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Mirrors > Home > MPE Home > Th. List > dchrzrh1 | Structured version Visualization version GIF version |
Description: Value of a Dirichlet character at one. (Contributed by Mario Carneiro, 4-May-2016.) |
Ref | Expression |
---|---|
dchrmhm.g | ⊢ 𝐺 = (DChr‘𝑁) |
dchrmhm.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
dchrmhm.b | ⊢ 𝐷 = (Base‘𝐺) |
dchrelbas4.l | ⊢ 𝐿 = (ℤRHom‘𝑍) |
dchrzrh1.x | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
Ref | Expression |
---|---|
dchrzrh1 | ⊢ (𝜑 → (𝑋‘(𝐿‘1)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchrzrh1.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
2 | dchrmhm.g | . . . . . . 7 ⊢ 𝐺 = (DChr‘𝑁) | |
3 | dchrmhm.b | . . . . . . 7 ⊢ 𝐷 = (Base‘𝐺) | |
4 | 2, 3 | dchrrcl 27160 | . . . . . 6 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
6 | 5 | nnnn0d 12554 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
7 | dchrmhm.z | . . . . 5 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
8 | 7 | zncrng 21465 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing) |
9 | crngring 20176 | . . . 4 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring) | |
10 | dchrelbas4.l | . . . . 5 ⊢ 𝐿 = (ℤRHom‘𝑍) | |
11 | eqid 2727 | . . . . 5 ⊢ (1r‘𝑍) = (1r‘𝑍) | |
12 | 10, 11 | zrh1 21425 | . . . 4 ⊢ (𝑍 ∈ Ring → (𝐿‘1) = (1r‘𝑍)) |
13 | 6, 8, 9, 12 | 4syl 19 | . . 3 ⊢ (𝜑 → (𝐿‘1) = (1r‘𝑍)) |
14 | 13 | fveq2d 6895 | . 2 ⊢ (𝜑 → (𝑋‘(𝐿‘1)) = (𝑋‘(1r‘𝑍))) |
15 | 2, 7, 3 | dchrmhm 27161 | . . . 4 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) |
16 | 15, 1 | sselid 3976 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) |
17 | eqid 2727 | . . . . 5 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍) | |
18 | 17, 11 | ringidval 20114 | . . . 4 ⊢ (1r‘𝑍) = (0g‘(mulGrp‘𝑍)) |
19 | eqid 2727 | . . . . 5 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
20 | cnfld1 21308 | . . . . 5 ⊢ 1 = (1r‘ℂfld) | |
21 | 19, 20 | ringidval 20114 | . . . 4 ⊢ 1 = (0g‘(mulGrp‘ℂfld)) |
22 | 18, 21 | mhm0 18742 | . . 3 ⊢ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) → (𝑋‘(1r‘𝑍)) = 1) |
23 | 16, 22 | syl 17 | . 2 ⊢ (𝜑 → (𝑋‘(1r‘𝑍)) = 1) |
24 | 14, 23 | eqtrd 2767 | 1 ⊢ (𝜑 → (𝑋‘(𝐿‘1)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6542 (class class class)co 7414 1c1 11131 ℕcn 12234 ℕ0cn0 12494 Basecbs 17171 MndHom cmhm 18729 mulGrpcmgp 20065 1rcur 20112 Ringcrg 20164 CRingccrg 20165 ℂfldccnfld 21266 ℤRHomczrh 21412 ℤ/nℤczn 21415 DChrcdchr 27152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-addf 11209 ax-mulf 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-ec 8720 df-qs 8724 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9457 df-inf 9458 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-seq 13991 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-0g 17414 df-imas 17481 df-qus 17482 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-grp 18884 df-minusg 18885 df-sbg 18886 df-mulg 19015 df-subg 19069 df-nsg 19070 df-eqg 19071 df-ghm 19159 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-cring 20167 df-oppr 20262 df-rhm 20400 df-subrng 20472 df-subrg 20497 df-lmod 20734 df-lss 20805 df-lsp 20845 df-sra 21047 df-rgmod 21048 df-lidl 21093 df-rsp 21094 df-2idl 21133 df-cnfld 21267 df-zring 21360 df-zrh 21416 df-zn 21419 df-dchr 27153 |
This theorem is referenced by: dchrmusum2 27414 dchrvmasum2lem 27416 |
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