Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| 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 27203 | . . . . . 6 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
| 7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 8 | 1 | dchrabl 27217 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) |
| 9 | ablgrp 19760 | . . . . 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 18996 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → (𝑋(-g‘𝐺)𝑌) ∈ 𝐷) |
| 14 | 10, 5, 11, 13 | syl3anc 1374 | . . 3 ⊢ (𝜑 → (𝑋(-g‘𝐺)𝑌) ∈ 𝐷) |
| 15 | dchr2sum.b | . . 3 ⊢ 𝐵 = (Base‘𝑍) | |
| 16 | 1, 2, 3, 4, 14, 15 | dchrsum 27232 | . 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 18961 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → (𝑋(-g‘𝐺)𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
| 22 | 17, 18, 21 | syl2anc 585 | . . . . . 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 18963 | . . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐷) → ((invg‘𝐺)‘𝑌) ∈ 𝐷) |
| 26 | 24, 18, 25 | syl2anc 585 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐷) |
| 27 | 1, 2, 3, 19, 17, 26 | dchrmul 27211 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) = (𝑋 ∘f · ((invg‘𝐺)‘𝑌))) |
| 28 | 22, 27 | eqtrd 2771 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑋(-g‘𝐺)𝑌) = (𝑋 ∘f · ((invg‘𝐺)‘𝑌))) |
| 29 | 28 | fveq1d 6842 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋(-g‘𝐺)𝑌)‘𝑎) = ((𝑋 ∘f · ((invg‘𝐺)‘𝑌))‘𝑎)) |
| 30 | 1, 2, 3, 15, 17 | dchrf 27205 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋:𝐵⟶ℂ) |
| 31 | 30 | ffnd 6669 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋 Fn 𝐵) |
| 32 | 1, 2, 3, 15, 26 | dchrf 27205 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((invg‘𝐺)‘𝑌):𝐵⟶ℂ) |
| 33 | 32 | ffnd 6669 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) Fn 𝐵) |
| 34 | 15 | fvexi 6854 | . . . . . 6 ⊢ 𝐵 ∈ V |
| 35 | 34 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 ∈ V) |
| 36 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵) | |
| 37 | fnfvof 7648 | . . . . 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 839 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋 ∘f · ((invg‘𝐺)‘𝑌))‘𝑎) = ((𝑋‘𝑎) · (((invg‘𝐺)‘𝑌)‘𝑎))) |
| 39 | 1, 3, 18, 20 | dchrinv 27224 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) = (∗ ∘ 𝑌)) |
| 40 | 39 | fveq1d 6842 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (((invg‘𝐺)‘𝑌)‘𝑎) = ((∗ ∘ 𝑌)‘𝑎)) |
| 41 | 1, 2, 3, 15, 18 | dchrf 27205 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑌:𝐵⟶ℂ) |
| 42 | fvco3 6939 | . . . . . . 7 ⊢ ((𝑌:𝐵⟶ℂ ∧ 𝑎 ∈ 𝐵) → ((∗ ∘ 𝑌)‘𝑎) = (∗‘(𝑌‘𝑎))) | |
| 43 | 41, 36, 42 | syl2anc 585 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((∗ ∘ 𝑌)‘𝑎) = (∗‘(𝑌‘𝑎))) |
| 44 | 40, 43 | eqtrd 2771 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (((invg‘𝐺)‘𝑌)‘𝑎) = (∗‘(𝑌‘𝑎))) |
| 45 | 44 | oveq2d 7383 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋‘𝑎) · (((invg‘𝐺)‘𝑌)‘𝑎)) = ((𝑋‘𝑎) · (∗‘(𝑌‘𝑎)))) |
| 46 | 29, 38, 45 | 3eqtrd 2775 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋(-g‘𝐺)𝑌)‘𝑎) = ((𝑋‘𝑎) · (∗‘(𝑌‘𝑎)))) |
| 47 | 46 | sumeq2dv 15664 | . 2 ⊢ (𝜑 → Σ𝑎 ∈ 𝐵 ((𝑋(-g‘𝐺)𝑌)‘𝑎) = Σ𝑎 ∈ 𝐵 ((𝑋‘𝑎) · (∗‘(𝑌‘𝑎)))) |
| 48 | 3, 4, 12 | grpsubeq0 19002 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → ((𝑋(-g‘𝐺)𝑌) = (0g‘𝐺) ↔ 𝑋 = 𝑌)) |
| 49 | 10, 5, 11, 48 | syl3anc 1374 | . . 3 ⊢ (𝜑 → ((𝑋(-g‘𝐺)𝑌) = (0g‘𝐺) ↔ 𝑋 = 𝑌)) |
| 50 | 49 | ifbid 4490 | . 2 ⊢ (𝜑 → if((𝑋(-g‘𝐺)𝑌) = (0g‘𝐺), (ϕ‘𝑁), 0) = if(𝑋 = 𝑌, (ϕ‘𝑁), 0)) |
| 51 | 16, 47, 50 | 3eqtr3d 2779 | 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 1542 ∈ wcel 2114 Vcvv 3429 ifcif 4466 ∘ ccom 5635 Fn wfn 6493 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ∘f cof 7629 ℂcc 11036 0cc0 11038 · cmul 11043 ℕcn 12174 ∗ccj 15058 Σcsu 15648 ϕcphi 16734 Basecbs 17179 +gcplusg 17220 0gc0g 17402 Grpcgrp 18909 invgcminusg 18910 -gcsg 18911 Abelcabl 19756 ℤ/nℤczn 21482 DChrcdchr 27195 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-disj 5053 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-omul 8410 df-er 8643 df-ec 8645 df-qs 8649 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-acn 9866 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-sin 16034 df-cos 16035 df-pi 16037 df-dvds 16222 df-gcd 16464 df-phi 16736 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-qus 17473 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-nsg 19100 df-eqg 19101 df-ghm 19188 df-cntz 19292 df-od 19503 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-cring 20217 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-dvr 20381 df-rhm 20452 df-subrng 20523 df-subrg 20547 df-drng 20708 df-lmod 20857 df-lss 20927 df-lsp 20967 df-sra 21168 df-rgmod 21169 df-lidl 21206 df-rsp 21207 df-2idl 21248 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-zring 21427 df-zrh 21483 df-zn 21486 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-log 26520 df-cxp 26521 df-dchr 27196 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |