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Theorem fourierdlem29 44463
Description: Explicit function value for 𝐾 applied to 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
fourierdlem29.1 𝐾 = (𝑠 ∈ (-Ο€[,]Ο€) ↦ if(𝑠 = 0, 1, (𝑠 / (2 Β· (sinβ€˜(𝑠 / 2))))))
Assertion
Ref Expression
fourierdlem29 (𝐴 ∈ (-Ο€[,]Ο€) β†’ (πΎβ€˜π΄) = if(𝐴 = 0, 1, (𝐴 / (2 Β· (sinβ€˜(𝐴 / 2))))))
Distinct variable group:   𝐴,𝑠
Allowed substitution hint:   𝐾(𝑠)

Proof of Theorem fourierdlem29
StepHypRef Expression
1 eqeq1 2737 . . 3 (𝑠 = 𝐴 β†’ (𝑠 = 0 ↔ 𝐴 = 0))
2 id 22 . . . 4 (𝑠 = 𝐴 β†’ 𝑠 = 𝐴)
3 fvoveq1 7381 . . . . 5 (𝑠 = 𝐴 β†’ (sinβ€˜(𝑠 / 2)) = (sinβ€˜(𝐴 / 2)))
43oveq2d 7374 . . . 4 (𝑠 = 𝐴 β†’ (2 Β· (sinβ€˜(𝑠 / 2))) = (2 Β· (sinβ€˜(𝐴 / 2))))
52, 4oveq12d 7376 . . 3 (𝑠 = 𝐴 β†’ (𝑠 / (2 Β· (sinβ€˜(𝑠 / 2)))) = (𝐴 / (2 Β· (sinβ€˜(𝐴 / 2)))))
61, 5ifbieq2d 4513 . 2 (𝑠 = 𝐴 β†’ if(𝑠 = 0, 1, (𝑠 / (2 Β· (sinβ€˜(𝑠 / 2))))) = if(𝐴 = 0, 1, (𝐴 / (2 Β· (sinβ€˜(𝐴 / 2))))))
7 fourierdlem29.1 . 2 𝐾 = (𝑠 ∈ (-Ο€[,]Ο€) ↦ if(𝑠 = 0, 1, (𝑠 / (2 Β· (sinβ€˜(𝑠 / 2))))))
8 1ex 11156 . . 3 1 ∈ V
9 ovex 7391 . . 3 (𝐴 / (2 Β· (sinβ€˜(𝐴 / 2)))) ∈ V
108, 9ifex 4537 . 2 if(𝐴 = 0, 1, (𝐴 / (2 Β· (sinβ€˜(𝐴 / 2))))) ∈ V
116, 7, 10fvmpt 6949 1 (𝐴 ∈ (-Ο€[,]Ο€) β†’ (πΎβ€˜π΄) = if(𝐴 = 0, 1, (𝐴 / (2 Β· (sinβ€˜(𝐴 / 2))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  ifcif 4487   ↦ cmpt 5189  β€˜cfv 6497  (class class class)co 7358  0cc0 11056  1c1 11057   Β· cmul 11061  -cneg 11391   / cdiv 11817  2c2 12213  [,]cicc 13273  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-sep 5257  ax-nul 5264  ax-pr 5385  ax-1cn 11114
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-rab 3407  df-v 3446  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-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-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361
This theorem is referenced by: (None)
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