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Theorem fourierdlem29 40870
 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 2775 . . 3 (𝑠 = 𝐴 → (𝑠 = 0 ↔ 𝐴 = 0))
2 id 22 . . . 4 (𝑠 = 𝐴𝑠 = 𝐴)
3 oveq1 6800 . . . . . 6 (𝑠 = 𝐴 → (𝑠 / 2) = (𝐴 / 2))
43fveq2d 6336 . . . . 5 (𝑠 = 𝐴 → (sin‘(𝑠 / 2)) = (sin‘(𝐴 / 2)))
54oveq2d 6809 . . . 4 (𝑠 = 𝐴 → (2 · (sin‘(𝑠 / 2))) = (2 · (sin‘(𝐴 / 2))))
62, 5oveq12d 6811 . . 3 (𝑠 = 𝐴 → (𝑠 / (2 · (sin‘(𝑠 / 2)))) = (𝐴 / (2 · (sin‘(𝐴 / 2)))))
71, 6ifbieq2d 4250 . 2 (𝑠 = 𝐴 → if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))) = if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2))))))
8 fourierdlem29.1 . 2 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2))))))
9 1ex 10237 . . 3 1 ∈ V
10 ovex 6823 . . 3 (𝐴 / (2 · (sin‘(𝐴 / 2)))) ∈ V
119, 10ifex 4295 . 2 if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2))))) ∈ V
127, 8, 11fvmpt 6424 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 1631   ∈ wcel 2145  ifcif 4225   ↦ cmpt 4863  ‘cfv 6031  (class class class)co 6793  0cc0 10138  1c1 10139   · cmul 10143  -cneg 10469   / cdiv 10886  2c2 11272  [,]cicc 12383  sincsin 15000  πcpi 15003 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-1cn 10196 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796 This theorem is referenced by: (None)
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