Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dirkerper Structured version   Visualization version   GIF version

Theorem dirkerper 44798
Description: the Dirichlet Kernel has period 2Ο€. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
dirkerper.1 𝐷 = (𝑛 ∈ β„• ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑛) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑦)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑦 / 2)))))))
dirkerper.2 𝑇 = (2 Β· Ο€)
Assertion
Ref Expression
dirkerper ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ ((π·β€˜π‘)β€˜(π‘₯ + 𝑇)) = ((π·β€˜π‘)β€˜π‘₯))
Distinct variable groups:   𝑦,𝑁   𝑦,𝑛
Allowed substitution hints:   𝐷(π‘₯,𝑦,𝑛)   𝑇(π‘₯,𝑦,𝑛)   𝑁(π‘₯,𝑛)

Proof of Theorem dirkerper
StepHypRef Expression
1 dirkerper.2 . . . . . . . . . . . . 13 𝑇 = (2 Β· Ο€)
21eqcomi 2741 . . . . . . . . . . . 12 (2 Β· Ο€) = 𝑇
32oveq2i 7416 . . . . . . . . . . 11 (1 Β· (2 Β· Ο€)) = (1 Β· 𝑇)
4 2re 12282 . . . . . . . . . . . . . . 15 2 ∈ ℝ
5 pire 25959 . . . . . . . . . . . . . . 15 Ο€ ∈ ℝ
64, 5remulcli 11226 . . . . . . . . . . . . . 14 (2 Β· Ο€) ∈ ℝ
71, 6eqeltri 2829 . . . . . . . . . . . . 13 𝑇 ∈ ℝ
87recni 11224 . . . . . . . . . . . 12 𝑇 ∈ β„‚
98mullidi 11215 . . . . . . . . . . 11 (1 Β· 𝑇) = 𝑇
103, 9eqtri 2760 . . . . . . . . . 10 (1 Β· (2 Β· Ο€)) = 𝑇
1110oveq2i 7416 . . . . . . . . 9 (π‘₯ + (1 Β· (2 Β· Ο€))) = (π‘₯ + 𝑇)
1211eqcomi 2741 . . . . . . . 8 (π‘₯ + 𝑇) = (π‘₯ + (1 Β· (2 Β· Ο€)))
1312oveq1i 7415 . . . . . . 7 ((π‘₯ + 𝑇) mod (2 Β· Ο€)) = ((π‘₯ + (1 Β· (2 Β· Ο€))) mod (2 Β· Ο€))
1413a1i 11 . . . . . 6 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) = ((π‘₯ + (1 Β· (2 Β· Ο€))) mod (2 Β· Ο€)))
15 id 22 . . . . . . . 8 (π‘₯ ∈ ℝ β†’ π‘₯ ∈ ℝ)
1615ad2antlr 725 . . . . . . 7 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ π‘₯ ∈ ℝ)
17 2rp 12975 . . . . . . . . 9 2 ∈ ℝ+
18 pirp 25962 . . . . . . . . 9 Ο€ ∈ ℝ+
19 rpmulcl 12993 . . . . . . . . 9 ((2 ∈ ℝ+ ∧ Ο€ ∈ ℝ+) β†’ (2 Β· Ο€) ∈ ℝ+)
2017, 18, 19mp2an 690 . . . . . . . 8 (2 Β· Ο€) ∈ ℝ+
2120a1i 11 . . . . . . 7 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ (2 Β· Ο€) ∈ ℝ+)
22 1z 12588 . . . . . . . 8 1 ∈ β„€
2322a1i 11 . . . . . . 7 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ 1 ∈ β„€)
24 modcyc 13867 . . . . . . 7 ((π‘₯ ∈ ℝ ∧ (2 Β· Ο€) ∈ ℝ+ ∧ 1 ∈ β„€) β†’ ((π‘₯ + (1 Β· (2 Β· Ο€))) mod (2 Β· Ο€)) = (π‘₯ mod (2 Β· Ο€)))
2516, 21, 23, 24syl3anc 1371 . . . . . 6 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((π‘₯ + (1 Β· (2 Β· Ο€))) mod (2 Β· Ο€)) = (π‘₯ mod (2 Β· Ο€)))
26 simpr 485 . . . . . 6 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ (π‘₯ mod (2 Β· Ο€)) = 0)
2714, 25, 263eqtrd 2776 . . . . 5 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) = 0)
2827iftrued 4535 . . . 4 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ if(((π‘₯ + 𝑇) mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + 𝑇) / 2))))) = (((2 Β· 𝑁) + 1) / (2 Β· Ο€)))
29 iftrue 4533 . . . . 5 ((π‘₯ mod (2 Β· Ο€)) = 0 β†’ if((π‘₯ mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))))) = (((2 Β· 𝑁) + 1) / (2 Β· Ο€)))
3029adantl 482 . . . 4 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ if((π‘₯ mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))))) = (((2 Β· 𝑁) + 1) / (2 Β· Ο€)))
3128, 30eqtr4d 2775 . . 3 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ if(((π‘₯ + 𝑇) mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + 𝑇) / 2))))) = if((π‘₯ mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))))))
32 iffalse 4536 . . . . 5 (Β¬ (π‘₯ mod (2 Β· Ο€)) = 0 β†’ if((π‘₯ mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))))) = ((sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2)))))
3332adantl 482 . . . 4 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ if((π‘₯ mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))))) = ((sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2)))))
34 nncn 12216 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„‚)
35 halfcn 12423 . . . . . . . . . . 11 (1 / 2) ∈ β„‚
3635a1i 11 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ (1 / 2) ∈ β„‚)
3734, 36addcld 11229 . . . . . . . . 9 (𝑁 ∈ β„• β†’ (𝑁 + (1 / 2)) ∈ β„‚)
3837adantr 481 . . . . . . . 8 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (𝑁 + (1 / 2)) ∈ β„‚)
39 recn 11196 . . . . . . . . 9 (π‘₯ ∈ ℝ β†’ π‘₯ ∈ β„‚)
4039adantl 482 . . . . . . . 8 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ π‘₯ ∈ β„‚)
4138, 40mulcld 11230 . . . . . . 7 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ ((𝑁 + (1 / 2)) Β· π‘₯) ∈ β„‚)
4241sincld 16069 . . . . . 6 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) ∈ β„‚)
4342adantr 481 . . . . 5 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ (sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) ∈ β„‚)
446recni 11224 . . . . . . . 8 (2 Β· Ο€) ∈ β„‚
4544a1i 11 . . . . . . 7 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (2 Β· Ο€) ∈ β„‚)
4640halfcld 12453 . . . . . . . 8 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (π‘₯ / 2) ∈ β„‚)
4746sincld 16069 . . . . . . 7 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (sinβ€˜(π‘₯ / 2)) ∈ β„‚)
4845, 47mulcld 11230 . . . . . 6 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))) ∈ β„‚)
4948adantr 481 . . . . 5 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))) ∈ β„‚)
50 dirkerdenne0 44795 . . . . . 6 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))) β‰  0)
5150adantll 712 . . . . 5 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))) β‰  0)
5243, 49, 51div2negd 12001 . . . 4 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ (-(sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / -((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2)))) = ((sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2)))))
5313a1i 11 . . . . . . . . . . 11 (π‘₯ ∈ ℝ β†’ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) = ((π‘₯ + (1 Β· (2 Β· Ο€))) mod (2 Β· Ο€)))
5420, 22, 24mp3an23 1453 . . . . . . . . . . 11 (π‘₯ ∈ ℝ β†’ ((π‘₯ + (1 Β· (2 Β· Ο€))) mod (2 Β· Ο€)) = (π‘₯ mod (2 Β· Ο€)))
5553, 54eqtrd 2772 . . . . . . . . . 10 (π‘₯ ∈ ℝ β†’ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) = (π‘₯ mod (2 Β· Ο€)))
5655adantr 481 . . . . . . . . 9 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) = (π‘₯ mod (2 Β· Ο€)))
57 simpr 485 . . . . . . . . . 10 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0)
5857neqned 2947 . . . . . . . . 9 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ (π‘₯ mod (2 Β· Ο€)) β‰  0)
5956, 58eqnetrd 3008 . . . . . . . 8 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) β‰  0)
6059neneqd 2945 . . . . . . 7 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ Β¬ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) = 0)
61 iffalse 4536 . . . . . . . 8 (Β¬ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) = 0 β†’ if(((π‘₯ + 𝑇) mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + 𝑇) / 2))))) = ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + 𝑇) / 2)))))
621oveq2i 7416 . . . . . . . . . . 11 (π‘₯ + 𝑇) = (π‘₯ + (2 Β· Ο€))
6362oveq2i 7416 . . . . . . . . . 10 ((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇)) = ((𝑁 + (1 / 2)) Β· (π‘₯ + (2 Β· Ο€)))
6463fveq2i 6891 . . . . . . . . 9 (sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇))) = (sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + (2 Β· Ο€))))
6562oveq1i 7415 . . . . . . . . . . 11 ((π‘₯ + 𝑇) / 2) = ((π‘₯ + (2 Β· Ο€)) / 2)
6665fveq2i 6891 . . . . . . . . . 10 (sinβ€˜((π‘₯ + 𝑇) / 2)) = (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2))
6766oveq2i 7416 . . . . . . . . 9 ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + 𝑇) / 2))) = ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2)))
6864, 67oveq12i 7417 . . . . . . . 8 ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + 𝑇) / 2)))) = ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + (2 Β· Ο€)))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2))))
6961, 68eqtrdi 2788 . . . . . . 7 (Β¬ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) = 0 β†’ if(((π‘₯ + 𝑇) mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + 𝑇) / 2))))) = ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + (2 Β· Ο€)))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2)))))
7060, 69syl 17 . . . . . 6 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ if(((π‘₯ + 𝑇) mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + 𝑇) / 2))))) = ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + (2 Β· Ο€)))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2)))))
7170adantll 712 . . . . 5 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ if(((π‘₯ + 𝑇) mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + 𝑇) / 2))))) = ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + (2 Β· Ο€)))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2)))))
7244a1i 11 . . . . . . . . . . . . . 14 (𝑁 ∈ β„• β†’ (2 Β· Ο€) ∈ β„‚)
7334, 36, 72adddird 11235 . . . . . . . . . . . . 13 (𝑁 ∈ β„• β†’ ((𝑁 + (1 / 2)) Β· (2 Β· Ο€)) = ((𝑁 Β· (2 Β· Ο€)) + ((1 / 2) Β· (2 Β· Ο€))))
74 ax-1cn 11164 . . . . . . . . . . . . . . . 16 1 ∈ β„‚
75 2cnne0 12418 . . . . . . . . . . . . . . . 16 (2 ∈ β„‚ ∧ 2 β‰  0)
76 2cn 12283 . . . . . . . . . . . . . . . . 17 2 ∈ β„‚
77 picn 25960 . . . . . . . . . . . . . . . . 17 Ο€ ∈ β„‚
7876, 77mulcli 11217 . . . . . . . . . . . . . . . 16 (2 Β· Ο€) ∈ β„‚
79 div32 11888 . . . . . . . . . . . . . . . 16 ((1 ∈ β„‚ ∧ (2 ∈ β„‚ ∧ 2 β‰  0) ∧ (2 Β· Ο€) ∈ β„‚) β†’ ((1 / 2) Β· (2 Β· Ο€)) = (1 Β· ((2 Β· Ο€) / 2)))
8074, 75, 78, 79mp3an 1461 . . . . . . . . . . . . . . 15 ((1 / 2) Β· (2 Β· Ο€)) = (1 Β· ((2 Β· Ο€) / 2))
81 2ne0 12312 . . . . . . . . . . . . . . . . . 18 2 β‰  0
8278, 76, 81divcli 11952 . . . . . . . . . . . . . . . . 17 ((2 Β· Ο€) / 2) ∈ β„‚
8382mullidi 11215 . . . . . . . . . . . . . . . 16 (1 Β· ((2 Β· Ο€) / 2)) = ((2 Β· Ο€) / 2)
8477, 76, 81divcan3i 11956 . . . . . . . . . . . . . . . 16 ((2 Β· Ο€) / 2) = Ο€
8583, 84eqtri 2760 . . . . . . . . . . . . . . 15 (1 Β· ((2 Β· Ο€) / 2)) = Ο€
8680, 85eqtri 2760 . . . . . . . . . . . . . 14 ((1 / 2) Β· (2 Β· Ο€)) = Ο€
8786oveq2i 7416 . . . . . . . . . . . . 13 ((𝑁 Β· (2 Β· Ο€)) + ((1 / 2) Β· (2 Β· Ο€))) = ((𝑁 Β· (2 Β· Ο€)) + Ο€)
8873, 87eqtrdi 2788 . . . . . . . . . . . 12 (𝑁 ∈ β„• β†’ ((𝑁 + (1 / 2)) Β· (2 Β· Ο€)) = ((𝑁 Β· (2 Β· Ο€)) + Ο€))
8988oveq2d 7421 . . . . . . . . . . 11 (𝑁 ∈ β„• β†’ (((𝑁 + (1 / 2)) Β· π‘₯) + ((𝑁 + (1 / 2)) Β· (2 Β· Ο€))) = (((𝑁 + (1 / 2)) Β· π‘₯) + ((𝑁 Β· (2 Β· Ο€)) + Ο€)))
9089adantr 481 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (((𝑁 + (1 / 2)) Β· π‘₯) + ((𝑁 + (1 / 2)) Β· (2 Β· Ο€))) = (((𝑁 + (1 / 2)) Β· π‘₯) + ((𝑁 Β· (2 Β· Ο€)) + Ο€)))
9138, 40, 45adddid 11234 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ ((𝑁 + (1 / 2)) Β· (π‘₯ + (2 Β· Ο€))) = (((𝑁 + (1 / 2)) Β· π‘₯) + ((𝑁 + (1 / 2)) Β· (2 Β· Ο€))))
9234, 72mulcld 11230 . . . . . . . . . . . 12 (𝑁 ∈ β„• β†’ (𝑁 Β· (2 Β· Ο€)) ∈ β„‚)
9392adantr 481 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (𝑁 Β· (2 Β· Ο€)) ∈ β„‚)
9477a1i 11 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ Ο€ ∈ β„‚)
9541, 93, 94addassd 11232 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ ((((𝑁 + (1 / 2)) Β· π‘₯) + (𝑁 Β· (2 Β· Ο€))) + Ο€) = (((𝑁 + (1 / 2)) Β· π‘₯) + ((𝑁 Β· (2 Β· Ο€)) + Ο€)))
9690, 91, 953eqtr4d 2782 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ ((𝑁 + (1 / 2)) Β· (π‘₯ + (2 Β· Ο€))) = ((((𝑁 + (1 / 2)) Β· π‘₯) + (𝑁 Β· (2 Β· Ο€))) + Ο€))
9796fveq2d 6892 . . . . . . . 8 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + (2 Β· Ο€)))) = (sinβ€˜((((𝑁 + (1 / 2)) Β· π‘₯) + (𝑁 Β· (2 Β· Ο€))) + Ο€)))
9841, 93addcld 11229 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (((𝑁 + (1 / 2)) Β· π‘₯) + (𝑁 Β· (2 Β· Ο€))) ∈ β„‚)
99 sinppi 25990 . . . . . . . . 9 ((((𝑁 + (1 / 2)) Β· π‘₯) + (𝑁 Β· (2 Β· Ο€))) ∈ β„‚ β†’ (sinβ€˜((((𝑁 + (1 / 2)) Β· π‘₯) + (𝑁 Β· (2 Β· Ο€))) + Ο€)) = -(sinβ€˜(((𝑁 + (1 / 2)) Β· π‘₯) + (𝑁 Β· (2 Β· Ο€)))))
10098, 99syl 17 . . . . . . . 8 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (sinβ€˜((((𝑁 + (1 / 2)) Β· π‘₯) + (𝑁 Β· (2 Β· Ο€))) + Ο€)) = -(sinβ€˜(((𝑁 + (1 / 2)) Β· π‘₯) + (𝑁 Β· (2 Β· Ο€)))))
101 simpl 483 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ 𝑁 ∈ β„•)
102101nnzd 12581 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ 𝑁 ∈ β„€)
103 sinper 25982 . . . . . . . . . 10 ((((𝑁 + (1 / 2)) Β· π‘₯) ∈ β„‚ ∧ 𝑁 ∈ β„€) β†’ (sinβ€˜(((𝑁 + (1 / 2)) Β· π‘₯) + (𝑁 Β· (2 Β· Ο€)))) = (sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)))
10441, 102, 103syl2anc 584 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (sinβ€˜(((𝑁 + (1 / 2)) Β· π‘₯) + (𝑁 Β· (2 Β· Ο€)))) = (sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)))
105104negeqd 11450 . . . . . . . 8 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ -(sinβ€˜(((𝑁 + (1 / 2)) Β· π‘₯) + (𝑁 Β· (2 Β· Ο€)))) = -(sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)))
10697, 100, 1053eqtrd 2776 . . . . . . 7 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + (2 Β· Ο€)))) = -(sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)))
10744a1i 11 . . . . . . . . . . . . 13 (π‘₯ ∈ ℝ β†’ (2 Β· Ο€) ∈ β„‚)
10876a1i 11 . . . . . . . . . . . . 13 (π‘₯ ∈ ℝ β†’ 2 ∈ β„‚)
10981a1i 11 . . . . . . . . . . . . 13 (π‘₯ ∈ ℝ β†’ 2 β‰  0)
11039, 107, 108, 109divdird 12024 . . . . . . . . . . . 12 (π‘₯ ∈ ℝ β†’ ((π‘₯ + (2 Β· Ο€)) / 2) = ((π‘₯ / 2) + ((2 Β· Ο€) / 2)))
11184a1i 11 . . . . . . . . . . . . 13 (π‘₯ ∈ ℝ β†’ ((2 Β· Ο€) / 2) = Ο€)
112111oveq2d 7421 . . . . . . . . . . . 12 (π‘₯ ∈ ℝ β†’ ((π‘₯ / 2) + ((2 Β· Ο€) / 2)) = ((π‘₯ / 2) + Ο€))
113110, 112eqtrd 2772 . . . . . . . . . . 11 (π‘₯ ∈ ℝ β†’ ((π‘₯ + (2 Β· Ο€)) / 2) = ((π‘₯ / 2) + Ο€))
114113fveq2d 6892 . . . . . . . . . 10 (π‘₯ ∈ ℝ β†’ (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2)) = (sinβ€˜((π‘₯ / 2) + Ο€)))
11539halfcld 12453 . . . . . . . . . . 11 (π‘₯ ∈ ℝ β†’ (π‘₯ / 2) ∈ β„‚)
116 sinppi 25990 . . . . . . . . . . 11 ((π‘₯ / 2) ∈ β„‚ β†’ (sinβ€˜((π‘₯ / 2) + Ο€)) = -(sinβ€˜(π‘₯ / 2)))
117115, 116syl 17 . . . . . . . . . 10 (π‘₯ ∈ ℝ β†’ (sinβ€˜((π‘₯ / 2) + Ο€)) = -(sinβ€˜(π‘₯ / 2)))
118114, 117eqtrd 2772 . . . . . . . . 9 (π‘₯ ∈ ℝ β†’ (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2)) = -(sinβ€˜(π‘₯ / 2)))
119118oveq2d 7421 . . . . . . . 8 (π‘₯ ∈ ℝ β†’ ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2))) = ((2 Β· Ο€) Β· -(sinβ€˜(π‘₯ / 2))))
120119adantl 482 . . . . . . 7 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2))) = ((2 Β· Ο€) Β· -(sinβ€˜(π‘₯ / 2))))
121106, 120oveq12d 7423 . . . . . 6 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + (2 Β· Ο€)))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2)))) = (-(sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / ((2 Β· Ο€) Β· -(sinβ€˜(π‘₯ / 2)))))
122121adantr 481 . . . . 5 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + (2 Β· Ο€)))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2)))) = (-(sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / ((2 Β· Ο€) Β· -(sinβ€˜(π‘₯ / 2)))))
123115sincld 16069 . . . . . . . 8 (π‘₯ ∈ ℝ β†’ (sinβ€˜(π‘₯ / 2)) ∈ β„‚)
124107, 123mulneg2d 11664 . . . . . . 7 (π‘₯ ∈ ℝ β†’ ((2 Β· Ο€) Β· -(sinβ€˜(π‘₯ / 2))) = -((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))))
125124oveq2d 7421 . . . . . 6 (π‘₯ ∈ ℝ β†’ (-(sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / ((2 Β· Ο€) Β· -(sinβ€˜(π‘₯ / 2)))) = (-(sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / -((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2)))))
126125ad2antlr 725 . . . . 5 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ (-(sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / ((2 Β· Ο€) Β· -(sinβ€˜(π‘₯ / 2)))) = (-(sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / -((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2)))))
12771, 122, 1263eqtrrd 2777 . . . 4 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ (-(sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / -((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2)))) = if(((π‘₯ + 𝑇) mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + 𝑇) / 2))))))
12833, 52, 1273eqtr2rd 2779 . . 3 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ if(((π‘₯ + 𝑇) mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + 𝑇) / 2))))) = if((π‘₯ mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))))))
12931, 128pm2.61dan 811 . 2 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ if(((π‘₯ + 𝑇) mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + 𝑇) / 2))))) = if((π‘₯ mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))))))
1307a1i 11 . . . 4 (π‘₯ ∈ ℝ β†’ 𝑇 ∈ ℝ)
13115, 130readdcld 11239 . . 3 (π‘₯ ∈ ℝ β†’ (π‘₯ + 𝑇) ∈ ℝ)
132 dirkerper.1 . . . 4 𝐷 = (𝑛 ∈ β„• ↦ (𝑦 ∈ ℝ ↦ if((𝑦 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑛) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑦)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑦 / 2)))))))
133132dirkerval2 44796 . . 3 ((𝑁 ∈ β„• ∧ (π‘₯ + 𝑇) ∈ ℝ) β†’ ((π·β€˜π‘)β€˜(π‘₯ + 𝑇)) = if(((π‘₯ + 𝑇) mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + 𝑇) / 2))))))
134131, 133sylan2 593 . 2 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ ((π·β€˜π‘)β€˜(π‘₯ + 𝑇)) = if(((π‘₯ + 𝑇) mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇))) / ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + 𝑇) / 2))))))
135132dirkerval2 44796 . 2 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ ((π·β€˜π‘)β€˜π‘₯) = if((π‘₯ mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) / ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))))))
136129, 134, 1353eqtr4d 2782 1 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ ((π·β€˜π‘)β€˜(π‘₯ + 𝑇)) = ((π·β€˜π‘)β€˜π‘₯))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  ifcif 4527   ↦ cmpt 5230  β€˜cfv 6540  (class class class)co 7405  β„‚cc 11104  β„cr 11105  0cc0 11106  1c1 11107   + caddc 11109   Β· cmul 11111  -cneg 11441   / cdiv 11867  β„•cn 12208  2c2 12263  β„€cz 12554  β„+crp 12970   mod cmo 13830  sincsin 16003  Ο€cpi 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:  fourierdlem111  44919
  Copyright terms: Public domain W3C validator