Definition df-tanh 44828

Description: Define the hyperbolic tangent function (tanh). We define it this way for cmpt 5138, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). (Contributed by David A. Wheeler, 10-May-2015.)

Assertion

Ref	Expression
df-tanh	⊢ tanh = (𝑥 ∈ (◡cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i))

Detailed syntax breakdown of Definition df-tanh

Step	Hyp	Ref	Expression
1		ctanh 44825	. 2 class tanh
2		vx	. . 3 setvar 𝑥
3		ccosh 44824	. . . . 5 class cosh
4	3	ccnv 5548	. . . 4 class ◡cosh
5		cc 10529	. . . . 5 class ℂ
6		cc0 10531	. . . . . 6 class 0
7	6	csn 4560	. . . . 5 class {0}
8	5, 7	cdif 3932	. . . 4 class (ℂ ∖ {0})
9	4, 8	cima 5552	. . 3 class (◡cosh “ (ℂ ∖ {0}))
10		ci 10533	. . . . . 6 class i
11	2	cv 1532	. . . . . 6 class 𝑥
12		cmul 10536	. . . . . 6 class ·
13	10, 11, 12	co 7150	. . . . 5 class (i · 𝑥)
14		ctan 15413	. . . . 5 class tan
15	13, 14	cfv 6349	. . . 4 class (tan‘(i · 𝑥))
16		cdiv 11291	. . . 4 class /
17	15, 10, 16	co 7150	. . 3 class ((tan‘(i · 𝑥)) / i)
18	2, 9, 17	cmpt 5138	. 2 class (𝑥 ∈ (◡cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i))
19	1, 18	wceq 1533	1 wff tanh = (𝑥 ∈ (◡cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i))

This definition is referenced by:  tanhval-named  44831