Definition df-tanh 51118

Description: Define the hyperbolic tangent function (tanh). We define it this way for cmpt 5190, which requires the form (𝑥 ∈ 𝐴 ↦ 𝐵). (Contributed by David A. Wheeler, 10-May-2015.) (Revised by Jon Pennant, 22-Oct-2026.)

Assertion

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

Detailed syntax breakdown of Definition df-tanh

Step	Hyp	Ref	Expression
1		ctanh 51115	. 2 class tanh
2		vx	. . 3 setvar 𝑥
3		ccosh 51114	. . . . 5 class cosh
4	3	ccnv 5655	. . . 4 class ◡cosh
5		cc 11228	. . . . 5 class ℂ
6		cc0 11230	. . . . . 6 class 0
7	6	csn 4588	. . . . 5 class {0}
8	5, 7	cdif 4048	. . . 4 class (ℂ ∖ {0})
9	4, 8	cima 5659	. . 3 class (◡cosh “ (ℂ ∖ {0}))
10	2	cv 1569	. . . . 5 class 𝑥
11		csinh 51113	. . . . 5 class sinh
12	10, 11	cfv 6561	. . . 4 class (sinh‘𝑥)
13	10, 3	cfv 6561	. . . 4 class (cosh‘𝑥)
14		cdiv 12003	. . . 4 class /
15	12, 13, 14	co 7446	. . 3 class ((sinh‘𝑥) / (cosh‘𝑥))
16	2, 9, 15	cmpt 5190	. 2 class (𝑥 ∈ (◡cosh “ (ℂ ∖ {0})) ↦ ((sinh‘𝑥) / (cosh‘𝑥)))
17	1, 16	wceq 1570	1 wff tanh = (𝑥 ∈ (◡cosh “ (ℂ ∖ {0})) ↦ ((sinh‘𝑥) / (cosh‘𝑥)))

This definition is used by:  tanhval  51131  tanhf  51133