Metamath Proof Explorer

Theorem rpcoshcl

Description: The hyperbolic cosine of a real number is a positive real. (Contributed by Mario Carneiro, 4-Apr-2015)

Ref Expression
Assertion rpcoshcl 
|- ( A e. RR -> ( cos ` ( _i x. A ) ) e. RR+ )

Proof

Step Hyp Ref Expression
1 recn 
 |-  ( A e. RR -> A e. CC )
2 coshval 
 |-  ( A e. CC -> ( cos ` ( _i x. A ) ) = ( ( ( exp ` A ) + ( exp ` -u A ) ) / 2 ) )
3 1 2 syl 
 |-  ( A e. RR -> ( cos ` ( _i x. A ) ) = ( ( ( exp ` A ) + ( exp ` -u A ) ) / 2 ) )
4 rpefcl 
 |-  ( A e. RR -> ( exp ` A ) e. RR+ )
5 renegcl 
 |-  ( A e. RR -> -u A e. RR )
6 5 rpefcld 
 |-  ( A e. RR -> ( exp ` -u A ) e. RR+ )
7 4 6 rpaddcld 
 |-  ( A e. RR -> ( ( exp ` A ) + ( exp ` -u A ) ) e. RR+ )
8 7 rphalfcld 
 |-  ( A e. RR -> ( ( ( exp ` A ) + ( exp ` -u A ) ) / 2 ) e. RR+ )
9 3 8 eqeltrd 
 |-  ( A e. RR -> ( cos ` ( _i x. A ) ) e. RR+ )