Metamath Proof Explorer


Definition df-sinh

Description: Define the hyperbolic sine function (sinh). We define it this way for cmpt , which requires the form ( x e. A |-> B ) . See sinhval-named for a simple way to evaluate it. We define this function by dividing by _i , which uses fewer operations than many conventional definitions (and thus is more convenient to use in set.mm). See sinh-conventional for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015)

Ref Expression
Assertion df-sinh
|- sinh = ( x e. CC |-> ( ( sin ` ( _i x. x ) ) / _i ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 csinh
 |-  sinh
1 vx
 |-  x
2 cc
 |-  CC
3 csin
 |-  sin
4 ci
 |-  _i
5 cmul
 |-  x.
6 1 cv
 |-  x
7 4 6 5 co
 |-  ( _i x. x )
8 7 3 cfv
 |-  ( sin ` ( _i x. x ) )
9 cdiv
 |-  /
10 8 4 9 co
 |-  ( ( sin ` ( _i x. x ) ) / _i )
11 1 2 10 cmpt
 |-  ( x e. CC |-> ( ( sin ` ( _i x. x ) ) / _i ) )
12 0 11 wceq
 |-  sinh = ( x e. CC |-> ( ( sin ` ( _i x. x ) ) / _i ) )