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Theorem dirkerper 45110
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 2739 . . . . . . . . . . . 12 (2 Β· Ο€) = 𝑇
32oveq2i 7422 . . . . . . . . . . 11 (1 Β· (2 Β· Ο€)) = (1 Β· 𝑇)
4 2re 12290 . . . . . . . . . . . . . . 15 2 ∈ ℝ
5 pire 26204 . . . . . . . . . . . . . . 15 Ο€ ∈ ℝ
64, 5remulcli 11234 . . . . . . . . . . . . . 14 (2 Β· Ο€) ∈ ℝ
71, 6eqeltri 2827 . . . . . . . . . . . . 13 𝑇 ∈ ℝ
87recni 11232 . . . . . . . . . . . 12 𝑇 ∈ β„‚
98mullidi 11223 . . . . . . . . . . 11 (1 Β· 𝑇) = 𝑇
103, 9eqtri 2758 . . . . . . . . . 10 (1 Β· (2 Β· Ο€)) = 𝑇
1110oveq2i 7422 . . . . . . . . 9 (π‘₯ + (1 Β· (2 Β· Ο€))) = (π‘₯ + 𝑇)
1211eqcomi 2739 . . . . . . . 8 (π‘₯ + 𝑇) = (π‘₯ + (1 Β· (2 Β· Ο€)))
1312oveq1i 7421 . . . . . . 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 723 . . . . . . 7 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ π‘₯ ∈ ℝ)
17 2rp 12983 . . . . . . . . 9 2 ∈ ℝ+
18 pirp 26207 . . . . . . . . 9 Ο€ ∈ ℝ+
19 rpmulcl 13001 . . . . . . . . 9 ((2 ∈ ℝ+ ∧ Ο€ ∈ ℝ+) β†’ (2 Β· Ο€) ∈ ℝ+)
2017, 18, 19mp2an 688 . . . . . . . 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 12596 . . . . . . . 8 1 ∈ β„€
2322a1i 11 . . . . . . 7 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ 1 ∈ β„€)
24 modcyc 13875 . . . . . . 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 483 . . . . . 6 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ (π‘₯ mod (2 Β· Ο€)) = 0)
2714, 25, 263eqtrd 2774 . . . . 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 480 . . . 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 2773 . . 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 480 . . . 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 12224 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„‚)
35 halfcn 12431 . . . . . . . . . . 11 (1 / 2) ∈ β„‚
3635a1i 11 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ (1 / 2) ∈ β„‚)
3734, 36addcld 11237 . . . . . . . . 9 (𝑁 ∈ β„• β†’ (𝑁 + (1 / 2)) ∈ β„‚)
3837adantr 479 . . . . . . . 8 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (𝑁 + (1 / 2)) ∈ β„‚)
39 recn 11202 . . . . . . . . 9 (π‘₯ ∈ ℝ β†’ π‘₯ ∈ β„‚)
4039adantl 480 . . . . . . . 8 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ π‘₯ ∈ β„‚)
4138, 40mulcld 11238 . . . . . . 7 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ ((𝑁 + (1 / 2)) Β· π‘₯) ∈ β„‚)
4241sincld 16077 . . . . . 6 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) ∈ β„‚)
4342adantr 479 . . . . 5 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ (sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)) ∈ β„‚)
446recni 11232 . . . . . . . 8 (2 Β· Ο€) ∈ β„‚
4544a1i 11 . . . . . . 7 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (2 Β· Ο€) ∈ β„‚)
4640halfcld 12461 . . . . . . . 8 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (π‘₯ / 2) ∈ β„‚)
4746sincld 16077 . . . . . . 7 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (sinβ€˜(π‘₯ / 2)) ∈ β„‚)
4845, 47mulcld 11238 . . . . . 6 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))) ∈ β„‚)
4948adantr 479 . . . . 5 (((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))) ∈ β„‚)
50 dirkerdenne0 45107 . . . . . 6 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))) β‰  0)
5150adantll 710 . . . . 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 12009 . . . 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 1451 . . . . . . . . . . 11 (π‘₯ ∈ ℝ β†’ ((π‘₯ + (1 Β· (2 Β· Ο€))) mod (2 Β· Ο€)) = (π‘₯ mod (2 Β· Ο€)))
5553, 54eqtrd 2770 . . . . . . . . . 10 (π‘₯ ∈ ℝ β†’ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) = (π‘₯ mod (2 Β· Ο€)))
5655adantr 479 . . . . . . . . 9 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) = (π‘₯ mod (2 Β· Ο€)))
57 simpr 483 . . . . . . . . . 10 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0)
5857neqned 2945 . . . . . . . . 9 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ (π‘₯ mod (2 Β· Ο€)) β‰  0)
5956, 58eqnetrd 3006 . . . . . . . 8 ((π‘₯ ∈ ℝ ∧ Β¬ (π‘₯ mod (2 Β· Ο€)) = 0) β†’ ((π‘₯ + 𝑇) mod (2 Β· Ο€)) β‰  0)
6059neneqd 2943 . . . . . . 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 7422 . . . . . . . . . . 11 (π‘₯ + 𝑇) = (π‘₯ + (2 Β· Ο€))
6362oveq2i 7422 . . . . . . . . . 10 ((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇)) = ((𝑁 + (1 / 2)) Β· (π‘₯ + (2 Β· Ο€)))
6463fveq2i 6893 . . . . . . . . 9 (sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + 𝑇))) = (sinβ€˜((𝑁 + (1 / 2)) Β· (π‘₯ + (2 Β· Ο€))))
6562oveq1i 7421 . . . . . . . . . . 11 ((π‘₯ + 𝑇) / 2) = ((π‘₯ + (2 Β· Ο€)) / 2)
6665fveq2i 6893 . . . . . . . . . 10 (sinβ€˜((π‘₯ + 𝑇) / 2)) = (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2))
6766oveq2i 7422 . . . . . . . . 9 ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + 𝑇) / 2))) = ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2)))
6864, 67oveq12i 7423 . . . . . . . 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 2786 . . . . . . 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 710 . . . . 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 11243 . . . . . . . . . . . . 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 11170 . . . . . . . . . . . . . . . 16 1 ∈ β„‚
75 2cnne0 12426 . . . . . . . . . . . . . . . 16 (2 ∈ β„‚ ∧ 2 β‰  0)
76 2cn 12291 . . . . . . . . . . . . . . . . 17 2 ∈ β„‚
77 picn 26205 . . . . . . . . . . . . . . . . 17 Ο€ ∈ β„‚
7876, 77mulcli 11225 . . . . . . . . . . . . . . . 16 (2 Β· Ο€) ∈ β„‚
79 div32 11896 . . . . . . . . . . . . . . . 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 1459 . . . . . . . . . . . . . . 15 ((1 / 2) Β· (2 Β· Ο€)) = (1 Β· ((2 Β· Ο€) / 2))
81 2ne0 12320 . . . . . . . . . . . . . . . . . 18 2 β‰  0
8278, 76, 81divcli 11960 . . . . . . . . . . . . . . . . 17 ((2 Β· Ο€) / 2) ∈ β„‚
8382mullidi 11223 . . . . . . . . . . . . . . . 16 (1 Β· ((2 Β· Ο€) / 2)) = ((2 Β· Ο€) / 2)
8477, 76, 81divcan3i 11964 . . . . . . . . . . . . . . . 16 ((2 Β· Ο€) / 2) = Ο€
8583, 84eqtri 2758 . . . . . . . . . . . . . . 15 (1 Β· ((2 Β· Ο€) / 2)) = Ο€
8680, 85eqtri 2758 . . . . . . . . . . . . . 14 ((1 / 2) Β· (2 Β· Ο€)) = Ο€
8786oveq2i 7422 . . . . . . . . . . . . 13 ((𝑁 Β· (2 Β· Ο€)) + ((1 / 2) Β· (2 Β· Ο€))) = ((𝑁 Β· (2 Β· Ο€)) + Ο€)
8873, 87eqtrdi 2786 . . . . . . . . . . . 12 (𝑁 ∈ β„• β†’ ((𝑁 + (1 / 2)) Β· (2 Β· Ο€)) = ((𝑁 Β· (2 Β· Ο€)) + Ο€))
8988oveq2d 7427 . . . . . . . . . . 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 479 . . . . . . . . . 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 11242 . . . . . . . . . 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 11238 . . . . . . . . . . . 12 (𝑁 ∈ β„• β†’ (𝑁 Β· (2 Β· Ο€)) ∈ β„‚)
9392adantr 479 . . . . . . . . . . 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 11240 . . . . . . . . . 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 2780 . . . . . . . . 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 6894 . . . . . . . 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 11237 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (((𝑁 + (1 / 2)) Β· π‘₯) + (𝑁 Β· (2 Β· Ο€))) ∈ β„‚)
99 sinppi 26235 . . . . . . . . 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 481 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ 𝑁 ∈ β„•)
102101nnzd 12589 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ 𝑁 ∈ β„€)
103 sinper 26227 . . . . . . . . . 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 582 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ π‘₯ ∈ ℝ) β†’ (sinβ€˜(((𝑁 + (1 / 2)) Β· π‘₯) + (𝑁 Β· (2 Β· Ο€)))) = (sinβ€˜((𝑁 + (1 / 2)) Β· π‘₯)))
105104negeqd 11458 . . . . . . . 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 2774 . . . . . . 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 12032 . . . . . . . . . . . 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 7427 . . . . . . . . . . . 12 (π‘₯ ∈ ℝ β†’ ((π‘₯ / 2) + ((2 Β· Ο€) / 2)) = ((π‘₯ / 2) + Ο€))
113110, 112eqtrd 2770 . . . . . . . . . . 11 (π‘₯ ∈ ℝ β†’ ((π‘₯ + (2 Β· Ο€)) / 2) = ((π‘₯ / 2) + Ο€))
114113fveq2d 6894 . . . . . . . . . 10 (π‘₯ ∈ ℝ β†’ (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2)) = (sinβ€˜((π‘₯ / 2) + Ο€)))
11539halfcld 12461 . . . . . . . . . . 11 (π‘₯ ∈ ℝ β†’ (π‘₯ / 2) ∈ β„‚)
116 sinppi 26235 . . . . . . . . . . 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 2770 . . . . . . . . 9 (π‘₯ ∈ ℝ β†’ (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2)) = -(sinβ€˜(π‘₯ / 2)))
119118oveq2d 7427 . . . . . . . 8 (π‘₯ ∈ ℝ β†’ ((2 Β· Ο€) Β· (sinβ€˜((π‘₯ + (2 Β· Ο€)) / 2))) = ((2 Β· Ο€) Β· -(sinβ€˜(π‘₯ / 2))))
120119adantl 480 . . . . . . 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 7429 . . . . . 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 479 . . . . 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 16077 . . . . . . . 8 (π‘₯ ∈ ℝ β†’ (sinβ€˜(π‘₯ / 2)) ∈ β„‚)
124107, 123mulneg2d 11672 . . . . . . 7 (π‘₯ ∈ ℝ β†’ ((2 Β· Ο€) Β· -(sinβ€˜(π‘₯ / 2))) = -((2 Β· Ο€) Β· (sinβ€˜(π‘₯ / 2))))
125124oveq2d 7427 . . . . . 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 723 . . . . 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 2775 . . . 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 2777 . . 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 809 . 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 11247 . . 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 45108 . . 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 591 . 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 45108 . 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 2780 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 394   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  ifcif 4527   ↦ cmpt 5230  β€˜cfv 6542  (class class class)co 7411  β„‚cc 11110  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115   Β· cmul 11117  -cneg 11449   / cdiv 11875  β„•cn 12216  2c2 12271  β„€cz 12562  β„+crp 12978   mod cmo 13838  sincsin 16011  Ο€cpi 16014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  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 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 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-q 12937  df-rp 12979  df-xneg 13096  df-xadd 13097  df-xmul 13098  df-ioo 13332  df-ioc 13333  df-ico 13334  df-icc 13335  df-fz 13489  df-fzo 13632  df-fl 13761  df-mod 13839  df-seq 13971  df-exp 14032  df-fac 14238  df-bc 14267  df-hash 14295  df-shft 15018  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-limsup 15419  df-clim 15436  df-rlim 15437  df-sum 15637  df-ef 16015  df-sin 16017  df-cos 16018  df-pi 16020  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-starv 17216  df-sca 17217  df-vsca 17218  df-ip 17219  df-tset 17220  df-ple 17221  df-ds 17223  df-unif 17224  df-hom 17225  df-cco 17226  df-rest 17372  df-topn 17373  df-0g 17391  df-gsum 17392  df-topgen 17393  df-pt 17394  df-prds 17397  df-xrs 17452  df-qtop 17457  df-imas 17458  df-xps 17460  df-mre 17534  df-mrc 17535  df-acs 17537  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18706  df-mulg 18987  df-cntz 19222  df-cmn 19691  df-psmet 21136  df-xmet 21137  df-met 21138  df-bl 21139  df-mopn 21140  df-fbas 21141  df-fg 21142  df-cnfld 21145  df-top 22616  df-topon 22633  df-topsp 22655  df-bases 22669  df-cld 22743  df-ntr 22744  df-cls 22745  df-nei 22822  df-lp 22860  df-perf 22861  df-cn 22951  df-cnp 22952  df-haus 23039  df-tx 23286  df-hmeo 23479  df-fil 23570  df-fm 23662  df-flim 23663  df-flf 23664  df-xms 24046  df-ms 24047  df-tms 24048  df-cncf 24618  df-limc 25615  df-dv 25616
This theorem is referenced by:  fourierdlem111  45231
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