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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 6493 | . . . 4 ⊢ (𝑦 = 𝐴 → (sin‘𝑦) = (sin‘𝐴)) | |
2 | 1 | neeq1d 3020 | . . 3 ⊢ (𝑦 = 𝐴 → ((sin‘𝑦) ≠ 0 ↔ (sin‘𝐴) ≠ 0)) |
3 | 2 | elrab 3589 | . 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↔ (𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0)) |
4 | fveq2 6493 | . . . 4 ⊢ (𝑥 = 𝐴 → (sin‘𝑥) = (sin‘𝐴)) | |
5 | 4 | oveq2d 6986 | . . 3 ⊢ (𝑥 = 𝐴 → (1 / (sin‘𝑥)) = (1 / (sin‘𝐴))) |
6 | df-csc 44151 | . . 3 ⊢ csc = (𝑥 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} ↦ (1 / (sin‘𝑥))) | |
7 | ovex 7002 | . . 3 ⊢ (1 / (sin‘𝐴)) ∈ V | |
8 | 5, 6, 7 | fvmpt 6589 | . 2 ⊢ (𝐴 ∈ {𝑦 ∈ ℂ ∣ (sin‘𝑦) ≠ 0} → (csc‘𝐴) = (1 / (sin‘𝐴))) |
9 | 3, 8 | sylbir 227 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (sin‘𝐴) ≠ 0) → (csc‘𝐴) = (1 / (sin‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 {crab 3086 ‘cfv 6182 (class class class)co 6970 ℂcc 10325 0cc0 10327 1c1 10328 / cdiv 11090 sincsin 15267 cscccsc 44148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pr 5180 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3678 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-iota 6146 df-fun 6184 df-fv 6190 df-ov 6973 df-csc 44151 |
This theorem is referenced by: csccl 44157 recsccl 44160 reccsc 44163 cotsqcscsq 44168 |
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