍))) |
31 | 14 | simpld 497 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) |
32 | 31 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) |
33 | 16 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝐴 ∈ 𝐵) |
34 | 9, 25, 8 | ringinvcl 19428 | . . . . . . 7 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ 𝑈) → ((invr‘𝑍)‘𝐴) ∈ 𝐵) |
35 | 24, 34 | sylan 582 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((invr‘𝑍)‘𝐴) ∈ 𝐵) |
36 | eqid 2823 | . . . . . . . 8 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍) | |
37 | 36, 8 | mgpbas 19247 | . . . . . . 7 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍)) |
38 | 36, 26 | mgpplusg 19245 | . . . . . . 7 ⊢ (.r‘𝑍) = (+g‘(mulGrp‘𝑍)) |
39 | eqid 2823 | . . . . . . . 8 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
40 | cnfldmul 20553 | . . . . . . . 8 ⊢ · = (.r‘ℂfld) | |
41 | 39, 40 | mgpplusg 19245 | . . . . . . 7 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
42 | 37, 38, 41 | mhmlin 17965 | . . . . . 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 1367 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘(𝐴(.r‘𝑍)((invr‘𝑍)‘𝐴))) = ((𝑋‘𝐴) · (𝑋‘((invr‘𝑍)‘𝐴)))) |
44 | 36, 27 | ringidval 19255 | . . . . . . 7 ⊢ (1r‘𝑍) = (0g‘(mulGrp‘𝑍)) |
45 | cnfld1 20572 | . . . . . . . 8 ⊢ 1 = (1r‘ℂfld) | |
46 | 39, 45 | ringidval 19255 | . . . . . . 7 ⊢ 1 = (0g‘(mulGrp‘ℂfld)) |
47 | 44, 46 | mhm0 17966 | . . . . . 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 2866 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((𝑋‘𝐴) · (𝑋‘((invr‘𝑍)‘𝐴))) = 1) |
50 | cnfldbas 20551 | . . . . . . . . 9 ⊢ ℂ = (Base‘ℂfld) | |
51 | 39, 50 | mgpbas 19247 | . . . . . . . 8 ⊢ ℂ = (Base‘(mulGrp‘ℂfld)) |
52 | 37, 51 | mhmf 17963 | . . . . . . 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 | ffvelrnd 6854 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘((invr‘𝑍)‘𝐴)) ∈ ℂ) |
55 | 54 | mul02d 10840 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (0 · (𝑋‘((invr‘𝑍)‘𝐴))) = 0) |
56 | 20, 49, 55 | 3netr4d 3095 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((𝑋‘𝐴) · (𝑋‘((invr‘𝑍)‘𝐴))) ≠ (0 · (𝑋‘((invr‘𝑍)‘𝐴)))) |
57 | oveq1 7165 | . . . 4 ⊢ ((𝑋‘𝐴) = 0 → ((𝑋‘𝐴) · (𝑋‘((invr‘𝑍)‘𝐴))) = (0 · (𝑋‘((invr‘𝑍)‘𝐴)))) | |
58 | 57 | necon3i 3050 | . . 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 799 | 1 ⊢ (𝜑 → ((𝑋‘𝐴) ≠ 0 ↔ 𝐴 ∈ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 · cmul 10544 ℕcn 11640 ℕ0cn0 11900 Basecbs 16485 .rcmulr 16568 MndHom cmhm 17956 mulGrpcmgp 19241 1rcur 19253 Ringcrg 19299 CRingccrg 19300 Unitcui 19391 invrcinvr 19423 ℂfldccnfld 20547 ℤ/nℤczn 20652 DChrcdchr 25810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-ec 8293 df-qs 8297 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-imas 16783 df-qus 16784 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-nsg 18279 df-eqg 18280 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-subrg 19535 df-lmod 19638 df-lss 19706 df-lsp 19746 df-sra 19946 df-rgmod 19947 df-lidl 19948 df-rsp 19949 df-2idl 20007 df-cnfld 20548 df-zring 20620 df-zn 20656 df-dchr 25811 |
This theorem is referenced by: dchrinvcl 25831 dchrfi 25833 dchrghm 25834 dchreq 25836 dchrabs 25838 dchrabs2 25840 dchr1re 25841 dchrpt 25845 dchrsum 25847 sum2dchr 25852 rpvmasumlem 26065 dchrisum0flblem1 26086 |
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