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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 6843 | . . . 4 β’ (π¦ = π΄ β (sinβπ¦) = (sinβπ΄)) | |
2 | 1 | neeq1d 3000 | . . 3 β’ (π¦ = π΄ β ((sinβπ¦) β 0 β (sinβπ΄) β 0)) |
3 | 2 | elrab 3646 | . 2 β’ (π΄ β {π¦ β β β£ (sinβπ¦) β 0} β (π΄ β β β§ (sinβπ΄) β 0)) |
4 | fveq2 6843 | . . . 4 β’ (π₯ = π΄ β (sinβπ₯) = (sinβπ΄)) | |
5 | 4 | oveq2d 7374 | . . 3 β’ (π₯ = π΄ β (1 / (sinβπ₯)) = (1 / (sinβπ΄))) |
6 | df-csc 47276 | . . 3 β’ csc = (π₯ β {π¦ β β β£ (sinβπ¦) β 0} β¦ (1 / (sinβπ₯))) | |
7 | ovex 7391 | . . 3 β’ (1 / (sinβπ΄)) β V | |
8 | 5, 6, 7 | fvmpt 6949 | . 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 397 = wceq 1542 β wcel 2107 β wne 2940 {crab 3406 βcfv 6497 (class class class)co 7358 βcc 11054 0cc0 11056 1c1 11057 / cdiv 11817 sincsin 15951 cscccsc 47273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-csc 47276 |
This theorem is referenced by: csccl 47282 recsccl 47285 reccsc 47288 cotsqcscsq 47293 |
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