Theorem tanhval-named 47783

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 47780. (Contributed by David A. Wheeler, 10-May-2015.)

Assertion

Ref	Expression
tanhval-named	⊢ (𝐴 ∈ (◡cosh “ (ℂ ∖ {0})) → (tanh‘𝐴) = ((tan‘(i · 𝐴)) / i))

Proof of Theorem tanhval-named

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7417	. . . 4 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
2	1	fveq2d 6896	. . 3 ⊢ (𝑥 = 𝐴 → (tan‘(i · 𝑥)) = (tan‘(i · 𝐴)))
3	2	oveq1d 7424	. 2 ⊢ (𝑥 = 𝐴 → ((tan‘(i · 𝑥)) / i) = ((tan‘(i · 𝐴)) / i))
4		df-tanh 47780	. 2 ⊢ tanh = (𝑥 ∈ (◡cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i))
5		ovex 7442	. 2 ⊢ ((tan‘(i · 𝐴)) / i) ∈ V
6	3, 4, 5	fvmpt 6999	1 ⊢ (𝐴 ∈ (◡cosh “ (ℂ ∖ {0})) → (tanh‘𝐴) = ((tan‘(i · 𝐴)) / i))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107   ∖ cdif 3946  {csn 4629  ^◡ccnv 5676   “ cima 5680  ‘cfv 6544  (class class class)co 7409  ℂcc 11108  0cc0 11110  ici 11112   · cmul 11115   / cdiv 11871  tanctan 16009  coshccosh 47776  tanhctanh 47777

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 5300  ax-nul 5307  ax-pr 5428

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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-tanh 47780

This theorem is referenced by: (None)