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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 6766 | . . . 4 ⊢ (𝑦 = 𝐴 → (cos‘𝑦) = (cos‘𝐴)) | |
2 | 1 | neeq1d 3003 | . . 3 ⊢ (𝑦 = 𝐴 → ((cos‘𝑦) ≠ 0 ↔ (cos‘𝐴) ≠ 0)) |
3 | 2 | elrab 3623 | . 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0)) |
4 | fveq2 6766 | . . . 4 ⊢ (𝑥 = 𝐴 → (cos‘𝑥) = (cos‘𝐴)) | |
5 | 4 | oveq2d 7283 | . . 3 ⊢ (𝑥 = 𝐴 → (1 / (cos‘𝑥)) = (1 / (cos‘𝐴))) |
6 | df-sec 46424 | . . 3 ⊢ sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥))) | |
7 | ovex 7300 | . . 3 ⊢ (1 / (cos‘𝐴)) ∈ V | |
8 | 5, 6, 7 | fvmpt 6867 | . 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 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 {crab 3068 ‘cfv 6426 (class class class)co 7267 ℂcc 10879 0cc0 10881 1c1 10882 / cdiv 11642 cosccos 15784 seccsec 46421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pr 5350 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-iota 6384 df-fun 6428 df-fv 6434 df-ov 7270 df-sec 46424 |
This theorem is referenced by: seccl 46430 reseccl 46433 recsec 46436 sec0 46440 onetansqsecsq 46441 |
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