Theorem tanhval-named 50435

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 50432. (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 7419	. . . 4 ⊢ (𝑥 = 𝐴 → (i · 𝑥) = (i · 𝐴))
2	1	fveq2d 6886	. . 3 ⊢ (𝑥 = 𝐴 → (tan‘(i · 𝑥)) = (tan‘(i · 𝐴)))
3	2	oveq1d 7426	. 2 ⊢ (𝑥 = 𝐴 → ((tan‘(i · 𝑥)) / i) = ((tan‘(i · 𝐴)) / i))
4		df-tanh 50432	. 2 ⊢ tanh = (𝑥 ∈ (◡cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i))
5		ovex 7444	. 2 ⊢ ((tan‘(i · 𝐴)) / i) ∈ V
6	3, 4, 5	fvmpt 6990	1 ⊢ (𝐴 ∈ (◡cosh “ (ℂ ∖ {0})) → (tanh‘𝐴) = ((tan‘(i · 𝐴)) / i))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149   ∖ cdif 3910  {csn 4594  ^◡ccnv 5661   “ cima 5665  ‘cfv 6537  (class class class)co 7411  ℂcc 11098  0cc0 11100  ici 11102   · cmul 11105   / cdiv 11871  tanctan 16119  coshccosh 50428  tanhctanh 50429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-tanh 50432

This theorem is referenced by: (None)