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Theorem tanhval-named 47871
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.)
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 7420 . . . 4 (π‘₯ = 𝐴 β†’ (i Β· π‘₯) = (i Β· 𝐴))
21fveq2d 6895 . . 3 (π‘₯ = 𝐴 β†’ (tanβ€˜(i Β· π‘₯)) = (tanβ€˜(i Β· 𝐴)))
32oveq1d 7427 . 2 (π‘₯ = 𝐴 β†’ ((tanβ€˜(i Β· π‘₯)) / i) = ((tanβ€˜(i Β· 𝐴)) / i))
4 df-tanh 47868 . 2 tanh = (π‘₯ ∈ (β—‘cosh β€œ (β„‚ βˆ– {0})) ↦ ((tanβ€˜(i Β· π‘₯)) / i))
5 ovex 7445 . 2 ((tanβ€˜(i Β· 𝐴)) / i) ∈ V
63, 4, 5fvmpt 6998 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 1540   ∈ wcel 2105   βˆ– cdif 3945  {csn 4628  β—‘ccnv 5675   β€œ cima 5679  β€˜cfv 6543  (class class class)co 7412  β„‚cc 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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