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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 2775 | . . 3 ⊢ (𝑠 = 𝐴 → (𝑠 = 0 ↔ 𝐴 = 0)) | |
2 | id 22 | . . . 4 ⊢ (𝑠 = 𝐴 → 𝑠 = 𝐴) | |
3 | oveq1 6801 | . . . . . 6 ⊢ (𝑠 = 𝐴 → (𝑠 / 2) = (𝐴 / 2)) | |
4 | 3 | fveq2d 6337 | . . . . 5 ⊢ (𝑠 = 𝐴 → (sin‘(𝑠 / 2)) = (sin‘(𝐴 / 2))) |
5 | 4 | oveq2d 6810 | . . . 4 ⊢ (𝑠 = 𝐴 → (2 · (sin‘(𝑠 / 2))) = (2 · (sin‘(𝐴 / 2)))) |
6 | 2, 5 | oveq12d 6812 | . . 3 ⊢ (𝑠 = 𝐴 → (𝑠 / (2 · (sin‘(𝑠 / 2)))) = (𝐴 / (2 · (sin‘(𝐴 / 2))))) |
7 | 1, 6 | ifbieq2d 4251 | . 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 10238 | . . 3 ⊢ 1 ∈ V | |
10 | ovex 6824 | . . 3 ⊢ (𝐴 / (2 · (sin‘(𝐴 / 2)))) ∈ V | |
11 | 9, 10 | ifex 4296 | . 2 ⊢ if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2))))) ∈ V |
12 | 7, 8, 11 | fvmpt 6425 | 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 1631 ∈ wcel 2145 ifcif 4226 ↦ cmpt 4864 ‘cfv 6032 (class class class)co 6794 0cc0 10139 1c1 10140 · cmul 10144 -cneg 10470 / cdiv 10887 2c2 11273 [,]cicc 12384 sincsin 15001 πcpi 15004 |
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 4916 ax-nul 4924 ax-pr 5035 ax-1cn 10197 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 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 3589 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-nul 4065 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-iota 5995 df-fun 6034 df-fv 6040 df-ov 6797 |
This theorem is referenced by: (None) |
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