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| Mirrors > Home > MPE Home > Th. List > dchr1cl | Structured version Visualization version GIF version | ||
| Description: Closure of the principal Dirichlet character. (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) |
| dchr1cl.o | ⊢ 1 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0)) |
| dchr1cl.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
| Ref | Expression |
|---|---|
| dchr1cl | ⊢ (𝜑 → 1 ∈ 𝐷) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dchr1cl.o | . 2 ⊢ 1 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0)) | |
| 2 | dchrmhm.g | . . 3 ⊢ 𝐺 = (DChr‘𝑁) | |
| 3 | dchrmhm.z | . . 3 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
| 4 | dchrn0.b | . . 3 ⊢ 𝐵 = (Base‘𝑍) | |
| 5 | dchrn0.u | . . 3 ⊢ 𝑈 = (Unit‘𝑍) | |
| 6 | dchr1cl.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
| 7 | dchrmhm.b | . . 3 ⊢ 𝐷 = (Base‘𝐺) | |
| 8 | eqidd 2738 | . . 3 ⊢ (𝑘 = 𝑥 → 1 = 1) | |
| 9 | eqidd 2738 | . . 3 ⊢ (𝑘 = 𝑦 → 1 = 1) | |
| 10 | eqidd 2738 | . . 3 ⊢ (𝑘 = (𝑥(.r‘𝑍)𝑦) → 1 = 1) | |
| 11 | eqidd 2738 | . . 3 ⊢ (𝑘 = (1r‘𝑍) → 1 = 1) | |
| 12 | 1cnd 11256 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 1 ∈ ℂ) | |
| 13 | 1t1e1 12428 | . . . . 5 ⊢ (1 · 1) = 1 | |
| 14 | 13 | eqcomi 2746 | . . . 4 ⊢ 1 = (1 · 1) |
| 15 | 14 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 1 = (1 · 1)) |
| 16 | eqidd 2738 | . . 3 ⊢ (𝜑 → 1 = 1) | |
| 17 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16 | dchrelbasd 27283 | . 2 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0)) ∈ 𝐷) |
| 18 | 1, 17 | eqeltrid 2845 | 1 ⊢ (𝜑 → 1 ∈ 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ifcif 4525 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 · cmul 11160 ℕcn 12266 Basecbs 17247 .rcmulr 17298 1rcur 20178 Unitcui 20355 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-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: dchrmullid 27296 dchrabl 27298 dchr1 27301 |
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