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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 6229 | . . . 4 ⊢ (𝑦 = 𝐴 → (sin‘𝑦) = (sin‘𝐴)) | |
2 | 1 | neeq1d 2882 | . . 3 ⊢ (𝑦 = 𝐴 → ((sin‘𝑦) ≠ 0 ↔ (sin‘𝐴) ≠ 0)) |
3 | 2 | elrab 3396 | . 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0)) |
4 | fveq2 6229 | . . . 4 ⊢ (𝑥 = 𝐴 → (cos‘𝑥) = (cos‘𝐴)) | |
5 | fveq2 6229 | . . . 4 ⊢ (𝑥 = 𝐴 → (sin‘𝑥) = (sin‘𝐴)) | |
6 | 4, 5 | oveq12d 6708 | . . 3 ⊢ (𝑥 = 𝐴 → ((cos‘𝑥) / (sin‘𝑥)) = ((cos‘𝐴) / (sin‘𝐴))) |
7 | df-cot 42815 | . . 3 ⊢ cot = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ ((cos‘𝑥) / (sin‘𝑥))) | |
8 | ovex 6718 | . . 3 ⊢ ((cos‘𝐴) / (sin‘𝐴)) ∈ V | |
9 | 6, 7, 8 | fvmpt 6321 | . 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴))) |
10 | 3, 9 | sylbir 225 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (cot‘𝐴) = ((cos‘𝐴) / (sin‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 {crab 2945 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 0cc0 9974 / cdiv 10722 sincsin 14838 cosccos 14839 cotccot 42812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-iota 5889 df-fun 5928 df-fv 5934 df-ov 6693 df-cot 42815 |
This theorem is referenced by: cotcl 42821 recotcl 42824 reccot 42827 rectan 42828 cotsqcscsq 42831 |
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