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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 47868. (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 7420 | . . . 4 β’ (π₯ = π΄ β (i Β· π₯) = (i Β· π΄)) | |
2 | 1 | fveq2d 6895 | . . 3 β’ (π₯ = π΄ β (tanβ(i Β· π₯)) = (tanβ(i Β· π΄))) |
3 | 2 | oveq1d 7427 | . 2 β’ (π₯ = π΄ β ((tanβ(i Β· π₯)) / i) = ((tanβ(i Β· π΄)) / i)) |
4 | df-tanh 47868 | . 2 β’ tanh = (π₯ β (β‘cosh β (β β {0})) β¦ ((tanβ(i Β· π₯)) / i)) | |
5 | ovex 7445 | . 2 β’ ((tanβ(i Β· π΄)) / i) β V | |
6 | 3, 4, 5 | fvmpt 6998 | 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 1540 β wcel 2105 β cdif 3945 {csn 4628 β‘ccnv 5675 β cima 5679 βcfv 6543 (class class class)co 7412 βcc 11112 0cc0 11114 ici 11116 Β· cmul 11119 / cdiv 11876 tanctan 16014 coshccosh 47864 tanhctanh 47865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-tanh 47868 |
This theorem is referenced by: (None) |
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