Metamath Proof Explorer

Theorem rmyfval

Description: Value of the Y sequence. Not used after rmxyval is proved. (Contributed by Stefan O'Rear, 21-Sep-2014)

Ref Expression
Assertion rmyfval 
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A rmY N ) = ( 2nd ` ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqrt ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( A + ( sqrt ` ( ( A ^ 2 ) - 1 ) ) ) ^ N ) ) ) )

Proof

Step Hyp Ref Expression
1 oveq1 
 |-  ( a = A -> ( a ^ 2 ) = ( A ^ 2 ) )
2 1 fvoveq1d 
 |-  ( a = A -> ( sqrt ` ( ( a ^ 2 ) - 1 ) ) = ( sqrt ` ( ( A ^ 2 ) - 1 ) ) )
3 2 oveq1d 
 |-  ( a = A -> ( ( sqrt ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) = ( ( sqrt ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) )
4 3 oveq2d 
 |-  ( a = A -> ( ( 1st ` b ) + ( ( sqrt ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) = ( ( 1st ` b ) + ( ( sqrt ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) )
5 4 mpteq2dv 
 |-  ( a = A -> ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqrt ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) = ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqrt ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) )
6 5 cnveqd 
 |-  ( a = A -> `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqrt ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) = `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqrt ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) )
7 6 adantr 
 |-  ( ( a = A /\ n = N ) -> `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqrt ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) = `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqrt ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) )
8 id 
 |-  ( a = A -> a = A )
9 8 2 oveq12d 
 |-  ( a = A -> ( a + ( sqrt ` ( ( a ^ 2 ) - 1 ) ) ) = ( A + ( sqrt ` ( ( A ^ 2 ) - 1 ) ) ) )
10 id 
 |-  ( n = N -> n = N )
11 9 10 oveqan12d 
 |-  ( ( a = A /\ n = N ) -> ( ( a + ( sqrt ` ( ( a ^ 2 ) - 1 ) ) ) ^ n ) = ( ( A + ( sqrt ` ( ( A ^ 2 ) - 1 ) ) ) ^ N ) )
12 7 11 fveq12d 
 |-  ( ( a = A /\ n = N ) -> ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqrt ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( a + ( sqrt ` ( ( a ^ 2 ) - 1 ) ) ) ^ n ) ) = ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqrt ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( A + ( sqrt ` ( ( A ^ 2 ) - 1 ) ) ) ^ N ) ) )
13 12 fveq2d 
 |-  ( ( a = A /\ n = N ) -> ( 2nd ` ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqrt ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( a + ( sqrt ` ( ( a ^ 2 ) - 1 ) ) ) ^ n ) ) ) = ( 2nd ` ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqrt ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( A + ( sqrt ` ( ( A ^ 2 ) - 1 ) ) ) ^ N ) ) ) )
14 df-rmy 
 |-  rmY = ( a e. ( ZZ>= ` 2 ) , n e. ZZ |-> ( 2nd ` ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqrt ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( a + ( sqrt ` ( ( a ^ 2 ) - 1 ) ) ) ^ n ) ) ) )
15 fvex 
 |-  ( 2nd ` ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqrt ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( A + ( sqrt ` ( ( A ^ 2 ) - 1 ) ) ) ^ N ) ) ) e. _V
16 13 14 15 ovmpoa 
 |-  ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A rmY N ) = ( 2nd ` ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqrt ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( A + ( sqrt ` ( ( A ^ 2 ) - 1 ) ) ) ^ N ) ) ) )