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Theorem fourierdlem29 42717
Description: Explicit function value for 𝐾 applied to 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
fourierdlem29.1 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
Assertion
Ref Expression
fourierdlem29 (𝐴 ∈ (-π[,]π) → (𝐾𝐴) = if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2))))))
Distinct variable group:   𝐴,𝑠
Allowed substitution hint:   𝐾(𝑠)

Proof of Theorem fourierdlem29
StepHypRef Expression
1 eqeq1 2826 . . 3 (𝑠 = 𝐴 → (𝑠 = 0 ↔ 𝐴 = 0))
2 id 22 . . . 4 (𝑠 = 𝐴𝑠 = 𝐴)
3 fvoveq1 7163 . . . . 5 (𝑠 = 𝐴 → (sin‘(𝑠 / 2)) = (sin‘(𝐴 / 2)))
43oveq2d 7156 . . . 4 (𝑠 = 𝐴 → (2 · (sin‘(𝑠 / 2))) = (2 · (sin‘(𝐴 / 2))))
52, 4oveq12d 7158 . . 3 (𝑠 = 𝐴 → (𝑠 / (2 · (sin‘(𝑠 / 2)))) = (𝐴 / (2 · (sin‘(𝐴 / 2)))))
61, 5ifbieq2d 4464 . 2 (𝑠 = 𝐴 → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) = if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2))))))
7 fourierdlem29.1 . 2 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
8 1ex 10626 . . 3 1 ∈ V
9 ovex 7173 . . 3 (𝐴 / (2 · (sin‘(𝐴 / 2)))) ∈ V
108, 9ifex 4487 . 2 if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2))))) ∈ V
116, 7, 10fvmpt 6750 1 (𝐴 ∈ (-π[,]π) → (𝐾𝐴) = if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  ifcif 4439  cmpt 5122  cfv 6334  (class class class)co 7140  0cc0 10526  1c1 10527   · cmul 10531  -cneg 10860   / cdiv 11286  2c2 11680  [,]cicc 12729  sincsin 15408  πcpi 15411
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-1cn 10584
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143
This theorem is referenced by: (None)
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