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Theorem dirkerval 44886
Description: The Nth Dirichlet Kernel. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
dirkerval.1 𝐷 = (𝑛 ∈ β„• ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑛) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))))))
Assertion
Ref Expression
dirkerval (𝑁 ∈ β„• β†’ (π·β€˜π‘) = (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))))))
Distinct variable groups:   𝑁,𝑠   𝑛,𝑠
Allowed substitution hints:   𝐷(𝑛,𝑠)   𝑁(𝑛)

Proof of Theorem dirkerval
Dummy variable π‘š is distinct from all other variables.
StepHypRef Expression
1 simpl 483 . . . . . . 7 ((π‘š = 𝑁 ∧ 𝑠 ∈ ℝ) β†’ π‘š = 𝑁)
21oveq2d 7427 . . . . . 6 ((π‘š = 𝑁 ∧ 𝑠 ∈ ℝ) β†’ (2 Β· π‘š) = (2 Β· 𝑁))
32oveq1d 7426 . . . . 5 ((π‘š = 𝑁 ∧ 𝑠 ∈ ℝ) β†’ ((2 Β· π‘š) + 1) = ((2 Β· 𝑁) + 1))
43oveq1d 7426 . . . 4 ((π‘š = 𝑁 ∧ 𝑠 ∈ ℝ) β†’ (((2 Β· π‘š) + 1) / (2 Β· Ο€)) = (((2 Β· 𝑁) + 1) / (2 Β· Ο€)))
51oveq1d 7426 . . . . . 6 ((π‘š = 𝑁 ∧ 𝑠 ∈ ℝ) β†’ (π‘š + (1 / 2)) = (𝑁 + (1 / 2)))
65fvoveq1d 7433 . . . . 5 ((π‘š = 𝑁 ∧ 𝑠 ∈ ℝ) β†’ (sinβ€˜((π‘š + (1 / 2)) Β· 𝑠)) = (sinβ€˜((𝑁 + (1 / 2)) Β· 𝑠)))
76oveq1d 7426 . . . 4 ((π‘š = 𝑁 ∧ 𝑠 ∈ ℝ) β†’ ((sinβ€˜((π‘š + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))) = ((sinβ€˜((𝑁 + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))))
84, 7ifeq12d 4549 . . 3 ((π‘š = 𝑁 ∧ 𝑠 ∈ ℝ) β†’ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· π‘š) + 1) / (2 Β· Ο€)), ((sinβ€˜((π‘š + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2))))) = if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2))))))
98mpteq2dva 5248 . 2 (π‘š = 𝑁 β†’ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· π‘š) + 1) / (2 Β· Ο€)), ((sinβ€˜((π‘š + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))))) = (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))))))
10 dirkerval.1 . . 3 𝐷 = (𝑛 ∈ β„• ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑛) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))))))
11 simpl 483 . . . . . . . . 9 ((𝑛 = π‘š ∧ 𝑠 ∈ ℝ) β†’ 𝑛 = π‘š)
1211oveq2d 7427 . . . . . . . 8 ((𝑛 = π‘š ∧ 𝑠 ∈ ℝ) β†’ (2 Β· 𝑛) = (2 Β· π‘š))
1312oveq1d 7426 . . . . . . 7 ((𝑛 = π‘š ∧ 𝑠 ∈ ℝ) β†’ ((2 Β· 𝑛) + 1) = ((2 Β· π‘š) + 1))
1413oveq1d 7426 . . . . . 6 ((𝑛 = π‘š ∧ 𝑠 ∈ ℝ) β†’ (((2 Β· 𝑛) + 1) / (2 Β· Ο€)) = (((2 Β· π‘š) + 1) / (2 Β· Ο€)))
1511oveq1d 7426 . . . . . . . 8 ((𝑛 = π‘š ∧ 𝑠 ∈ ℝ) β†’ (𝑛 + (1 / 2)) = (π‘š + (1 / 2)))
1615fvoveq1d 7433 . . . . . . 7 ((𝑛 = π‘š ∧ 𝑠 ∈ ℝ) β†’ (sinβ€˜((𝑛 + (1 / 2)) Β· 𝑠)) = (sinβ€˜((π‘š + (1 / 2)) Β· 𝑠)))
1716oveq1d 7426 . . . . . 6 ((𝑛 = π‘š ∧ 𝑠 ∈ ℝ) β†’ ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))) = ((sinβ€˜((π‘š + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))))
1814, 17ifeq12d 4549 . . . . 5 ((𝑛 = π‘š ∧ 𝑠 ∈ ℝ) β†’ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑛) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2))))) = if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· π‘š) + 1) / (2 Β· Ο€)), ((sinβ€˜((π‘š + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2))))))
1918mpteq2dva 5248 . . . 4 (𝑛 = π‘š β†’ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑛) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))))) = (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· π‘š) + 1) / (2 Β· Ο€)), ((sinβ€˜((π‘š + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))))))
2019cbvmptv 5261 . . 3 (𝑛 ∈ β„• ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑛) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑛 + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2))))))) = (π‘š ∈ β„• ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· π‘š) + 1) / (2 Β· Ο€)), ((sinβ€˜((π‘š + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))))))
2110, 20eqtri 2760 . 2 𝐷 = (π‘š ∈ β„• ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· π‘š) + 1) / (2 Β· Ο€)), ((sinβ€˜((π‘š + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))))))
22 reex 11203 . . 3 ℝ ∈ V
2322mptex 7227 . 2 (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))))) ∈ V
249, 21, 23fvmpt 6998 1 (𝑁 ∈ β„• β†’ (π·β€˜π‘) = (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 Β· Ο€)) = 0, (((2 Β· 𝑁) + 1) / (2 Β· Ο€)), ((sinβ€˜((𝑁 + (1 / 2)) Β· 𝑠)) / ((2 Β· Ο€) Β· (sinβ€˜(𝑠 / 2)))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ifcif 4528   ↦ cmpt 5231  β€˜cfv 6543  (class class class)co 7411  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115   Β· cmul 11117   / cdiv 11873  β„•cn 12214  2c2 12269   mod cmo 13836  sincsin 16009  Ο€cpi 16012
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 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-cnex 11168  ax-resscn 11169
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414
This theorem is referenced by:  dirkerval2  44889  dirkerf  44892  dirkertrigeq  44896  dirkercncflem2  44899  dirkercncflem4  44901
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