Theorem fourierdlem29 46964

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 2764	. . 3 ⊢ (𝑠 = 𝐴 → (𝑠 = 0 ↔ 𝐴 = 0))
2		id 23	. . . 4 ⊢ (𝑠 = 𝐴 → 𝑠 = 𝐴)
3		fvoveq1 7436	. . . . 5 ⊢ (𝑠 = 𝐴 → (sin‘(𝑠 / 2)) = (sin‘(𝐴 / 2)))
4	3	oveq2d 7429	. . . 4 ⊢ (𝑠 = 𝐴 → (2 · (sin‘(𝑠 / 2))) = (2 · (sin‘(𝐴 / 2))))
5	2, 4	oveq12d 7431	. . 3 ⊢ (𝑠 = 𝐴 → (𝑠 / (2 · (sin‘(𝑠 / 2)))) = (𝐴 / (2 · (sin‘(𝐴 / 2)))))
6	1, 5	ifbieq2d 4509	. 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 11227	. . 3 ⊢ 1 ∈ V
9		ovex 7446	. . 3 ⊢ (𝐴 / (2 · (sin‘(𝐴 / 2)))) ∈ V
10	8, 9	ifex 4533	. 2 ⊢ if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2))))) ∈ V
11	6, 7, 10	fvmpt 6986	1 ⊢ (𝐴 ∈ (-π[,]π) → (𝐾‘𝐴) = if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2))))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ifcif 4482   ↦ cmpt 5186  ‘cfv 6533  (class class class)co 7413  0cc0 11124  1c1 11125   · cmul 11129  -cneg 11466   / cdiv 11895  2c2 12319  [,]cicc 13401  sincsin 16149  πcpi 16152

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-1cn 11182

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7416

This theorem is used by: (None)