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Definition df-tanh 46108
Description: Define the hyperbolic tangent function (tanh). We define it this way for cmpt 5135, which requires the form (𝑥𝐴𝐵). (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 46105 . 2 class tanh
2 vx . . 3 setvar 𝑥
3 ccosh 46104 . . . . 5 class cosh
43ccnv 5550 . . . 4 class cosh
5 cc 10727 . . . . 5 class
6 cc0 10729 . . . . . 6 class 0
76csn 4541 . . . . 5 class {0}
85, 7cdif 3863 . . . 4 class (ℂ ∖ {0})
94, 8cima 5554 . . 3 class (cosh “ (ℂ ∖ {0}))
10 ci 10731 . . . . . 6 class i
112cv 1542 . . . . . 6 class 𝑥
12 cmul 10734 . . . . . 6 class ·
1310, 11, 12co 7213 . . . . 5 class (i · 𝑥)
14 ctan 15627 . . . . 5 class tan
1513, 14cfv 6380 . . . 4 class (tan‘(i · 𝑥))
16 cdiv 11489 . . . 4 class /
1715, 10, 16co 7213 . . 3 class ((tan‘(i · 𝑥)) / i)
182, 9, 17cmpt 5135 . 2 class (𝑥 ∈ (cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i))
191, 18wceq 1543 1 wff tanh = (𝑥 ∈ (cosh “ (ℂ ∖ {0})) ↦ ((tan‘(i · 𝑥)) / i))
Colors of variables: wff setvar class
This definition is referenced by:  tanhval-named  46111
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