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Metamath Proof Explorer

Theorem asinsinb

Description: Relationship between sine and arcsine. (Contributed by Mario Carneiro, 2-Apr-2015)

Ref Expression
Assertion asinsinb 
|- ( ( A e. CC /\ B e. CC /\ ( Re ` B ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) ) -> ( ( arcsin ` A ) = B <-> ( sin ` B ) = A ) )

Proof

Step Hyp Ref Expression
1 sinasin 
 |-  ( A e. CC -> ( sin ` ( arcsin ` A ) ) = A )
2 1 3ad2ant1 
 |-  ( ( A e. CC /\ B e. CC /\ ( Re ` B ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) ) -> ( sin ` ( arcsin ` A ) ) = A )
3 fveqeq2 
 |-  ( ( arcsin ` A ) = B -> ( ( sin ` ( arcsin ` A ) ) = A <-> ( sin ` B ) = A ) )
4 2 3 syl5ibcom 
 |-  ( ( A e. CC /\ B e. CC /\ ( Re ` B ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) ) -> ( ( arcsin ` A ) = B -> ( sin ` B ) = A ) )
5 asinsin 
 |-  ( ( B e. CC /\ ( Re ` B ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) ) -> ( arcsin ` ( sin ` B ) ) = B )
6 5 3adant1 
 |-  ( ( A e. CC /\ B e. CC /\ ( Re ` B ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) ) -> ( arcsin ` ( sin ` B ) ) = B )
7 fveqeq2 
 |-  ( ( sin ` B ) = A -> ( ( arcsin ` ( sin ` B ) ) = B <-> ( arcsin ` A ) = B ) )
8 6 7 syl5ibcom 
 |-  ( ( A e. CC /\ B e. CC /\ ( Re ` B ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) ) -> ( ( sin ` B ) = A -> ( arcsin ` A ) = B ) )
9 4 8 impbid 
 |-  ( ( A e. CC /\ B e. CC /\ ( Re ` B ) e. ( -u ( _pi / 2 ) (,) ( _pi / 2 ) ) ) -> ( ( arcsin ` A ) = B <-> ( sin ` B ) = A ) )