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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dirkerper Structured version   Visualization version   GIF version

Theorem dirkerper 44423
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 2742 . . . . . . . . . . . 12 (2 Β· Ο€) = 𝑇
32oveq2i 7369 . . . . . . . . . . 11 (1 Β· (2 Β· Ο€)) = (1 Β· 𝑇)
4 2re 12232 . . . . . . . . . . . . . . 15 2 ∈ ℝ
5 pire 25831 . . . . . . . . . . . . . . 15 Ο€ ∈ ℝ
64, 5remulcli 11176 . . . . . . . . . . . . . 14 (2 Β· Ο€) ∈ ℝ
71, 6eqeltri 2830 . . . . . . . . . . . . 13 𝑇 ∈ ℝ
87recni 11174 . . . . . . . . . . . 12 𝑇 ∈ β„‚
98mulid2i 11165 . . . . . . . . . . 11 (1 Β· 𝑇) = 𝑇
103, 9eqtri 2761 . . . . . . . . . 10 (1 Β· (2 Β· Ο€)) = 𝑇
1110oveq2i 7369 . . . . . . . . 9 (π‘₯ + (1 Β· (2 Β· Ο€))) = (π‘₯ + 𝑇)
1211eqcomi 2742 . . . . . . . 8 (π‘₯ + 𝑇) = (π‘₯ + (1 Β· (2 Β· Ο€)))
1312oveq1i 7368 . . . . . . 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 726 . . . . . . 7 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ π‘₯ ∈ ℝ)
17 2rp 12925 . . . . . . . . 9 2 ∈ ℝ+
18 pirp 25834 . . . . . . . . 9 Ο€ ∈ ℝ+
19 rpmulcl 12943 . . . . . . . . 9 ((2 ∈ ℝ+ ∧ Ο€ ∈ ℝ+) β†’ (2 Β· Ο€) ∈ ℝ+)
2017, 18, 19mp2an 691 . . . . . . . 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 12538 . . . . . . . 8 1 ∈ β„€
2322a1i 11 . . . . . . 7 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ 1 ∈ β„€)
24 modcyc 13817 . . . . . . 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 1372 . . . . . 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 486 . . . . . 6 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ (π‘₯ mod (2 Β· Ο€)) = 0)
2714, 25, 263eqtrd 2777 . . . . 5 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) = 0)
2827iftrued 4495 . . . 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 4493 . . . . 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 483 . . . 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 2776 . . 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 4496 . . . . 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 483 . . . 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 12166 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„‚)
35 halfcn 12373 . . . . . . . . . . 11 (1 / 2) ∈ β„‚
3635a1i 11 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ (1 / 2) ∈ β„‚)
3734, 36addcld 11179 . . . . . . . . 9 (𝑁 ∈ β„• β†’ (𝑁 + (1 / 2)) ∈ β„‚)
3837adantr 482 . . . . . . . 8 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (𝑁 + (1 / 2)) ∈ β„‚)
39 recn 11146 . . . . . . . . 9 (π‘₯ ∈ ℝ β†’ π‘₯ ∈ β„‚)
4039adantl 483 . . . . . . . 8 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ π‘₯ ∈ β„‚)
4138, 40mulcld 11180 . . . . . . 7 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ ((𝑁 + (1 / 2)) Β· π‘₯) ∈ β„‚)
4241sincld 16017 . . . . . 6 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) ∈ β„‚)
4342adantr 482 . . . . 5 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ (sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) ∈ β„‚)
446recni 11174 . . . . . . . 8 (2 Β· Ο€) ∈ β„‚
4544a1i 11 . . . . . . 7 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (2 Β· Ο€) ∈ β„‚)
4640halfcld 12403 . . . . . . . 8 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (π‘₯ / 2) ∈ β„‚)
4746sincld 16017 . . . . . . 7 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (sinβ€˜(π‘₯ / 2)) ∈ β„‚)
4845, 47mulcld 11180 . . . . . 6 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))) ∈ β„‚)
4948adantr 482 . . . . 5 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))) ∈ β„‚)
50 dirkerdenne0 44420 . . . . . 6 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))) β‰  0)
5150adantll 713 . . . . 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 11951 . . . 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 1454 . . . . . . . . . . 11 (π‘₯ ∈ ℝ β†’ ((π‘₯ + (1 Β· (2 Β· Ο€))) mod (2 Β· Ο€)) = (π‘₯ mod (2 Β· Ο€)))
5553, 54eqtrd 2773 . . . . . . . . . 10 (π‘₯ ∈ ℝ β†’ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) = (π‘₯ mod (2 Β· Ο€)))
5655adantr 482 . . . . . . . . 9 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) = (π‘₯ mod (2 Β· Ο€)))
57 simpr 486 . . . . . . . . . 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 4496 . . . . . . . 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 7369 . . . . . . . . . . 11 (π‘₯ + 𝑇) = (π‘₯ + (2 Β· Ο€))
6362oveq2i 7369 . . . . . . . . . 10 ((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇)) = ((𝑁 + (1 / 2)) Β· (π‘₯ + (2 Β· Ο€)))
6463fveq2i 6846 . . . . . . . . 9 (sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇))) = (sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + (2 Β· Ο€))))
6562oveq1i 7368 . . . . . . . . . . 11 ((π‘₯ + 𝑇) / 2) = ((π‘₯ + (2 Β· Ο€)) / 2)
6665fveq2i 6846 . . . . . . . . . 10 (sinβ€˜((π‘₯ + 𝑇) / 2)) = (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2))
6766oveq2i 7369 . . . . . . . . 9 ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + 𝑇) / 2))) = ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2)))
6864, 67oveq12i 7370 . . . . . . . 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 2789 . . . . . . 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 713 . . . . 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 11185 . . . . . . . . . . . . 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 11114 . . . . . . . . . . . . . . . 16 1 ∈ β„‚
75 2cnne0 12368 . . . . . . . . . . . . . . . 16 (2 ∈ β„‚ ∧ 2 β‰  0)
76 2cn 12233 . . . . . . . . . . . . . . . . 17 2 ∈ β„‚
77 picn 25832 . . . . . . . . . . . . . . . . 17 Ο€ ∈ β„‚
7876, 77mulcli 11167 . . . . . . . . . . . . . . . 16 (2 Β· Ο€) ∈ β„‚
79 div32 11838 . . . . . . . . . . . . . . . 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 1462 . . . . . . . . . . . . . . 15 ((1 / 2) Β· (2 Β· Ο€)) = (1 Β· ((2 Β· Ο€) / 2))
81 2ne0 12262 . . . . . . . . . . . . . . . . . 18 2 β‰  0
8278, 76, 81divcli 11902 . . . . . . . . . . . . . . . . 17 ((2 Β· Ο€) / 2) ∈ β„‚
8382mulid2i 11165 . . . . . . . . . . . . . . . 16 (1 Β· ((2 Β· Ο€) / 2)) = ((2 Β· Ο€) / 2)
8477, 76, 81divcan3i 11906 . . . . . . . . . . . . . . . 16 ((2 Β· Ο€) / 2) = Ο€
8583, 84eqtri 2761 . . . . . . . . . . . . . . 15 (1 Β· ((2 Β· Ο€) / 2)) = Ο€
8680, 85eqtri 2761 . . . . . . . . . . . . . 14 ((1 / 2) Β· (2 Β· Ο€)) = Ο€
8786oveq2i 7369 . . . . . . . . . . . . 13 ((𝑁 Β· (2 Β· Ο€)) + ((1 / 2) Β· (2 Β· Ο€))) = ((𝑁 Β· (2 Β· Ο€)) + Ο€)
8873, 87eqtrdi 2789 . . . . . . . . . . . 12 (𝑁 ∈ β„• β†’ ((𝑁 + (1 / 2)) Β· (2 Β· Ο€)) = ((𝑁 Β· (2 Β· Ο€)) + Ο€))
8988oveq2d 7374 . . . . . . . . . . 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 482 . . . . . . . . . 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 11184 . . . . . . . . . 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 11180 . . . . . . . . . . . 12 (𝑁 ∈ β„• β†’ (𝑁 Β· (2 Β· Ο€)) ∈ β„‚)
9392adantr 482 . . . . . . . . . . 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 11182 . . . . . . . . . 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 2783 . . . . . . . . 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 6847 . . . . . . . 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 11179 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (((𝑁 + (1 / 2)) Β· π‘₯) + (𝑁 Β· (2 Β· Ο€))) ∈ β„‚)
99 sinppi 25862 . . . . . . . . 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 484 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ 𝑁 ∈ β„•)
102101nnzd 12531 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ 𝑁 ∈ β„€)
103 sinper 25854 . . . . . . . . . 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 585 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (sinβ€˜(((𝑁 + (1 / 2)) Β· π‘₯) + (𝑁 Β· (2 Β· Ο€)))) = (sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)))
105104negeqd 11400 . . . . . . . 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 2777 . . . . . . 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 11974 . . . . . . . . . . . 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 7374 . . . . . . . . . . . 12 (π‘₯ ∈ ℝ β†’ ((π‘₯ / 2) + ((2 Β· Ο€) / 2)) = ((π‘₯ / 2) + Ο€))
113110, 112eqtrd 2773 . . . . . . . . . . 11 (π‘₯ ∈ ℝ β†’ ((π‘₯ + (2 Β· Ο€)) / 2) = ((π‘₯ / 2) + Ο€))
114113fveq2d 6847 . . . . . . . . . 10 (π‘₯ ∈ ℝ β†’ (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2)) = (sinβ€˜((π‘₯ / 2) + Ο€)))
11539halfcld 12403 . . . . . . . . . . 11 (π‘₯ ∈ ℝ β†’ (π‘₯ / 2) ∈ β„‚)
116 sinppi 25862 . . . . . . . . . . 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 2773 . . . . . . . . 9 (π‘₯ ∈ ℝ β†’ (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2)) = -(sinβ€˜(π‘₯ / 2)))
119118oveq2d 7374 . . . . . . . 8 (π‘₯ ∈ ℝ β†’ ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2))) = ((2 Β· Ο€) Β· -(sinβ€˜(π‘₯ / 2))))
120119adantl 483 . . . . . . 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 7376 . . . . . 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 482 . . . . 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 16017 . . . . . . . 8 (π‘₯ ∈ ℝ β†’ (sinβ€˜(π‘₯ / 2)) ∈ β„‚)
124107, 123mulneg2d 11614 . . . . . . 7 (π‘₯ ∈ ℝ β†’ ((2 Β· Ο€) Β· -(sinβ€˜(π‘₯ / 2))) = -((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))))
125124oveq2d 7374 . . . . . 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 726 . . . . 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 2778 . . . 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 2780 . . 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 812 . 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 11189 . . 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 44421 . . 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 594 . 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 44421 . 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 2783 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  ifcif 4487   ↦ cmpt 5189  β€˜cfv 6497  (class class class)co 7358  β„‚cc 11054  β„cr 11055  0cc0 11056  1c1 11057   + caddc 11059   Β· cmul 11061  -cneg 11391   / cdiv 11817  β„•cn 12158  2c2 12213  β„€cz 12504  β„+crp 12920   mod cmo 13780  sincsin 15951  Ο€cpi 15954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134  ax-addf 11135  ax-mulf 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-om 7804  df-1st 7922  df-2nd 7923  df-supp 8094  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-er 8651  df-map 8770  df-pm 8771  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9309  df-fi 9352  df-sup 9383  df-inf 9384  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-q 12879  df-rp 12921  df-xneg 13038  df-xadd 13039  df-xmul 13040  df-ioo 13274  df-ioc 13275  df-ico 13276  df-icc 13277  df-fz 13431  df-fzo 13574  df-fl 13703  df-mod 13781  df-seq 13913  df-exp 13974  df-fac 14180  df-bc 14209  df-hash 14237  df-shft 14958  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-limsup 15359  df-clim 15376  df-rlim 15377  df-sum 15577  df-ef 15955  df-sin 15957  df-cos 15958  df-pi 15960  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-mulr 17152  df-starv 17153  df-sca 17154  df-vsca 17155  df-ip 17156  df-tset 17157  df-ple 17158  df-ds 17160  df-unif 17161  df-hom 17162  df-cco 17163  df-rest 17309  df-topn 17310  df-0g 17328  df-gsum 17329  df-topgen 17330  df-pt 17331  df-prds 17334  df-xrs 17389  df-qtop 17394  df-imas 17395  df-xps 17397  df-mre 17471  df-mrc 17472  df-acs 17474  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-submnd 18607  df-mulg 18878  df-cntz 19102  df-cmn 19569  df-psmet 20804  df-xmet 20805  df-met 20806  df-bl 20807  df-mopn 20808  df-fbas 20809  df-fg 20810  df-cnfld 20813  df-top 22259  df-topon 22276  df-topsp 22298  df-bases 22312  df-cld 22386  df-ntr 22387  df-cls 22388  df-nei 22465  df-lp 22503  df-perf 22504  df-cn 22594  df-cnp 22595  df-haus 22682  df-tx 22929  df-hmeo 23122  df-fil 23213  df-fm 23305  df-flim 23306  df-flf 23307  df-xms 23689  df-ms 23690  df-tms 23691  df-cncf 24257  df-limc 25246  df-dv 25247
This theorem is referenced by:  fourierdlem111  44544
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