Metamath Proof Explorer

Theorem chtval

Description: Value of the Chebyshev function. (Contributed by Mario Carneiro, 15-Sep-2014)

Ref Expression
Assertion chtval 
|- ( A e. RR -> ( theta ` A ) = sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` p ) )

Proof

Step Hyp Ref Expression
1 oveq2 
 |-  ( x = A -> ( 0 [,] x ) = ( 0 [,] A ) )
2 1 ineq1d 
 |-  ( x = A -> ( ( 0 [,] x ) i^i Prime ) = ( ( 0 [,] A ) i^i Prime ) )
3 2 sumeq1d 
 |-  ( x = A -> sum_ p e. ( ( 0 [,] x ) i^i Prime ) ( log ` p ) = sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` p ) )
4 df-cht 
 |-  theta = ( x e. RR |-> sum_ p e. ( ( 0 [,] x ) i^i Prime ) ( log ` p ) )
5 sumex 
 |-  sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` p ) e. _V
6 3 4 5 fvmpt 
 |-  ( A e. RR -> ( theta ` A ) = sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` p ) )