Theorem fourierdlem29 46296

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) → (𝐾‘𝐴) = if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2))))))

Distinct variable group:   𝐴,𝑠

Allowed substitution hint:   𝐾(𝑠)

Proof of Theorem fourierdlem29

Step	Hyp	Ref	Expression
1		eqeq1 2737	. . 3 ⊢ (𝑠 = 𝐴 → (𝑠 = 0 ↔ 𝐴 = 0))
2		id 22	. . . 4 ⊢ (𝑠 = 𝐴 → 𝑠 = 𝐴)
3		fvoveq1 7378	. . . . 5 ⊢ (𝑠 = 𝐴 → (sin‘(𝑠 / 2)) = (sin‘(𝐴 / 2)))
4	3	oveq2d 7371	. . . 4 ⊢ (𝑠 = 𝐴 → (2 · (sin‘(𝑠 / 2))) = (2 · (sin‘(𝐴 / 2))))
5	2, 4	oveq12d 7373	. . 3 ⊢ (𝑠 = 𝐴 → (𝑠 / (2 · (sin‘(𝑠 / 2)))) = (𝐴 / (2 · (sin‘(𝐴 / 2)))))
6	1, 5	ifbieq2d 4503	. 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 11119	. . 3 ⊢ 1 ∈ V
9		ovex 7388	. . 3 ⊢ (𝐴 / (2 · (sin‘(𝐴 / 2)))) ∈ V
10	8, 9	ifex 4527	. 2 ⊢ if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2))))) ∈ V
11	6, 7, 10	fvmpt 6938	1 ⊢ (𝐴 ∈ (-π[,]π) → (𝐾‘𝐴) = if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2))))))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ifcif 4476   ↦ cmpt 5176  ‘cfv 6489  (class class class)co 7355  0cc0 11017  1c1 11018   · cmul 11022  -cneg 11356   / cdiv 11785  2c2 12191  [,]cicc 13255  sincsin 15977  πcpi 15980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-1cn 11075

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7358

This theorem is referenced by: (None)