Theorem tanhval-named 48357

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 48354. (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 7427	. . . 4 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
2	1	fveq2d 6900	. . 3 ⊢ (𝑥 = 𝐴 → (tan‘(i · 𝑥)) = (tan‘(i · 𝐴)))
3	2	oveq1d 7434	. 2 ⊢ (𝑥 = 𝐴 → ((tan‘(i · 𝑥)) / i) = ((tan‘(i · 𝐴)) / i))
4		df-tanh 48354	. 2 ⊢ tanh = (𝑥 ∈ (◡cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i))
5		ovex 7452	. 2 ⊢ ((tan‘(i · 𝐴)) / i) ∈ V
6	3, 4, 5	fvmpt 7004	1 ⊢ (𝐴 ∈ (◡cosh “ (ℂ ∖ {0})) → (tanh‘𝐴) = ((tan‘(i · 𝐴)) / i))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   ∖ cdif 3941  {csn 4630  ^◡ccnv 5677   “ cima 5681  ‘cfv 6549  (class class class)co 7419  ℂcc 11143  0cc0 11145  ici 11147   · cmul 11150   / cdiv 11908  tanctan 16050  coshccosh 48350  tanhctanh 48351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-tanh 48354

This theorem is referenced by: (None)