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Mirrors > Home > MPE Home > Th. List > Mathboxes > cscval | Structured version Visualization version GIF version |
Description: Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.) |
Ref | Expression |
---|---|
cscval | ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) = (1 / (sin‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6663 | . . . 4 ⊢ (𝑦 = 𝐴 → (sin‘𝑦) = (sin‘𝐴)) | |
2 | 1 | neeq1d 3072 | . . 3 ⊢ (𝑦 = 𝐴 → ((sin‘𝑦) ≠ 0 ↔ (sin‘𝐴) ≠ 0)) |
3 | 2 | elrab 3677 | . 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0)) |
4 | fveq2 6663 | . . . 4 ⊢ (𝑥 = 𝐴 → (sin‘𝑥) = (sin‘𝐴)) | |
5 | 4 | oveq2d 7161 | . . 3 ⊢ (𝑥 = 𝐴 → (1 / (sin‘𝑥)) = (1 / (sin‘𝐴))) |
6 | df-csc 44772 | . . 3 ⊢ csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥))) | |
7 | ovex 7178 | . . 3 ⊢ (1 / (sin‘𝐴)) ∈ V | |
8 | 5, 6, 7 | fvmpt 6761 | . 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} → (csc‘𝐴) = (1 / (sin‘𝐴))) |
9 | 3, 8 | sylbir 236 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) = (1 / (sin‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 {crab 3139 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 / cdiv 11285 sincsin 15405 cscccsc 44769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-csc 44772 |
This theorem is referenced by: csccl 44778 recsccl 44781 reccsc 44784 cotsqcscsq 44789 |
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