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Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem29 | Structured version Visualization version GIF version |
Description: Explicit function value for 𝐾 applied to 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fourierdlem29.1 | ⊢ 𝐾 = (𝑠 ∈ (-π[,]π) ↦ if(𝑠 = 0, 1, (𝑠 / (2 · (sin‘(𝑠 / 2)))))) |
Ref | Expression |
---|---|
fourierdlem29 | ⊢ (𝐴 ∈ (-π[,]π) → (𝐾‘𝐴) = if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2614 | . . 3 ⊢ (𝑠 = 𝐴 → (𝑠 = 0 ↔ 𝐴 = 0)) | |
2 | id 22 | . . . 4 ⊢ (𝑠 = 𝐴 → 𝑠 = 𝐴) | |
3 | oveq1 6556 | . . . . . 6 ⊢ (𝑠 = 𝐴 → (𝑠 / 2) = (𝐴 / 2)) | |
4 | 3 | fveq2d 6107 | . . . . 5 ⊢ (𝑠 = 𝐴 → (sin‘(𝑠 / 2)) = (sin‘(𝐴 / 2))) |
5 | 4 | oveq2d 6565 | . . . 4 ⊢ (𝑠 = 𝐴 → (2 · (sin‘(𝑠 / 2))) = (2 · (sin‘(𝐴 / 2)))) |
6 | 2, 5 | oveq12d 6567 | . . 3 ⊢ (𝑠 = 𝐴 → (𝑠 / (2 · (sin‘(𝑠 / 2)))) = (𝐴 / (2 · (sin‘(𝐴 / 2))))) |
7 | 1, 6 | ifbieq2d 4061 | . 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 9914 | . . 3 ⊢ 1 ∈ V | |
10 | ovex 6577 | . . 3 ⊢ (𝐴 / (2 · (sin‘(𝐴 / 2)))) ∈ V | |
11 | 9, 10 | ifex 4106 | . 2 ⊢ if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2))))) ∈ V |
12 | 7, 8, 11 | fvmpt 6191 | 1 ⊢ (𝐴 ∈ (-π[,]π) → (𝐾‘𝐴) = if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ifcif 4036 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 · cmul 9820 -cneg 10146 / cdiv 10563 2c2 10947 [,]cicc 12049 sincsin 14633 πcpi 14636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-1cn 9873 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 |
This theorem is referenced by: (None) |
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