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| Mirrors > Home > MPE Home > Th. List > dchrmullid | Structured version Visualization version GIF version | ||
| Description: Left identity for 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)) |
| dchrmullid.t | ⊢ · = (+g‘𝐺) |
| dchrmullid.x | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
| Ref | Expression |
|---|---|
| dchrmullid | ⊢ (𝜑 → ( 1 · 𝑋) = 𝑋) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dchrmhm.g | . . 3 ⊢ 𝐺 = (DChr‘𝑁) | |
| 2 | dchrmhm.z | . . 3 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
| 3 | dchrmhm.b | . . 3 ⊢ 𝐷 = (Base‘𝐺) | |
| 4 | dchrmullid.t | . . 3 ⊢ · = (+g‘𝐺) | |
| 5 | dchrn0.b | . . . 4 ⊢ 𝐵 = (Base‘𝑍) | |
| 6 | dchrn0.u | . . . 4 ⊢ 𝑈 = (Unit‘𝑍) | |
| 7 | dchr1cl.o | . . . 4 ⊢ 1 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0)) | |
| 8 | dchrmullid.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
| 9 | 1, 3 | dchrrcl 27284 | . . . . 5 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
| 10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 11 | 1, 2, 3, 5, 6, 7, 10 | dchr1cl 27295 | . . 3 ⊢ (𝜑 → 1 ∈ 𝐷) |
| 12 | 1, 2, 3, 4, 11, 8 | dchrmul 27292 | . 2 ⊢ (𝜑 → ( 1 · 𝑋) = ( 1 ∘f · 𝑋)) |
| 13 | oveq1 7438 | . . . . . 6 ⊢ (1 = if(𝑘 ∈ 𝑈, 1, 0) → (1 · (𝑋‘𝑘)) = (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) | |
| 14 | 13 | eqeq1d 2739 | . . . . 5 ⊢ (1 = if(𝑘 ∈ 𝑈, 1, 0) → ((1 · (𝑋‘𝑘)) = (𝑋‘𝑘) ↔ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘))) |
| 15 | oveq1 7438 | . . . . . 6 ⊢ (0 = if(𝑘 ∈ 𝑈, 1, 0) → (0 · (𝑋‘𝑘)) = (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) | |
| 16 | 15 | eqeq1d 2739 | . . . . 5 ⊢ (0 = if(𝑘 ∈ 𝑈, 1, 0) → ((0 · (𝑋‘𝑘)) = (𝑋‘𝑘) ↔ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘))) |
| 17 | 1, 2, 3, 5, 8 | dchrf 27286 | . . . . . . . 8 ⊢ (𝜑 → 𝑋:𝐵⟶ℂ) |
| 18 | 17 | ffvelcdmda 7104 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑋‘𝑘) ∈ ℂ) |
| 19 | 18 | adantr 480 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) ∈ ℂ) |
| 20 | 19 | mullidd 11279 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝑈) → (1 · (𝑋‘𝑘)) = (𝑋‘𝑘)) |
| 21 | 0cn 11253 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
| 22 | 21 | mul02i 11450 | . . . . . 6 ⊢ (0 · 0) = 0 |
| 23 | 1, 2, 5, 6, 10, 3 | dchrelbas2 27281 | . . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)))) |
| 24 | 8, 23 | mpbid 232 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈))) |
| 25 | 24 | simprd 495 | . . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)) |
| 26 | 25 | r19.21bi 3251 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ((𝑋‘𝑘) ≠ 0 → 𝑘 ∈ 𝑈)) |
| 27 | 26 | necon1bd 2958 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝑈 → (𝑋‘𝑘) = 0)) |
| 28 | 27 | imp 406 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (𝑋‘𝑘) = 0) |
| 29 | 28 | oveq2d 7447 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (0 · (𝑋‘𝑘)) = (0 · 0)) |
| 30 | 22, 29, 28 | 3eqtr4a 2803 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝑈) → (0 · (𝑋‘𝑘)) = (𝑋‘𝑘)) |
| 31 | 14, 16, 20, 30 | ifbothda 4564 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)) = (𝑋‘𝑘)) |
| 32 | 31 | mpteq2dva 5242 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘))) = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘))) |
| 33 | 5 | fvexi 6920 | . . . . 5 ⊢ 𝐵 ∈ V |
| 34 | 33 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ V) |
| 35 | ax-1cn 11213 | . . . . . 6 ⊢ 1 ∈ ℂ | |
| 36 | 35, 21 | ifcli 4573 | . . . . 5 ⊢ if(𝑘 ∈ 𝑈, 1, 0) ∈ ℂ |
| 37 | 36 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → if(𝑘 ∈ 𝑈, 1, 0) ∈ ℂ) |
| 38 | 7 | a1i 11 | . . . 4 ⊢ (𝜑 → 1 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0))) |
| 39 | 17 | feqmptd 6977 | . . . 4 ⊢ (𝜑 → 𝑋 = (𝑘 ∈ 𝐵 ↦ (𝑋‘𝑘))) |
| 40 | 34, 37, 18, 38, 39 | offval2 7717 | . . 3 ⊢ (𝜑 → ( 1 ∘f · 𝑋) = (𝑘 ∈ 𝐵 ↦ (if(𝑘 ∈ 𝑈, 1, 0) · (𝑋‘𝑘)))) |
| 41 | 32, 40, 39 | 3eqtr4d 2787 | . 2 ⊢ (𝜑 → ( 1 ∘f · 𝑋) = 𝑋) |
| 42 | 12, 41 | eqtrd 2777 | 1 ⊢ (𝜑 → ( 1 · 𝑋) = 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 Vcvv 3480 ifcif 4525 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 ℂcc 11153 0cc0 11155 1c1 11156 · cmul 11160 ℕcn 12266 Basecbs 17247 +gcplusg 17297 MndHom cmhm 18794 mulGrpcmgp 20137 Unitcui 20355 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-of 7697 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: dchrabl 27298 dchr1 27301 |
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