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Metamath Proof Explorer

Theorem asinlem2

Description: The argument to the logarithm in df-asin has the property that replacing A with -u A in the expression gives the reciprocal. (Contributed by Mario Carneiro, 1-Apr-2015)

Ref Expression
Assertion asinlem2 
|- ( A e. CC -> ( ( ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) x. ( ( _i x. -u A ) + ( sqrt ` ( 1 - ( -u A ^ 2 ) ) ) ) ) = 1 )

Proof

Step Hyp Ref Expression
1 ax-icn 
 |-  _i e. CC
2 mulcl 
 |-  ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
3 1 2 mpan 
 |-  ( A e. CC -> ( _i x. A ) e. CC )
4 ax-1cn 
 |-  1 e. CC
5 sqcl 
 |-  ( A e. CC -> ( A ^ 2 ) e. CC )
6 subcl 
 |-  ( ( 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( 1 - ( A ^ 2 ) ) e. CC )
7 4 5 6 sylancr 
 |-  ( A e. CC -> ( 1 - ( A ^ 2 ) ) e. CC )
8 7 sqrtcld 
 |-  ( A e. CC -> ( sqrt ` ( 1 - ( A ^ 2 ) ) ) e. CC )
9 3 8 addcomd 
 |-  ( A e. CC -> ( ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) = ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) + ( _i x. A ) ) )
10 mulneg2 
 |-  ( ( _i e. CC /\ A e. CC ) -> ( _i x. -u A ) = -u ( _i x. A ) )
11 1 10 mpan 
 |-  ( A e. CC -> ( _i x. -u A ) = -u ( _i x. A ) )
12 sqneg 
 |-  ( A e. CC -> ( -u A ^ 2 ) = ( A ^ 2 ) )
13 12 oveq2d 
 |-  ( A e. CC -> ( 1 - ( -u A ^ 2 ) ) = ( 1 - ( A ^ 2 ) ) )
14 13 fveq2d 
 |-  ( A e. CC -> ( sqrt ` ( 1 - ( -u A ^ 2 ) ) ) = ( sqrt ` ( 1 - ( A ^ 2 ) ) ) )
15 11 14 oveq12d 
 |-  ( A e. CC -> ( ( _i x. -u A ) + ( sqrt ` ( 1 - ( -u A ^ 2 ) ) ) ) = ( -u ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) )
16 3 negcld 
 |-  ( A e. CC -> -u ( _i x. A ) e. CC )
17 16 8 addcomd 
 |-  ( A e. CC -> ( -u ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) = ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) + -u ( _i x. A ) ) )
18 8 3 negsubd 
 |-  ( A e. CC -> ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) + -u ( _i x. A ) ) = ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) - ( _i x. A ) ) )
19 15 17 18 3eqtrd 
 |-  ( A e. CC -> ( ( _i x. -u A ) + ( sqrt ` ( 1 - ( -u A ^ 2 ) ) ) ) = ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) - ( _i x. A ) ) )
20 9 19 oveq12d 
 |-  ( A e. CC -> ( ( ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) x. ( ( _i x. -u A ) + ( sqrt ` ( 1 - ( -u A ^ 2 ) ) ) ) ) = ( ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) + ( _i x. A ) ) x. ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) - ( _i x. A ) ) ) )
21 7 sqsqrtd 
 |-  ( A e. CC -> ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) = ( 1 - ( A ^ 2 ) ) )
22 sqmul 
 |-  ( ( _i e. CC /\ A e. CC ) -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
23 1 22 mpan 
 |-  ( A e. CC -> ( ( _i x. A ) ^ 2 ) = ( ( _i ^ 2 ) x. ( A ^ 2 ) ) )
24 i2 
 |-  ( _i ^ 2 ) = -u 1
25 24 oveq1i 
 |-  ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = ( -u 1 x. ( A ^ 2 ) )
26 5 mulm1d 
 |-  ( A e. CC -> ( -u 1 x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
27 25 26 eqtrid 
 |-  ( A e. CC -> ( ( _i ^ 2 ) x. ( A ^ 2 ) ) = -u ( A ^ 2 ) )
28 23 27 eqtrd 
 |-  ( A e. CC -> ( ( _i x. A ) ^ 2 ) = -u ( A ^ 2 ) )
29 21 28 oveq12d 
 |-  ( A e. CC -> ( ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) - ( ( _i x. A ) ^ 2 ) ) = ( ( 1 - ( A ^ 2 ) ) - -u ( A ^ 2 ) ) )
30 subsq 
 |-  ( ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) e. CC /\ ( _i x. A ) e. CC ) -> ( ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) - ( ( _i x. A ) ^ 2 ) ) = ( ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) + ( _i x. A ) ) x. ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) - ( _i x. A ) ) ) )
31 8 3 30 syl2anc 
 |-  ( A e. CC -> ( ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ^ 2 ) - ( ( _i x. A ) ^ 2 ) ) = ( ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) + ( _i x. A ) ) x. ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) - ( _i x. A ) ) ) )
32 7 5 subnegd 
 |-  ( A e. CC -> ( ( 1 - ( A ^ 2 ) ) - -u ( A ^ 2 ) ) = ( ( 1 - ( A ^ 2 ) ) + ( A ^ 2 ) ) )
33 29 31 32 3eqtr3d 
 |-  ( A e. CC -> ( ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) + ( _i x. A ) ) x. ( ( sqrt ` ( 1 - ( A ^ 2 ) ) ) - ( _i x. A ) ) ) = ( ( 1 - ( A ^ 2 ) ) + ( A ^ 2 ) ) )
34 npcan 
 |-  ( ( 1 e. CC /\ ( A ^ 2 ) e. CC ) -> ( ( 1 - ( A ^ 2 ) ) + ( A ^ 2 ) ) = 1 )
35 4 5 34 sylancr 
 |-  ( A e. CC -> ( ( 1 - ( A ^ 2 ) ) + ( A ^ 2 ) ) = 1 )
36 20 33 35 3eqtrd 
 |-  ( A e. CC -> ( ( ( _i x. A ) + ( sqrt ` ( 1 - ( A ^ 2 ) ) ) ) x. ( ( _i x. -u A ) + ( sqrt ` ( 1 - ( -u A ^ 2 ) ) ) ) ) = 1 )