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Theorem tanhval-named 47870
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 47867. (Contributed by David A. Wheeler, 10-May-2015.)
Assertion
Ref Expression
tanhval-named (𝐴 ∈ (β—‘cosh β€œ (β„‚ βˆ– {0})) β†’ (tanhβ€˜π΄) = ((tanβ€˜(i Β· 𝐴)) / i))

Proof of Theorem tanhval-named
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 oveq2 7419 . . . 4 (π‘₯ = 𝐴 β†’ (i Β· π‘₯) = (i Β· 𝐴))
21fveq2d 6894 . . 3 (π‘₯ = 𝐴 β†’ (tanβ€˜(i Β· π‘₯)) = (tanβ€˜(i Β· 𝐴)))
32oveq1d 7426 . 2 (π‘₯ = 𝐴 β†’ ((tanβ€˜(i Β· π‘₯)) / i) = ((tanβ€˜(i Β· 𝐴)) / i))
4 df-tanh 47867 . 2 tanh = (π‘₯ ∈ (β—‘cosh β€œ (β„‚ βˆ– {0})) ↦ ((tanβ€˜(i Β· π‘₯)) / i))
5 ovex 7444 . 2 ((tanβ€˜(i Β· 𝐴)) / i) ∈ V
63, 4, 5fvmpt 6997 1 (𝐴 ∈ (β—‘cosh β€œ (β„‚ βˆ– {0})) β†’ (tanhβ€˜π΄) = ((tanβ€˜(i Β· 𝐴)) / i))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1539   ∈ wcel 2104   βˆ– cdif 3944  {csn 4627  β—‘ccnv 5674   β€œ cima 5678  β€˜cfv 6542  (class class class)co 7411  β„‚cc 11110  0cc0 11112  ici 11114   Β· cmul 11117   / cdiv 11875  tanctan 16013  coshccosh 47863  tanhctanh 47864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-tanh 47867
This theorem is referenced by: (None)
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