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| Mirrors > Home > MPE Home > Th. List > dchr2sum | Structured version Visualization version GIF version | ||
| Description: An orthogonality relation for Dirichlet characters: the sum of 𝑋(𝑎) · ∗𝑌(𝑎) over all 𝑎 is nonzero only when 𝑋 = 𝑌. Part of Theorem 6.5.2 of [Shapiro] p. 232. (Contributed by Mario Carneiro, 28-Apr-2016.) | 
| Ref | Expression | 
|---|---|
| dchr2sum.g | ⊢ 𝐺 = (DChr‘𝑁) | 
| dchr2sum.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | 
| dchr2sum.d | ⊢ 𝐷 = (Base‘𝐺) | 
| dchr2sum.b | ⊢ 𝐵 = (Base‘𝑍) | 
| dchr2sum.x | ⊢ (𝜑 → 𝑋 ∈ 𝐷) | 
| dchr2sum.y | ⊢ (𝜑 → 𝑌 ∈ 𝐷) | 
| Ref | Expression | 
|---|---|
| dchr2sum | ⊢ (𝜑 → Σ𝑎 ∈ 𝐵 ((𝑋‘𝑎) · (∗‘(𝑌‘𝑎))) = if(𝑋 = 𝑌, (ϕ‘𝑁), 0)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dchr2sum.g | . . 3 ⊢ 𝐺 = (DChr‘𝑁) | |
| 2 | dchr2sum.z | . . 3 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
| 3 | dchr2sum.d | . . 3 ⊢ 𝐷 = (Base‘𝐺) | |
| 4 | eqid 2736 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 5 | dchr2sum.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
| 6 | 1, 3 | dchrrcl 27285 | . . . . . 6 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) | 
| 7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) | 
| 8 | 1 | dchrabl 27299 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) | 
| 9 | ablgrp 19804 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
| 10 | 7, 8, 9 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) | 
| 11 | dchr2sum.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐷) | |
| 12 | eqid 2736 | . . . . 5 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
| 13 | 3, 12 | grpsubcl 19039 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → (𝑋(-g‘𝐺)𝑌) ∈ 𝐷) | 
| 14 | 10, 5, 11, 13 | syl3anc 1372 | . . 3 ⊢ (𝜑 → (𝑋(-g‘𝐺)𝑌) ∈ 𝐷) | 
| 15 | dchr2sum.b | . . 3 ⊢ 𝐵 = (Base‘𝑍) | |
| 16 | 1, 2, 3, 4, 14, 15 | dchrsum 27314 | . 2 ⊢ (𝜑 → Σ𝑎 ∈ 𝐵 ((𝑋(-g‘𝐺)𝑌)‘𝑎) = if((𝑋(-g‘𝐺)𝑌) = (0g‘𝐺), (ϕ‘𝑁), 0)) | 
| 17 | 5 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋 ∈ 𝐷) | 
| 18 | 11 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑌 ∈ 𝐷) | 
| 19 | eqid 2736 | . . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 20 | eqid 2736 | . . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
| 21 | 3, 19, 20, 12 | grpsubval 19004 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → (𝑋(-g‘𝐺)𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) | 
| 22 | 17, 18, 21 | syl2anc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑋(-g‘𝐺)𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) | 
| 23 | 7 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑁 ∈ ℕ) | 
| 24 | 23, 8, 9 | 3syl 18 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐺 ∈ Grp) | 
| 25 | 3, 20 | grpinvcl 19006 | . . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐷) → ((invg‘𝐺)‘𝑌) ∈ 𝐷) | 
| 26 | 24, 18, 25 | syl2anc 584 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐷) | 
| 27 | 1, 2, 3, 19, 17, 26 | dchrmul 27293 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) = (𝑋 ∘f · ((invg‘𝐺)‘𝑌))) | 
| 28 | 22, 27 | eqtrd 2776 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑋(-g‘𝐺)𝑌) = (𝑋 ∘f · ((invg‘𝐺)‘𝑌))) | 
| 29 | 28 | fveq1d 6907 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋(-g‘𝐺)𝑌)‘𝑎) = ((𝑋 ∘f · ((invg‘𝐺)‘𝑌))‘𝑎)) | 
| 30 | 1, 2, 3, 15, 17 | dchrf 27287 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋:𝐵⟶ℂ) | 
| 31 | 30 | ffnd 6736 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋 Fn 𝐵) | 
| 32 | 1, 2, 3, 15, 26 | dchrf 27287 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((invg‘𝐺)‘𝑌):𝐵⟶ℂ) | 
| 33 | 32 | ffnd 6736 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) Fn 𝐵) | 
| 34 | 15 | fvexi 6919 | . . . . . 6 ⊢ 𝐵 ∈ V | 
| 35 | 34 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 ∈ V) | 
| 36 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵) | |
| 37 | fnfvof 7715 | . . . . 5 ⊢ (((𝑋 Fn 𝐵 ∧ ((invg‘𝐺)‘𝑌) Fn 𝐵) ∧ (𝐵 ∈ V ∧ 𝑎 ∈ 𝐵)) → ((𝑋 ∘f · ((invg‘𝐺)‘𝑌))‘𝑎) = ((𝑋‘𝑎) · (((invg‘𝐺)‘𝑌)‘𝑎))) | |
| 38 | 31, 33, 35, 36, 37 | syl22anc 838 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋 ∘f · ((invg‘𝐺)‘𝑌))‘𝑎) = ((𝑋‘𝑎) · (((invg‘𝐺)‘𝑌)‘𝑎))) | 
| 39 | 1, 3, 18, 20 | dchrinv 27306 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) = (∗ ∘ 𝑌)) | 
| 40 | 39 | fveq1d 6907 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (((invg‘𝐺)‘𝑌)‘𝑎) = ((∗ ∘ 𝑌)‘𝑎)) | 
| 41 | 1, 2, 3, 15, 18 | dchrf 27287 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑌:𝐵⟶ℂ) | 
| 42 | fvco3 7007 | . . . . . . 7 ⊢ ((𝑌:𝐵⟶ℂ ∧ 𝑎 ∈ 𝐵) → ((∗ ∘ 𝑌)‘𝑎) = (∗‘(𝑌‘𝑎))) | |
| 43 | 41, 36, 42 | syl2anc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((∗ ∘ 𝑌)‘𝑎) = (∗‘(𝑌‘𝑎))) | 
| 44 | 40, 43 | eqtrd 2776 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (((invg‘𝐺)‘𝑌)‘𝑎) = (∗‘(𝑌‘𝑎))) | 
| 45 | 44 | oveq2d 7448 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋‘𝑎) · (((invg‘𝐺)‘𝑌)‘𝑎)) = ((𝑋‘𝑎) · (∗‘(𝑌‘𝑎)))) | 
| 46 | 29, 38, 45 | 3eqtrd 2780 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋(-g‘𝐺)𝑌)‘𝑎) = ((𝑋‘𝑎) · (∗‘(𝑌‘𝑎)))) | 
| 47 | 46 | sumeq2dv 15739 | . 2 ⊢ (𝜑 → Σ𝑎 ∈ 𝐵 ((𝑋(-g‘𝐺)𝑌)‘𝑎) = Σ𝑎 ∈ 𝐵 ((𝑋‘𝑎) · (∗‘(𝑌‘𝑎)))) | 
| 48 | 3, 4, 12 | grpsubeq0 19045 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → ((𝑋(-g‘𝐺)𝑌) = (0g‘𝐺) ↔ 𝑋 = 𝑌)) | 
| 49 | 10, 5, 11, 48 | syl3anc 1372 | . . 3 ⊢ (𝜑 → ((𝑋(-g‘𝐺)𝑌) = (0g‘𝐺) ↔ 𝑋 = 𝑌)) | 
| 50 | 49 | ifbid 4548 | . 2 ⊢ (𝜑 → if((𝑋(-g‘𝐺)𝑌) = (0g‘𝐺), (ϕ‘𝑁), 0) = if(𝑋 = 𝑌, (ϕ‘𝑁), 0)) | 
| 51 | 16, 47, 50 | 3eqtr3d 2784 | 1 ⊢ (𝜑 → Σ𝑎 ∈ 𝐵 ((𝑋‘𝑎) · (∗‘(𝑌‘𝑎))) = if(𝑋 = 𝑌, (ϕ‘𝑁), 0)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ifcif 4524 ∘ ccom 5688 Fn wfn 6555 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ∘f cof 7696 ℂcc 11154 0cc0 11156 · cmul 11161 ℕcn 12267 ∗ccj 15136 Σcsu 15723 ϕcphi 16802 Basecbs 17248 +gcplusg 17298 0gc0g 17485 Grpcgrp 18952 invgcminusg 18953 -gcsg 18954 Abelcabl 19800 ℤ/nℤczn 21514 DChrcdchr 27277 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-disj 5110 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-omul 8512 df-er 8746 df-ec 8748 df-qs 8752 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-acn 9983 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-xnn0 12602 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-shft 15107 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-ef 16104 df-sin 16106 df-cos 16107 df-pi 16109 df-dvds 16292 df-gcd 16533 df-phi 16804 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-qus 17555 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-nsg 19143 df-eqg 19144 df-ghm 19232 df-cntz 19336 df-od 19547 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-cring 20234 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-dvr 20402 df-rhm 20473 df-subrng 20547 df-subrg 20571 df-drng 20732 df-lmod 20861 df-lss 20931 df-lsp 20971 df-sra 21173 df-rgmod 21174 df-lidl 21219 df-rsp 21220 df-2idl 21261 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-zring 21459 df-zrh 21515 df-zn 21518 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-haus 23324 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-limc 25902 df-dv 25903 df-log 26599 df-cxp 26600 df-dchr 27278 | 
| This theorem is referenced by: (None) | 
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