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Mirrors > Home > MPE Home > Th. List > Mathboxes > cotval | Structured version Visualization version GIF version |
Description: Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014.) |
Ref | Expression |
---|---|
cotval | ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6896 | . . . 4 ⊢ (𝑦 = 𝐴 → (sin‘𝑦) = (sin‘𝐴)) | |
2 | 1 | neeq1d 2989 | . . 3 ⊢ (𝑦 = 𝐴 → ((sin‘𝑦) ≠ 0 ↔ (sin‘𝐴) ≠ 0)) |
3 | 2 | elrab 3679 | . 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0)) |
4 | fveq2 6896 | . . . 4 ⊢ (𝑥 = 𝐴 → (cos‘𝑥) = (cos‘𝐴)) | |
5 | fveq2 6896 | . . . 4 ⊢ (𝑥 = 𝐴 → (sin‘𝑥) = (sin‘𝐴)) | |
6 | 4, 5 | oveq12d 7437 | . . 3 ⊢ (𝑥 = 𝐴 → ((cos‘𝑥) / (sin‘𝑥)) = ((cos‘𝐴) / (sin‘𝐴))) |
7 | df-cot 48360 | . . 3 ⊢ cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥))) | |
8 | ovex 7452 | . . 3 ⊢ ((cos‘𝐴) / (sin‘𝐴)) ∈ V | |
9 | 6, 7, 8 | fvmpt 7004 | . 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴))) |
10 | 3, 9 | sylbir 234 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 {crab 3418 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 0cc0 11140 / cdiv 11903 sincsin 16043 cosccos 16044 cotccot 48357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-ov 7422 df-cot 48360 |
This theorem is referenced by: cotcl 48366 recotcl 48369 reccot 48372 rectan 48373 cotsqcscsq 48376 |
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