Description: Theorem 13. of RosserSchoenfeld p. 71. Theorem chpchtlim states that the psi and theta function are asymtotic to each other; this axiom postulates an upper bound for their difference. This is stated as an axiom until a formal proof can be provided. (Contributed by Thierry Arnoux, 28-Dec-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | ax-ros336 | |- A. x e. RR+ ( ( psi ` x ) - ( theta ` x ) ) < ( ( 1 . _ 4 _ 2 _ 6 2 ) x. ( sqrt ` x ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | vx | |- x |
|
| 1 | crp | |- RR+ |
|
| 2 | cchp | |- psi |
|
| 3 | 0 | cv | |- x |
| 4 | 3 2 | cfv | |- ( psi ` x ) |
| 5 | cmin | |- - |
|
| 6 | ccht | |- theta |
|
| 7 | 3 6 | cfv | |- ( theta ` x ) |
| 8 | 4 7 5 | co | |- ( ( psi ` x ) - ( theta ` x ) ) |
| 9 | clt | |- < |
|
| 10 | c1 | |- 1 |
|
| 11 | cdp | |- . |
|
| 12 | c4 | |- 4 |
|
| 13 | c2 | |- 2 |
|
| 14 | c6 | |- 6 |
|
| 15 | 14 13 | cdp2 | |- _ 6 2 |
| 16 | 13 15 | cdp2 | |- _ 2 _ 6 2 |
| 17 | 12 16 | cdp2 | |- _ 4 _ 2 _ 6 2 |
| 18 | 10 17 11 | co | |- ( 1 . _ 4 _ 2 _ 6 2 ) |
| 19 | cmul | |- x. |
|
| 20 | csqrt | |- sqrt |
|
| 21 | 3 20 | cfv | |- ( sqrt ` x ) |
| 22 | 18 21 19 | co | |- ( ( 1 . _ 4 _ 2 _ 6 2 ) x. ( sqrt ` x ) ) |
| 23 | 8 22 9 | wbr | |- ( ( psi ` x ) - ( theta ` x ) ) < ( ( 1 . _ 4 _ 2 _ 6 2 ) x. ( sqrt ` x ) ) |
| 24 | 23 0 1 | wral | |- A. x e. RR+ ( ( psi ` x ) - ( theta ` x ) ) < ( ( 1 . _ 4 _ 2 _ 6 2 ) x. ( sqrt ` x ) ) |