Metamath Proof Explorer

Theorem rmyluc2

Description: Lucas sequence property of Y with better output ordering. (Contributed by Stefan O'Rear, 16-Oct-2014)

Ref Expression
Assertion rmyluc2 
|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A rmY ( N + 1 ) ) = ( ( ( 2 x. A ) x. ( A rmY N ) ) - ( A rmY ( N - 1 ) ) ) )

Proof

Step Hyp Ref Expression
1 rmyluc 
 |-  ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A rmY ( N + 1 ) ) = ( ( 2 x. ( ( A rmY N ) x. A ) ) - ( A rmY ( N - 1 ) ) ) )
2 frmy 
 |-  rmY : ( ( ZZ>= ` 2 ) X. ZZ ) --> ZZ
3 2 fovcl 
 |-  ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A rmY N ) e. ZZ )
4 3 zcnd 
 |-  ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A rmY N ) e. CC )
5 eluzelcn 
 |-  ( A e. ( ZZ>= ` 2 ) -> A e. CC )
6 5 adantr 
 |-  ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> A e. CC )
7 4 6 mulcomd 
 |-  ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( ( A rmY N ) x. A ) = ( A x. ( A rmY N ) ) )
8 7 oveq2d 
 |-  ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( 2 x. ( ( A rmY N ) x. A ) ) = ( 2 x. ( A x. ( A rmY N ) ) ) )
9 2cnd 
 |-  ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> 2 e. CC )
10 9 6 4 mulassd 
 |-  ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( ( 2 x. A ) x. ( A rmY N ) ) = ( 2 x. ( A x. ( A rmY N ) ) ) )
11 8 10 eqtr4d 
 |-  ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( 2 x. ( ( A rmY N ) x. A ) ) = ( ( 2 x. A ) x. ( A rmY N ) ) )
12 11 oveq1d 
 |-  ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( ( 2 x. ( ( A rmY N ) x. A ) ) - ( A rmY ( N - 1 ) ) ) = ( ( ( 2 x. A ) x. ( A rmY N ) ) - ( A rmY ( N - 1 ) ) ) )
13 1 12 eqtrd 
 |-  ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A rmY ( N + 1 ) ) = ( ( ( 2 x. A ) x. ( A rmY N ) ) - ( A rmY ( N - 1 ) ) ) )