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

Theorem dirkerper 45407
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 2736 . . . . . . . . . . . 12 (2 Β· Ο€) = 𝑇
32oveq2i 7425 . . . . . . . . . . 11 (1 Β· (2 Β· Ο€)) = (1 Β· 𝑇)
4 2re 12308 . . . . . . . . . . . . . . 15 2 ∈ ℝ
5 pire 26380 . . . . . . . . . . . . . . 15 Ο€ ∈ ℝ
64, 5remulcli 11252 . . . . . . . . . . . . . 14 (2 Β· Ο€) ∈ ℝ
71, 6eqeltri 2824 . . . . . . . . . . . . 13 𝑇 ∈ ℝ
87recni 11250 . . . . . . . . . . . 12 𝑇 ∈ β„‚
98mullidi 11241 . . . . . . . . . . 11 (1 Β· 𝑇) = 𝑇
103, 9eqtri 2755 . . . . . . . . . 10 (1 Β· (2 Β· Ο€)) = 𝑇
1110oveq2i 7425 . . . . . . . . 9 (π‘₯ + (1 Β· (2 Β· Ο€))) = (π‘₯ + 𝑇)
1211eqcomi 2736 . . . . . . . 8 (π‘₯ + 𝑇) = (π‘₯ + (1 Β· (2 Β· Ο€)))
1312oveq1i 7424 . . . . . . 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 13003 . . . . . . . . 9 2 ∈ ℝ+
18 pirp 26383 . . . . . . . . 9 Ο€ ∈ ℝ+
19 rpmulcl 13021 . . . . . . . . 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 12614 . . . . . . . 8 1 ∈ β„€
2322a1i 11 . . . . . . 7 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ 1 ∈ β„€)
24 modcyc 13895 . . . . . . 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 1369 . . . . . 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 484 . . . . . 6 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ (π‘₯ mod (2 Β· Ο€)) = 0)
2714, 25, 263eqtrd 2771 . . . . 5 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) = 0)
2827iftrued 4532 . . . 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 4530 . . . . 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 481 . . . 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 2770 . . 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 4533 . . . . 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 481 . . . 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 12242 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„‚)
35 halfcn 12449 . . . . . . . . . . 11 (1 / 2) ∈ β„‚
3635a1i 11 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ (1 / 2) ∈ β„‚)
3734, 36addcld 11255 . . . . . . . . 9 (𝑁 ∈ β„• β†’ (𝑁 + (1 / 2)) ∈ β„‚)
3837adantr 480 . . . . . . . 8 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (𝑁 + (1 / 2)) ∈ β„‚)
39 recn 11220 . . . . . . . . 9 (π‘₯ ∈ ℝ β†’ π‘₯ ∈ β„‚)
4039adantl 481 . . . . . . . 8 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ π‘₯ ∈ β„‚)
4138, 40mulcld 11256 . . . . . . 7 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ ((𝑁 + (1 / 2)) Β· π‘₯) ∈ β„‚)
4241sincld 16098 . . . . . 6 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) ∈ β„‚)
4342adantr 480 . . . . 5 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ (sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) ∈ β„‚)
446recni 11250 . . . . . . . 8 (2 Β· Ο€) ∈ β„‚
4544a1i 11 . . . . . . 7 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (2 Β· Ο€) ∈ β„‚)
4640halfcld 12479 . . . . . . . 8 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (π‘₯ / 2) ∈ β„‚)
4746sincld 16098 . . . . . . 7 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (sinβ€˜(π‘₯ / 2)) ∈ β„‚)
4845, 47mulcld 11256 . . . . . 6 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))) ∈ β„‚)
4948adantr 480 . . . . 5 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))) ∈ β„‚)
50 dirkerdenne0 45404 . . . . . 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 12027 . . . 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 1450 . . . . . . . . . . 11 (π‘₯ ∈ ℝ β†’ ((π‘₯ + (1 Β· (2 Β· Ο€))) mod (2 Β· Ο€)) = (π‘₯ mod (2 Β· Ο€)))
5553, 54eqtrd 2767 . . . . . . . . . 10 (π‘₯ ∈ ℝ β†’ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) = (π‘₯ mod (2 Β· Ο€)))
5655adantr 480 . . . . . . . . 9 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) = (π‘₯ mod (2 Β· Ο€)))
57 simpr 484 . . . . . . . . . 10 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0)
5857neqned 2942 . . . . . . . . 9 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ (π‘₯ mod (2 Β· Ο€)) β‰  0)
5956, 58eqnetrd 3003 . . . . . . . 8 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) β‰  0)
6059neneqd 2940 . . . . . . 7 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ Β¬ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) = 0)
61 iffalse 4533 . . . . . . . 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 7425 . . . . . . . . . . 11 (π‘₯ + 𝑇) = (π‘₯ + (2 Β· Ο€))
6362oveq2i 7425 . . . . . . . . . 10 ((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇)) = ((𝑁 + (1 / 2)) Β· (π‘₯ + (2 Β· Ο€)))
6463fveq2i 6894 . . . . . . . . 9 (sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇))) = (sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + (2 Β· Ο€))))
6562oveq1i 7424 . . . . . . . . . . 11 ((π‘₯ + 𝑇) / 2) = ((π‘₯ + (2 Β· Ο€)) / 2)
6665fveq2i 6894 . . . . . . . . . 10 (sinβ€˜((π‘₯ + 𝑇) / 2)) = (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2))
6766oveq2i 7425 . . . . . . . . 9 ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + 𝑇) / 2))) = ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2)))
6864, 67oveq12i 7426 . . . . . . . 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 2783 . . . . . . 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 11261 . . . . . . . . . . . . 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 11188 . . . . . . . . . . . . . . . 16 1 ∈ β„‚
75 2cnne0 12444 . . . . . . . . . . . . . . . 16 (2 ∈ β„‚ ∧ 2 β‰  0)
76 2cn 12309 . . . . . . . . . . . . . . . . 17 2 ∈ β„‚
77 picn 26381 . . . . . . . . . . . . . . . . 17 Ο€ ∈ β„‚
7876, 77mulcli 11243 . . . . . . . . . . . . . . . 16 (2 Β· Ο€) ∈ β„‚
79 div32 11914 . . . . . . . . . . . . . . . 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 1458 . . . . . . . . . . . . . . 15 ((1 / 2) Β· (2 Β· Ο€)) = (1 Β· ((2 Β· Ο€) / 2))
81 2ne0 12338 . . . . . . . . . . . . . . . . . 18 2 β‰  0
8278, 76, 81divcli 11978 . . . . . . . . . . . . . . . . 17 ((2 Β· Ο€) / 2) ∈ β„‚
8382mullidi 11241 . . . . . . . . . . . . . . . 16 (1 Β· ((2 Β· Ο€) / 2)) = ((2 Β· Ο€) / 2)
8477, 76, 81divcan3i 11982 . . . . . . . . . . . . . . . 16 ((2 Β· Ο€) / 2) = Ο€
8583, 84eqtri 2755 . . . . . . . . . . . . . . 15 (1 Β· ((2 Β· Ο€) / 2)) = Ο€
8680, 85eqtri 2755 . . . . . . . . . . . . . 14 ((1 / 2) Β· (2 Β· Ο€)) = Ο€
8786oveq2i 7425 . . . . . . . . . . . . 13 ((𝑁 Β· (2 Β· Ο€)) + ((1 / 2) Β· (2 Β· Ο€))) = ((𝑁 Β· (2 Β· Ο€)) + Ο€)
8873, 87eqtrdi 2783 . . . . . . . . . . . 12 (𝑁 ∈ β„• β†’ ((𝑁 + (1 / 2)) Β· (2 Β· Ο€)) = ((𝑁 Β· (2 Β· Ο€)) + Ο€))
8988oveq2d 7430 . . . . . . . . . . 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 480 . . . . . . . . . 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 11260 . . . . . . . . . 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 11256 . . . . . . . . . . . 12 (𝑁 ∈ β„• β†’ (𝑁 Β· (2 Β· Ο€)) ∈ β„‚)
9392adantr 480 . . . . . . . . . . 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 11258 . . . . . . . . . 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 2777 . . . . . . . . 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 6895 . . . . . . . 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 11255 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (((𝑁 + (1 / 2)) Β· π‘₯) + (𝑁 Β· (2 Β· Ο€))) ∈ β„‚)
99 sinppi 26411 . . . . . . . . 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 482 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ 𝑁 ∈ β„•)
102101nnzd 12607 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ 𝑁 ∈ β„€)
103 sinper 26403 . . . . . . . . . 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 583 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (sinβ€˜(((𝑁 + (1 / 2)) Β· π‘₯) + (𝑁 Β· (2 Β· Ο€)))) = (sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)))
105104negeqd 11476 . . . . . . . 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 2771 . . . . . . 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 12050 . . . . . . . . . . . 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 7430 . . . . . . . . . . . 12 (π‘₯ ∈ ℝ β†’ ((π‘₯ / 2) + ((2 Β· Ο€) / 2)) = ((π‘₯ / 2) + Ο€))
113110, 112eqtrd 2767 . . . . . . . . . . 11 (π‘₯ ∈ ℝ β†’ ((π‘₯ + (2 Β· Ο€)) / 2) = ((π‘₯ / 2) + Ο€))
114113fveq2d 6895 . . . . . . . . . 10 (π‘₯ ∈ ℝ β†’ (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2)) = (sinβ€˜((π‘₯ / 2) + Ο€)))
11539halfcld 12479 . . . . . . . . . . 11 (π‘₯ ∈ ℝ β†’ (π‘₯ / 2) ∈ β„‚)
116 sinppi 26411 . . . . . . . . . . 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 2767 . . . . . . . . 9 (π‘₯ ∈ ℝ β†’ (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2)) = -(sinβ€˜(π‘₯ / 2)))
119118oveq2d 7430 . . . . . . . 8 (π‘₯ ∈ ℝ β†’ ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2))) = ((2 Β· Ο€) Β· -(sinβ€˜(π‘₯ / 2))))
120119adantl 481 . . . . . . 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 7432 . . . . . 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 480 . . . . 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 16098 . . . . . . . 8 (π‘₯ ∈ ℝ β†’ (sinβ€˜(π‘₯ / 2)) ∈ β„‚)
124107, 123mulneg2d 11690 . . . . . . 7 (π‘₯ ∈ ℝ β†’ ((2 Β· Ο€) Β· -(sinβ€˜(π‘₯ / 2))) = -((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))))
125124oveq2d 7430 . . . . . 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 2772 . . . 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 2774 . . 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 11265 . . 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 45405 . . 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 592 . 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 45405 . 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 2777 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 395   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  ifcif 4524   ↦ cmpt 5225  β€˜cfv 6542  (class class class)co 7414  β„‚cc 11128  β„cr 11129  0cc0 11130  1c1 11131   + caddc 11133   Β· cmul 11135  -cneg 11467   / cdiv 11893  β„•cn 12234  2c2 12289  β„€cz 12580  β„+crp 12998   mod cmo 13858  sincsin 16031  Ο€cpi 16034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208  ax-addf 11209
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679  df-om 7865  df-1st 7987  df-2nd 7988  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8718  df-map 8838  df-pm 8839  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9378  df-fi 9426  df-sup 9457  df-inf 9458  df-oi 9525  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-9 12304  df-n0 12495  df-z 12581  df-dec 12700  df-uz 12845  df-q 12955  df-rp 12999  df-xneg 13116  df-xadd 13117  df-xmul 13118  df-ioo 13352  df-ioc 13353  df-ico 13354  df-icc 13355  df-fz 13509  df-fzo 13652  df-fl 13781  df-mod 13859  df-seq 13991  df-exp 14051  df-fac 14257  df-bc 14286  df-hash 14314  df-shft 15038  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-limsup 15439  df-clim 15456  df-rlim 15457  df-sum 15657  df-ef 16035  df-sin 16037  df-cos 16038  df-pi 16040  df-struct 17107  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-ress 17201  df-plusg 17237  df-mulr 17238  df-starv 17239  df-sca 17240  df-vsca 17241  df-ip 17242  df-tset 17243  df-ple 17244  df-ds 17246  df-unif 17247  df-hom 17248  df-cco 17249  df-rest 17395  df-topn 17396  df-0g 17414  df-gsum 17415  df-topgen 17416  df-pt 17417  df-prds 17420  df-xrs 17475  df-qtop 17480  df-imas 17481  df-xps 17483  df-mre 17557  df-mrc 17558  df-acs 17560  df-mgm 18591  df-sgrp 18670  df-mnd 18686  df-submnd 18732  df-mulg 19015  df-cntz 19259  df-cmn 19728  df-psmet 21258  df-xmet 21259  df-met 21260  df-bl 21261  df-mopn 21262  df-fbas 21263  df-fg 21264  df-cnfld 21267  df-top 22783  df-topon 22800  df-topsp 22822  df-bases 22836  df-cld 22910  df-ntr 22911  df-cls 22912  df-nei 22989  df-lp 23027  df-perf 23028  df-cn 23118  df-cnp 23119  df-haus 23206  df-tx 23453  df-hmeo 23646  df-fil 23737  df-fm 23829  df-flim 23830  df-flf 23831  df-xms 24213  df-ms 24214  df-tms 24215  df-cncf 24785  df-limc 25782  df-dv 25783
This theorem is referenced by:  fourierdlem111  45528
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