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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 2725 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
5 | dchr2sum.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
6 | 1, 3 | dchrrcl 27218 | . . . . . 6 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
8 | 1 | dchrabl 27232 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) |
9 | ablgrp 19752 | . . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp) | |
10 | 7, 8, 9 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp) |
11 | dchr2sum.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐷) | |
12 | eqid 2725 | . . . . 5 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
13 | 3, 12 | grpsubcl 18984 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → (𝑋(-g‘𝐺)𝑌) ∈ 𝐷) |
14 | 10, 5, 11, 13 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝑋(-g‘𝐺)𝑌) ∈ 𝐷) |
15 | dchr2sum.b | . . 3 ⊢ 𝐵 = (Base‘𝑍) | |
16 | 1, 2, 3, 4, 14, 15 | dchrsum 27247 | . 2 ⊢ (𝜑 → Σ𝑎 ∈ 𝐵 ((𝑋(-g‘𝐺)𝑌)‘𝑎) = if((𝑋(-g‘𝐺)𝑌) = (0g‘𝐺), (ϕ‘𝑁), 0)) |
17 | 5 | adantr 479 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋 ∈ 𝐷) |
18 | 11 | adantr 479 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑌 ∈ 𝐷) |
19 | eqid 2725 | . . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
20 | eqid 2725 | . . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
21 | 3, 19, 20, 12 | grpsubval 18950 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → (𝑋(-g‘𝐺)𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
22 | 17, 18, 21 | syl2anc 582 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑋(-g‘𝐺)𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
23 | 7 | adantr 479 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑁 ∈ ℕ) |
24 | 23, 8, 9 | 3syl 18 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐺 ∈ Grp) |
25 | 3, 20 | grpinvcl 18952 | . . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐷) → ((invg‘𝐺)‘𝑌) ∈ 𝐷) |
26 | 24, 18, 25 | syl2anc 582 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐷) |
27 | 1, 2, 3, 19, 17, 26 | dchrmul 27226 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) = (𝑋 ∘f · ((invg‘𝐺)‘𝑌))) |
28 | 22, 27 | eqtrd 2765 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑋(-g‘𝐺)𝑌) = (𝑋 ∘f · ((invg‘𝐺)‘𝑌))) |
29 | 28 | fveq1d 6898 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋(-g‘𝐺)𝑌)‘𝑎) = ((𝑋 ∘f · ((invg‘𝐺)‘𝑌))‘𝑎)) |
30 | 1, 2, 3, 15, 17 | dchrf 27220 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋:𝐵⟶ℂ) |
31 | 30 | ffnd 6724 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋 Fn 𝐵) |
32 | 1, 2, 3, 15, 26 | dchrf 27220 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((invg‘𝐺)‘𝑌):𝐵⟶ℂ) |
33 | 32 | ffnd 6724 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) Fn 𝐵) |
34 | 15 | fvexi 6910 | . . . . . 6 ⊢ 𝐵 ∈ V |
35 | 34 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 ∈ V) |
36 | simpr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵) | |
37 | fnfvof 7702 | . . . . 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 837 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋 ∘f · ((invg‘𝐺)‘𝑌))‘𝑎) = ((𝑋‘𝑎) · (((invg‘𝐺)‘𝑌)‘𝑎))) |
39 | 1, 3, 18, 20 | dchrinv 27239 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) = (∗ ∘ 𝑌)) |
40 | 39 | fveq1d 6898 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (((invg‘𝐺)‘𝑌)‘𝑎) = ((∗ ∘ 𝑌)‘𝑎)) |
41 | 1, 2, 3, 15, 18 | dchrf 27220 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑌:𝐵⟶ℂ) |
42 | fvco3 6996 | . . . . . . 7 ⊢ ((𝑌:𝐵⟶ℂ ∧ 𝑎 ∈ 𝐵) → ((∗ ∘ 𝑌)‘𝑎) = (∗‘(𝑌‘𝑎))) | |
43 | 41, 36, 42 | syl2anc 582 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((∗ ∘ 𝑌)‘𝑎) = (∗‘(𝑌‘𝑎))) |
44 | 40, 43 | eqtrd 2765 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (((invg‘𝐺)‘𝑌)‘𝑎) = (∗‘(𝑌‘𝑎))) |
45 | 44 | oveq2d 7435 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋‘𝑎) · (((invg‘𝐺)‘𝑌)‘𝑎)) = ((𝑋‘𝑎) · (∗‘(𝑌‘𝑎)))) |
46 | 29, 38, 45 | 3eqtrd 2769 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋(-g‘𝐺)𝑌)‘𝑎) = ((𝑋‘𝑎) · (∗‘(𝑌‘𝑎)))) |
47 | 46 | sumeq2dv 15685 | . 2 ⊢ (𝜑 → Σ𝑎 ∈ 𝐵 ((𝑋(-g‘𝐺)𝑌)‘𝑎) = Σ𝑎 ∈ 𝐵 ((𝑋‘𝑎) · (∗‘(𝑌‘𝑎)))) |
48 | 3, 4, 12 | grpsubeq0 18990 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → ((𝑋(-g‘𝐺)𝑌) = (0g‘𝐺) ↔ 𝑋 = 𝑌)) |
49 | 10, 5, 11, 48 | syl3anc 1368 | . . 3 ⊢ (𝜑 → ((𝑋(-g‘𝐺)𝑌) = (0g‘𝐺) ↔ 𝑋 = 𝑌)) |
50 | 49 | ifbid 4553 | . 2 ⊢ (𝜑 → if((𝑋(-g‘𝐺)𝑌) = (0g‘𝐺), (ϕ‘𝑁), 0) = if(𝑋 = 𝑌, (ϕ‘𝑁), 0)) |
51 | 16, 47, 50 | 3eqtr3d 2773 | 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 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ifcif 4530 ∘ ccom 5682 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ∘f cof 7683 ℂcc 11138 0cc0 11140 · cmul 11145 ℕcn 12245 ∗ccj 15079 Σcsu 15668 ϕcphi 16736 Basecbs 17183 +gcplusg 17236 0gc0g 17424 Grpcgrp 18898 invgcminusg 18899 -gcsg 18900 Abelcabl 19748 ℤ/nℤczn 21445 DChrcdchr 27210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219 ax-mulf 11220 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-omul 8492 df-er 8725 df-ec 8727 df-qs 8731 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-acn 9967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-xnn0 12578 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-ioo 13363 df-ioc 13364 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-fl 13793 df-mod 13871 df-seq 14003 df-exp 14063 df-fac 14269 df-bc 14298 df-hash 14326 df-shft 15050 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-limsup 15451 df-clim 15468 df-rlim 15469 df-sum 15669 df-ef 16047 df-sin 16049 df-cos 16050 df-pi 16052 df-dvds 16235 df-gcd 16473 df-phi 16738 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-hom 17260 df-cco 17261 df-rest 17407 df-topn 17408 df-0g 17426 df-gsum 17427 df-topgen 17428 df-pt 17429 df-prds 17432 df-xrs 17487 df-qtop 17492 df-imas 17493 df-qus 17494 df-xps 17495 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18743 df-submnd 18744 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19032 df-subg 19086 df-nsg 19087 df-eqg 19088 df-ghm 19176 df-cntz 19280 df-od 19495 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-cring 20188 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-rhm 20423 df-subrng 20495 df-subrg 20520 df-drng 20638 df-lmod 20757 df-lss 20828 df-lsp 20868 df-sra 21070 df-rgmod 21071 df-lidl 21116 df-rsp 21117 df-2idl 21157 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-zring 21390 df-zrh 21446 df-zn 21449 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24270 df-ms 24271 df-tms 24272 df-cncf 24842 df-limc 25839 df-dv 25840 df-log 26535 df-cxp 26536 df-dchr 27211 |
This theorem is referenced by: (None) |
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