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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 6882 | . . . 4 β’ (π¦ = π΄ β (sinβπ¦) = (sinβπ΄)) | |
2 | 1 | neeq1d 2992 | . . 3 β’ (π¦ = π΄ β ((sinβπ¦) β 0 β (sinβπ΄) β 0)) |
3 | 2 | elrab 3676 | . 2 β’ (π΄ β {π¦ β β β£ (sinβπ¦) β 0} β (π΄ β β β§ (sinβπ΄) β 0)) |
4 | fveq2 6882 | . . . 4 β’ (π₯ = π΄ β (sinβπ₯) = (sinβπ΄)) | |
5 | 4 | oveq2d 7418 | . . 3 β’ (π₯ = π΄ β (1 / (sinβπ₯)) = (1 / (sinβπ΄))) |
6 | df-csc 48038 | . . 3 β’ csc = (π₯ β {π¦ β β β£ (sinβπ¦) β 0} β¦ (1 / (sinβπ₯))) | |
7 | ovex 7435 | . . 3 β’ (1 / (sinβπ΄)) β V | |
8 | 5, 6, 7 | fvmpt 6989 | . 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 1533 β wcel 2098 β wne 2932 {crab 3424 βcfv 6534 (class class class)co 7402 βcc 11105 0cc0 11107 1c1 11108 / cdiv 11870 sincsin 16009 cscccsc 48035 |
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-csc 48038 |
This theorem is referenced by: csccl 48044 recsccl 48047 reccsc 48050 cotsqcscsq 48055 |
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