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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dirkerval2 | Structured version Visualization version GIF version |
Description: The Nth Dirichlet Kernel evaluated at a specific point 𝑆. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
dirkerval2.1 | ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2))))))) |
Ref | Expression |
---|---|
dirkerval2 | ⊢ ((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) → ((𝐷‘𝑁)‘𝑆) = if((𝑆 mod (2 · π)) = 0, (((2 · 𝑁) + 1) / (2 · π)), ((sin‘((𝑁 + (1 / 2)) · 𝑆)) / ((2 · π) · (sin‘(𝑆 / 2)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dirkerval2.1 | . . . . 5 ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2))))))) | |
2 | 1 | dirkerval 44793 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁) = (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑁) + 1) / (2 · π)), ((sin‘((𝑁 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2))))))) |
3 | oveq1 7412 | . . . . . . 7 ⊢ (𝑠 = 𝑡 → (𝑠 mod (2 · π)) = (𝑡 mod (2 · π))) | |
4 | 3 | eqeq1d 2734 | . . . . . 6 ⊢ (𝑠 = 𝑡 → ((𝑠 mod (2 · π)) = 0 ↔ (𝑡 mod (2 · π)) = 0)) |
5 | oveq2 7413 | . . . . . . . 8 ⊢ (𝑠 = 𝑡 → ((𝑁 + (1 / 2)) · 𝑠) = ((𝑁 + (1 / 2)) · 𝑡)) | |
6 | 5 | fveq2d 6892 | . . . . . . 7 ⊢ (𝑠 = 𝑡 → (sin‘((𝑁 + (1 / 2)) · 𝑠)) = (sin‘((𝑁 + (1 / 2)) · 𝑡))) |
7 | fvoveq1 7428 | . . . . . . . 8 ⊢ (𝑠 = 𝑡 → (sin‘(𝑠 / 2)) = (sin‘(𝑡 / 2))) | |
8 | 7 | oveq2d 7421 | . . . . . . 7 ⊢ (𝑠 = 𝑡 → ((2 · π) · (sin‘(𝑠 / 2))) = ((2 · π) · (sin‘(𝑡 / 2)))) |
9 | 6, 8 | oveq12d 7423 | . . . . . 6 ⊢ (𝑠 = 𝑡 → ((sin‘((𝑁 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))) = ((sin‘((𝑁 + (1 / 2)) · 𝑡)) / ((2 · π) · (sin‘(𝑡 / 2))))) |
10 | 4, 9 | ifbieq2d 4553 | . . . . 5 ⊢ (𝑠 = 𝑡 → if((𝑠 mod (2 · π)) = 0, (((2 · 𝑁) + 1) / (2 · π)), ((sin‘((𝑁 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2))))) = if((𝑡 mod (2 · π)) = 0, (((2 · 𝑁) + 1) / (2 · π)), ((sin‘((𝑁 + (1 / 2)) · 𝑡)) / ((2 · π) · (sin‘(𝑡 / 2)))))) |
11 | 10 | cbvmptv 5260 | . . . 4 ⊢ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑁) + 1) / (2 · π)), ((sin‘((𝑁 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))) = (𝑡 ∈ ℝ ↦ if((𝑡 mod (2 · π)) = 0, (((2 · 𝑁) + 1) / (2 · π)), ((sin‘((𝑁 + (1 / 2)) · 𝑡)) / ((2 · π) · (sin‘(𝑡 / 2)))))) |
12 | 2, 11 | eqtrdi 2788 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝐷‘𝑁) = (𝑡 ∈ ℝ ↦ if((𝑡 mod (2 · π)) = 0, (((2 · 𝑁) + 1) / (2 · π)), ((sin‘((𝑁 + (1 / 2)) · 𝑡)) / ((2 · π) · (sin‘(𝑡 / 2))))))) |
13 | 12 | adantr 481 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) → (𝐷‘𝑁) = (𝑡 ∈ ℝ ↦ if((𝑡 mod (2 · π)) = 0, (((2 · 𝑁) + 1) / (2 · π)), ((sin‘((𝑁 + (1 / 2)) · 𝑡)) / ((2 · π) · (sin‘(𝑡 / 2))))))) |
14 | simpr 485 | . . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) ∧ 𝑡 = 𝑆) → 𝑡 = 𝑆) | |
15 | 14 | oveq1d 7420 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) ∧ 𝑡 = 𝑆) → (𝑡 mod (2 · π)) = (𝑆 mod (2 · π))) |
16 | 15 | eqeq1d 2734 | . . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) ∧ 𝑡 = 𝑆) → ((𝑡 mod (2 · π)) = 0 ↔ (𝑆 mod (2 · π)) = 0)) |
17 | 14 | oveq2d 7421 | . . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) ∧ 𝑡 = 𝑆) → ((𝑁 + (1 / 2)) · 𝑡) = ((𝑁 + (1 / 2)) · 𝑆)) |
18 | 17 | fveq2d 6892 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) ∧ 𝑡 = 𝑆) → (sin‘((𝑁 + (1 / 2)) · 𝑡)) = (sin‘((𝑁 + (1 / 2)) · 𝑆))) |
19 | 14 | fvoveq1d 7427 | . . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) ∧ 𝑡 = 𝑆) → (sin‘(𝑡 / 2)) = (sin‘(𝑆 / 2))) |
20 | 19 | oveq2d 7421 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) ∧ 𝑡 = 𝑆) → ((2 · π) · (sin‘(𝑡 / 2))) = ((2 · π) · (sin‘(𝑆 / 2)))) |
21 | 18, 20 | oveq12d 7423 | . . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) ∧ 𝑡 = 𝑆) → ((sin‘((𝑁 + (1 / 2)) · 𝑡)) / ((2 · π) · (sin‘(𝑡 / 2)))) = ((sin‘((𝑁 + (1 / 2)) · 𝑆)) / ((2 · π) · (sin‘(𝑆 / 2))))) |
22 | 16, 21 | ifbieq2d 4553 | . 2 ⊢ (((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) ∧ 𝑡 = 𝑆) → if((𝑡 mod (2 · π)) = 0, (((2 · 𝑁) + 1) / (2 · π)), ((sin‘((𝑁 + (1 / 2)) · 𝑡)) / ((2 · π) · (sin‘(𝑡 / 2))))) = if((𝑆 mod (2 · π)) = 0, (((2 · 𝑁) + 1) / (2 · π)), ((sin‘((𝑁 + (1 / 2)) · 𝑆)) / ((2 · π) · (sin‘(𝑆 / 2)))))) |
23 | simpr 485 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) → 𝑆 ∈ ℝ) | |
24 | 2re 12282 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
25 | 24 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℝ) |
26 | nnre 12215 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ) | |
27 | 25, 26 | remulcld 11240 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (2 · 𝑁) ∈ ℝ) |
28 | 1red 11211 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℝ) | |
29 | 27, 28 | readdcld 11239 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((2 · 𝑁) + 1) ∈ ℝ) |
30 | pire 25959 | . . . . . . 7 ⊢ π ∈ ℝ | |
31 | 30 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → π ∈ ℝ) |
32 | 25, 31 | remulcld 11240 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (2 · π) ∈ ℝ) |
33 | 2cnd 12286 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ) | |
34 | 31 | recnd 11238 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → π ∈ ℂ) |
35 | 2pos 12311 | . . . . . . . 8 ⊢ 0 < 2 | |
36 | 35 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < 2) |
37 | 36 | gt0ne0d 11774 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 2 ≠ 0) |
38 | pipos 25961 | . . . . . . . 8 ⊢ 0 < π | |
39 | 38 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 0 < π) |
40 | 39 | gt0ne0d 11774 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → π ≠ 0) |
41 | 33, 34, 37, 40 | mulne0d 11862 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (2 · π) ≠ 0) |
42 | 29, 32, 41 | redivcld 12038 | . . . 4 ⊢ (𝑁 ∈ ℕ → (((2 · 𝑁) + 1) / (2 · π)) ∈ ℝ) |
43 | 42 | ad2antrr 724 | . . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) ∧ (𝑆 mod (2 · π)) = 0) → (((2 · 𝑁) + 1) / (2 · π)) ∈ ℝ) |
44 | dirker2re 44794 | . . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) ∧ ¬ (𝑆 mod (2 · π)) = 0) → ((sin‘((𝑁 + (1 / 2)) · 𝑆)) / ((2 · π) · (sin‘(𝑆 / 2)))) ∈ ℝ) | |
45 | 43, 44 | ifclda 4562 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) → if((𝑆 mod (2 · π)) = 0, (((2 · 𝑁) + 1) / (2 · π)), ((sin‘((𝑁 + (1 / 2)) · 𝑆)) / ((2 · π) · (sin‘(𝑆 / 2))))) ∈ ℝ) |
46 | 13, 22, 23, 45 | fvmptd 7002 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℝ) → ((𝐷‘𝑁)‘𝑆) = if((𝑆 mod (2 · π)) = 0, (((2 · 𝑁) + 1) / (2 · π)), ((sin‘((𝑁 + (1 / 2)) · 𝑆)) / ((2 · π) · (sin‘(𝑆 / 2)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ifcif 4527 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6540 (class class class)co 7405 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11244 / cdiv 11867 ℕcn 12208 2c2 12263 mod cmo 13830 sincsin 16003 πcpi 16006 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375 |
This theorem is referenced by: dirkerre 44797 dirkerper 44798 dirkerf 44799 dirkercncflem2 44806 fourierdlem66 44874 |
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