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Mathbox for David A. Wheeler |
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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 | β’ ((π΄ β β β§ (sinβπ΄) β 0) β (cscβπ΄) = (1 / (sinβπ΄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6897 | . . . 4 β’ (π¦ = π΄ β (sinβπ¦) = (sinβπ΄)) | |
2 | 1 | neeq1d 2997 | . . 3 β’ (π¦ = π΄ β ((sinβπ¦) β 0 β (sinβπ΄) β 0)) |
3 | 2 | elrab 3682 | . 2 β’ (π΄ β {π¦ β β β£ (sinβπ¦) β 0} β (π΄ β β β§ (sinβπ΄) β 0)) |
4 | fveq2 6897 | . . . 4 β’ (π₯ = π΄ β (sinβπ₯) = (sinβπ΄)) | |
5 | 4 | oveq2d 7436 | . . 3 β’ (π₯ = π΄ β (1 / (sinβπ₯)) = (1 / (sinβπ΄))) |
6 | df-csc 48176 | . . 3 β’ csc = (π₯ β {π¦ β β β£ (sinβπ¦) β 0} β¦ (1 / (sinβπ₯))) | |
7 | ovex 7453 | . . 3 β’ (1 / (sinβπ΄)) β V | |
8 | 5, 6, 7 | fvmpt 7005 | . 2 β’ (π΄ β {π¦ β β β£ (sinβπ¦) β 0} β (cscβπ΄) = (1 / (sinβπ΄))) |
9 | 3, 8 | sylbir 234 | 1 β’ ((π΄ β β β§ (sinβπ΄) β 0) β (cscβπ΄) = (1 / (sinβπ΄))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wne 2937 {crab 3429 βcfv 6548 (class class class)co 7420 βcc 11137 0cc0 11139 1c1 11140 / cdiv 11902 sincsin 16040 cscccsc 48173 |
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-csc 48176 |
This theorem is referenced by: csccl 48182 recsccl 48185 reccsc 48188 cotsqcscsq 48193 |
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