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Metamath Proof Explorer

Theorem chpge0

Description: The second Chebyshev function is nonnegative. (Contributed by Mario Carneiro, 7-Apr-2016)

Ref Expression
Assertion chpge0 
|- ( A e. RR -> 0 <_ ( psi ` A ) )

Proof

Step Hyp Ref Expression
1 ef0 
 |-  ( exp ` 0 ) = 1
2 efchpcl 
 |-  ( A e. RR -> ( exp ` ( psi ` A ) ) e. NN )
3 2 nnge1d 
 |-  ( A e. RR -> 1 <_ ( exp ` ( psi ` A ) ) )
4 1 3 eqbrtrid 
 |-  ( A e. RR -> ( exp ` 0 ) <_ ( exp ` ( psi ` A ) ) )
5 0re 
 |-  0 e. RR
6 chpcl 
 |-  ( A e. RR -> ( psi ` A ) e. RR )
7 efle 
 |-  ( ( 0 e. RR /\ ( psi ` A ) e. RR ) -> ( 0 <_ ( psi ` A ) <-> ( exp ` 0 ) <_ ( exp ` ( psi ` A ) ) ) )
8 5 6 7 sylancr 
 |-  ( A e. RR -> ( 0 <_ ( psi ` A ) <-> ( exp ` 0 ) <_ ( exp ` ( psi ` A ) ) ) )
9 4 8 mpbird 
 |-  ( A e. RR -> 0 <_ ( psi ` A ) )