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Theorem dirkerval 44418
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 484 . . . . . . 7 ((π‘š = 𝑁 ∧ 𝑠 ∈ ℝ) β†’ π‘š = 𝑁)
21oveq2d 7374 . . . . . 6 ((π‘š = 𝑁 ∧ 𝑠 ∈ ℝ) β†’ (2 Β· π‘š) = (2 Β· 𝑁))
32oveq1d 7373 . . . . 5 ((π‘š = 𝑁 ∧ 𝑠 ∈ ℝ) β†’ ((2 Β· π‘š) + 1) = ((2 Β· 𝑁) + 1))
43oveq1d 7373 . . . 4 ((π‘š = 𝑁 ∧ 𝑠 ∈ ℝ) β†’ (((2 Β· π‘š) + 1) / (2 Β· Ο€)) = (((2 Β· 𝑁) + 1) / (2 Β· Ο€)))
51oveq1d 7373 . . . . . 6 ((π‘š = 𝑁 ∧ 𝑠 ∈ ℝ) β†’ (π‘š + (1 / 2)) = (𝑁 + (1 / 2)))
65fvoveq1d 7380 . . . . 5 ((π‘š = 𝑁 ∧ 𝑠 ∈ ℝ) β†’ (sinβ€˜((π‘š + (1 / 2)) Β· 𝑠)) = (sinβ€˜((𝑁 + (1 / 2)) Β· 𝑠)))
76oveq1d 7373 . . . 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 4508 . . 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 5206 . 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 484 . . . . . . . . 9 ((𝑛 = π‘š ∧ 𝑠 ∈ ℝ) β†’ 𝑛 = π‘š)
1211oveq2d 7374 . . . . . . . 8 ((𝑛 = π‘š ∧ 𝑠 ∈ ℝ) β†’ (2 Β· 𝑛) = (2 Β· π‘š))
1312oveq1d 7373 . . . . . . 7 ((𝑛 = π‘š ∧ 𝑠 ∈ ℝ) β†’ ((2 Β· 𝑛) + 1) = ((2 Β· π‘š) + 1))
1413oveq1d 7373 . . . . . 6 ((𝑛 = π‘š ∧ 𝑠 ∈ ℝ) β†’ (((2 Β· 𝑛) + 1) / (2 Β· Ο€)) = (((2 Β· π‘š) + 1) / (2 Β· Ο€)))
1511oveq1d 7373 . . . . . . . 8 ((𝑛 = π‘š ∧ 𝑠 ∈ ℝ) β†’ (𝑛 + (1 / 2)) = (π‘š + (1 / 2)))
1615fvoveq1d 7380 . . . . . . 7 ((𝑛 = π‘š ∧ 𝑠 ∈ ℝ) β†’ (sinβ€˜((𝑛 + (1 / 2)) Β· 𝑠)) = (sinβ€˜((π‘š + (1 / 2)) Β· 𝑠)))
1716oveq1d 7373 . . . . . 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 4508 . . . . 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 5206 . . . 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 5219 . . 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 2761 . 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 11147 . . 3 ℝ ∈ V
2322mptex 7174 . 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 6949 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 397   = wceq 1542   ∈ wcel 2107  ifcif 4487   ↦ cmpt 5189  β€˜cfv 6497  (class class class)co 7358  β„cr 11055  0cc0 11056  1c1 11057   + caddc 11059   Β· cmul 11061   / cdiv 11817  β„•cn 12158  2c2 12213   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-pr 5385  ax-cnex 11112  ax-resscn 11113
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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-ral 3062  df-rex 3071  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-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  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-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361
This theorem is referenced by:  dirkerval2  44421  dirkerf  44424  dirkertrigeq  44428  dirkercncflem2  44431  dirkercncflem4  44433
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