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Mirrors > Home > MPE Home > Th. List > Mathboxes > secval | Structured version Visualization version GIF version |
Description: Value of the secant function. (Contributed by David A. Wheeler, 14-Mar-2014.) |
Ref | Expression |
---|---|
secval | ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (sec‘𝐴) = (1 / (cos‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6897 | . . . 4 ⊢ (𝑦 = 𝐴 → (cos‘𝑦) = (cos‘𝐴)) | |
2 | 1 | neeq1d 2997 | . . 3 ⊢ (𝑦 = 𝐴 → ((cos‘𝑦) ≠ 0 ↔ (cos‘𝐴) ≠ 0)) |
3 | 2 | elrab 3682 | . 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0)) |
4 | fveq2 6897 | . . . 4 ⊢ (𝑥 = 𝐴 → (cos‘𝑥) = (cos‘𝐴)) | |
5 | 4 | oveq2d 7436 | . . 3 ⊢ (𝑥 = 𝐴 → (1 / (cos‘𝑥)) = (1 / (cos‘𝐴))) |
6 | df-sec 48175 | . . 3 ⊢ sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥))) | |
7 | ovex 7453 | . . 3 ⊢ (1 / (cos‘𝐴)) ∈ V | |
8 | 5, 6, 7 | fvmpt 7005 | . 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} → (sec‘𝐴) = (1 / (cos‘𝐴))) |
9 | 3, 8 | sylbir 234 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0) → (sec‘𝐴) = (1 / (cos‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 {crab 3429 ‘cfv 6548 (class class class)co 7420 ℂcc 11137 0cc0 11139 1c1 11140 / cdiv 11902 cosccos 16041 seccsec 48172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-sec 48175 |
This theorem is referenced by: seccl 48181 reseccl 48184 recsec 48187 sec0 48191 onetansqsecsq 48192 |
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