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Mirrors > Home > MPE Home > Th. List > Mathboxes > tanhval-named | Structured version Visualization version GIF version |
Description: Value of the named tanh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-tanh 47266. (Contributed by David A. Wheeler, 10-May-2015.) |
Ref | Expression |
---|---|
tanhval-named | β’ (π΄ β (β‘cosh β (β β {0})) β (tanhβπ΄) = ((tanβ(i Β· π΄)) / i)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7366 | . . . 4 β’ (π₯ = π΄ β (i Β· π₯) = (i Β· π΄)) | |
2 | 1 | fveq2d 6847 | . . 3 β’ (π₯ = π΄ β (tanβ(i Β· π₯)) = (tanβ(i Β· π΄))) |
3 | 2 | oveq1d 7373 | . 2 β’ (π₯ = π΄ β ((tanβ(i Β· π₯)) / i) = ((tanβ(i Β· π΄)) / i)) |
4 | df-tanh 47266 | . 2 β’ tanh = (π₯ β (β‘cosh β (β β {0})) β¦ ((tanβ(i Β· π₯)) / i)) | |
5 | ovex 7391 | . 2 β’ ((tanβ(i Β· π΄)) / i) β V | |
6 | 3, 4, 5 | fvmpt 6949 | 1 β’ (π΄ β (β‘cosh β (β β {0})) β (tanhβπ΄) = ((tanβ(i Β· π΄)) / i)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β cdif 3908 {csn 4587 β‘ccnv 5633 β cima 5637 βcfv 6497 (class class class)co 7358 βcc 11054 0cc0 11056 ici 11058 Β· cmul 11061 / cdiv 11817 tanctan 15953 coshccosh 47262 tanhctanh 47263 |
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-tanh 47266 |
This theorem is referenced by: (None) |
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