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Metamath Proof Explorer

Theorem resinhcl

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

Ref Expression
Assertion resinhcl 
|- ( A e. RR -> ( ( sin ` ( _i x. A ) ) / _i ) e. RR )

Proof

Step Hyp Ref Expression
1 recn 
 |-  ( A e. RR -> A e. CC )
2 sinhval 
 |-  ( A e. CC -> ( ( sin ` ( _i x. A ) ) / _i ) = ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) )
3 1 2 syl 
 |-  ( A e. RR -> ( ( sin ` ( _i x. A ) ) / _i ) = ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) )
4 reefcl 
 |-  ( A e. RR -> ( exp ` A ) e. RR )
5 renegcl 
 |-  ( A e. RR -> -u A e. RR )
6 5 reefcld 
 |-  ( A e. RR -> ( exp ` -u A ) e. RR )
7 4 6 resubcld 
 |-  ( A e. RR -> ( ( exp ` A ) - ( exp ` -u A ) ) e. RR )
8 7 rehalfcld 
 |-  ( A e. RR -> ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) e. RR )
9 3 8 eqeltrd 
 |-  ( A e. RR -> ( ( sin ` ( _i x. A ) ) / _i ) e. RR )