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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 2828 | . . 3 ⊢ (𝑠 = 𝐴 → (𝑠 = 0 ↔ 𝐴 = 0)) | |
2 | id 22 | . . . 4 ⊢ (𝑠 = 𝐴 → 𝑠 = 𝐴) | |
3 | fvoveq1 7182 | . . . . 5 ⊢ (𝑠 = 𝐴 → (sin‘(𝑠 / 2)) = (sin‘(𝐴 / 2))) | |
4 | 3 | oveq2d 7175 | . . . 4 ⊢ (𝑠 = 𝐴 → (2 · (sin‘(𝑠 / 2))) = (2 · (sin‘(𝐴 / 2)))) |
5 | 2, 4 | oveq12d 7177 | . . 3 ⊢ (𝑠 = 𝐴 → (𝑠 / (2 · (sin‘(𝑠 / 2)))) = (𝐴 / (2 · (sin‘(𝐴 / 2))))) |
6 | 1, 5 | ifbieq2d 4495 | . 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 10640 | . . 3 ⊢ 1 ∈ V | |
9 | ovex 7192 | . . 3 ⊢ (𝐴 / (2 · (sin‘(𝐴 / 2)))) ∈ V | |
10 | 8, 9 | ifex 4518 | . 2 ⊢ if(𝐴 = 0, 1, (𝐴 / (2 · (sin‘(𝐴 / 2))))) ∈ V |
11 | 6, 7, 10 | fvmpt 6771 | 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 1536 ∈ wcel 2113 ifcif 4470 ↦ cmpt 5149 ‘cfv 6358 (class class class)co 7159 0cc0 10540 1c1 10541 · cmul 10545 -cneg 10874 / cdiv 11300 2c2 11695 [,]cicc 12744 sincsin 15420 πcpi 15423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 ax-1cn 10598 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 |
This theorem is referenced by: (None) |
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