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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 6882 | . . . 4 ⊢ (𝑦 = 𝐴 → (cos‘𝑦) = (cos‘𝐴)) | |
2 | 1 | neeq1d 2992 | . . 3 ⊢ (𝑦 = 𝐴 → ((cos‘𝑦) ≠ 0 ↔ (cos‘𝐴) ≠ 0)) |
3 | 2 | elrab 3676 | . 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (cos‘𝐴) ≠ 0)) |
4 | fveq2 6882 | . . . 4 ⊢ (𝑥 = 𝐴 → (cos‘𝑥) = (cos‘𝐴)) | |
5 | 4 | oveq2d 7418 | . . 3 ⊢ (𝑥 = 𝐴 → (1 / (cos‘𝑥)) = (1 / (cos‘𝐴))) |
6 | df-sec 48037 | . . 3 ⊢ sec = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (cos‘𝑦) ≠ 0} ↦ (1 / (cos‘𝑥))) | |
7 | ovex 7435 | . . 3 ⊢ (1 / (cos‘𝐴)) ∈ V | |
8 | 5, 6, 7 | fvmpt 6989 | . 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 1533 ∈ wcel 2098 ≠ wne 2932 {crab 3424 ‘cfv 6534 (class class class)co 7402 ℂcc 11105 0cc0 11107 1c1 11108 / cdiv 11870 cosccos 16010 seccsec 48034 |
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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-iota 6486 df-fun 6536 df-fv 6542 df-ov 7405 df-sec 48037 |
This theorem is referenced by: seccl 48043 reseccl 48046 recsec 48049 sec0 48053 onetansqsecsq 48054 |
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