Metamath Proof Explorer

Theorem acosrecl

Description: The arccosine function is real in its principal domain. (Contributed by Mario Carneiro, 2-Apr-2015)

Ref Expression
Assertion acosrecl 
|- ( A e. ( -u 1 [,] 1 ) -> ( arccos ` A ) e. RR )

Proof

Step Hyp Ref Expression
1 neg1rr 
 |-  -u 1 e. RR
2 1re 
 |-  1 e. RR
3 iccssre 
 |-  ( ( -u 1 e. RR /\ 1 e. RR ) -> ( -u 1 [,] 1 ) C_ RR )
4 1 2 3 mp2an 
 |-  ( -u 1 [,] 1 ) C_ RR
5 4 sseli 
 |-  ( A e. ( -u 1 [,] 1 ) -> A e. RR )
6 5 recnd 
 |-  ( A e. ( -u 1 [,] 1 ) -> A e. CC )
7 acosval 
 |-  ( A e. CC -> ( arccos ` A ) = ( ( _pi / 2 ) - ( arcsin ` A ) ) )
8 6 7 syl 
 |-  ( A e. ( -u 1 [,] 1 ) -> ( arccos ` A ) = ( ( _pi / 2 ) - ( arcsin ` A ) ) )
9 halfpire 
 |-  ( _pi / 2 ) e. RR
10 asinrecl 
 |-  ( A e. ( -u 1 [,] 1 ) -> ( arcsin ` A ) e. RR )
11 resubcl 
 |-  ( ( ( _pi / 2 ) e. RR /\ ( arcsin ` A ) e. RR ) -> ( ( _pi / 2 ) - ( arcsin ` A ) ) e. RR )
12 9 10 11 sylancr 
 |-  ( A e. ( -u 1 [,] 1 ) -> ( ( _pi / 2 ) - ( arcsin ` A ) ) e. RR )
13 8 12 eqeltrd 
 |-  ( A e. ( -u 1 [,] 1 ) -> ( arccos ` A ) e. RR )