Definition df-tanh 42476

Description: Define the hyperbolic tangent function (tanh). We define it this way for cmpt 4729, which requires the form ( x e. A |-> B ). (Contributed by David A. Wheeler, 10-May-2015.)

Assertion

Ref	Expression
df-tanh	|- tanh = ( x e. ( `'cosh " ( CC \ { 0 } ) ) |-> ( ( tan ` ( _i x. x ) ) / _i ) )

Detailed syntax breakdown of Definition df-tanh

Step	Hyp	Ref	Expression
1		ctanh 42473	. 2 class tanh
2		vx	. . 3 setvar x
3		ccosh 42472	. . . . 5 class cosh
4	3	ccnv 5113	. . . 4 class `'cosh
5		cc 9934	. . . . 5 class CC
6		cc0 9936	. . . . . 6 class 0
7	6	csn 4177	. . . . 5 class { 0 }
8	5, 7	cdif 3571	. . . 4 class ( CC \ { 0 } )
9	4, 8	cima 5117	. . 3 class ( `'cosh " ( CC \ { 0 } ) )
10		ci 9938	. . . . . 6 class _i
11	2	cv 1482	. . . . . 6 class x
12		cmul 9941	. . . . . 6 class x.
13	10, 11, 12	co 6650	. . . . 5 class ( _i x. x )
14		ctan 14796	. . . . 5 class tan
15	13, 14	cfv 5888	. . . 4 class ( tan ` ( _i x. x ) )
16		cdiv 10684	. . . 4 class /
17	15, 10, 16	co 6650	. . 3 class ( ( tan ` ( _i x. x ) ) / _i )
18	2, 9, 17	cmpt 4729	. 2 class ( x e. ( `'cosh " ( CC \ { 0 } ) ) |-> ( ( tan ` ( _i x. x ) ) / _i ) )
19	1, 18	wceq 1483	1 wff tanh = ( x e. ( `'cosh " ( CC \ { 0 } ) ) |-> ( ( tan ` ( _i x. x ) ) / _i ) )

This definition is referenced by:  tanhval-named  42479