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Definition df-tanh 45302
 Description: Define the hyperbolic tangent function (tanh). We define it this way for cmpt 5111, 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
StepHypRef Expression
1 ctanh 45299 . 2 class tanh
2 vx . . 3 setvar 𝑥
3 ccosh 45298 . . . . 5 class cosh
43ccnv 5519 . . . 4 class cosh
5 cc 10527 . . . . 5 class
6 cc0 10529 . . . . . 6 class 0
76csn 4525 . . . . 5 class {0}
85, 7cdif 3878 . . . 4 class (ℂ ∖ {0})
94, 8cima 5523 . . 3 class (cosh “ (ℂ ∖ {0}))
10 ci 10531 . . . . . 6 class i
112cv 1537 . . . . . 6 class 𝑥
12 cmul 10534 . . . . . 6 class ·
1310, 11, 12co 7136 . . . . 5 class (i · 𝑥)
14 ctan 15414 . . . . 5 class tan
1513, 14cfv 6325 . . . 4 class (tan‘(i · 𝑥))
16 cdiv 11289 . . . 4 class /
1715, 10, 16co 7136 . . 3 class ((tan‘(i · 𝑥)) / i)
182, 9, 17cmpt 5111 . 2 class (𝑥 ∈ (cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i))
191, 18wceq 1538 1 wff tanh = (𝑥 ∈ (cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i))
 Colors of variables: wff setvar class This definition is referenced by:  tanhval-named  45305
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