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Theorem fourierdlem29 46092
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 2739 . . 3 (𝑠 = 𝐴 → (𝑠 = 0 ↔ 𝐴 = 0))
2 id 22 . . . 4 (𝑠 = 𝐴𝑠 = 𝐴)
3 fvoveq1 7454 . . . . 5 (𝑠 = 𝐴 → (sin‘(𝑠 / 2)) = (sin‘(𝐴 / 2)))
43oveq2d 7447 . . . 4 (𝑠 = 𝐴 → (2 · (sin‘(𝑠 / 2))) = (2 · (sin‘(𝐴 / 2))))
52, 4oveq12d 7449 . . 3 (𝑠 = 𝐴 → (𝑠 / (2 · (sin‘(𝑠 / 2)))) = (𝐴 / (2 · (sin‘(𝐴 / 2)))))
61, 5ifbieq2d 4557 . 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 11255 . . 3 1 ∈ V
9 ovex 7464 . . 3 (𝐴 / (2 · (sin‘(𝐴 / 2)))) ∈ V
108, 9ifex 4581 . 2 if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2))))) ∈ V
116, 7, 10fvmpt 7016 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 1537  wcel 2106  ifcif 4531  cmpt 5231  cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154   · cmul 11158  -cneg 11491   / cdiv 11918  2c2 12319  [,]cicc 13387  sincsin 16096  πcpi 16099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-1cn 11211
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434
This theorem is referenced by: (None)
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