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Definition df-tanh 47266
Description: Define the hyperbolic tangent function (tanh). We define it this way for cmpt 5189, which requires the form (π‘₯ ∈ 𝐴 ↦ 𝐡). (Contributed by David A. Wheeler, 10-May-2015.)
Assertion
Ref Expression
df-tanh tanh = (π‘₯ ∈ (β—‘cosh β€œ (β„‚ βˆ– {0})) ↦ ((tanβ€˜(i Β· π‘₯)) / i))

Detailed syntax breakdown of Definition df-tanh
StepHypRef Expression
1 ctanh 47263 . 2 class tanh
2 vx . . 3 setvar π‘₯
3 ccosh 47262 . . . . 5 class cosh
43ccnv 5633 . . . 4 class β—‘cosh
5 cc 11054 . . . . 5 class β„‚
6 cc0 11056 . . . . . 6 class 0
76csn 4587 . . . . 5 class {0}
85, 7cdif 3908 . . . 4 class (β„‚ βˆ– {0})
94, 8cima 5637 . . 3 class (β—‘cosh β€œ (β„‚ βˆ– {0}))
10 ci 11058 . . . . . 6 class i
112cv 1541 . . . . . 6 class π‘₯
12 cmul 11061 . . . . . 6 class Β·
1310, 11, 12co 7358 . . . . 5 class (i Β· π‘₯)
14 ctan 15953 . . . . 5 class tan
1513, 14cfv 6497 . . . 4 class (tanβ€˜(i Β· π‘₯))
16 cdiv 11817 . . . 4 class /
1715, 10, 16co 7358 . . 3 class ((tanβ€˜(i Β· π‘₯)) / i)
182, 9, 17cmpt 5189 . 2 class (π‘₯ ∈ (β—‘cosh β€œ (β„‚ βˆ– {0})) ↦ ((tanβ€˜(i Β· π‘₯)) / i))
191, 18wceq 1542 1 wff tanh = (π‘₯ ∈ (β—‘cosh β€œ (β„‚ βˆ– {0})) ↦ ((tanβ€˜(i Β· π‘₯)) / i))
Colors of variables: wff setvar class
This definition is referenced by:  tanhval-named  47269
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