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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ax-ros336 | Structured version Visualization version GIF version |
Description: Theorem 13. of [RosserSchoenfeld] p. 71. Theorem chpchtlim 25582 states that the ψ and θ 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 |
---|---|
ax-ros336 | ⊢ ∀𝑥 ∈ ℝ+ ((ψ‘𝑥) − (θ‘𝑥)) < ((1._4_2_62) · (√‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vx | . . . . . 6 setvar 𝑥 | |
2 | 1 | cv 1657 | . . . . 5 class 𝑥 |
3 | cchp 25233 | . . . . 5 class ψ | |
4 | 2, 3 | cfv 6124 | . . . 4 class (ψ‘𝑥) |
5 | ccht 25231 | . . . . 5 class θ | |
6 | 2, 5 | cfv 6124 | . . . 4 class (θ‘𝑥) |
7 | cmin 10586 | . . . 4 class − | |
8 | 4, 6, 7 | co 6906 | . . 3 class ((ψ‘𝑥) − (θ‘𝑥)) |
9 | c1 10254 | . . . . 5 class 1 | |
10 | c4 11409 | . . . . . 6 class 4 | |
11 | c2 11407 | . . . . . . 7 class 2 | |
12 | c6 11411 | . . . . . . . 8 class 6 | |
13 | 12, 11 | cdp2 30125 | . . . . . . 7 class _62 |
14 | 11, 13 | cdp2 30125 | . . . . . 6 class _2_62 |
15 | 10, 14 | cdp2 30125 | . . . . 5 class _4_2_62 |
16 | cdp 30142 | . . . . 5 class . | |
17 | 9, 15, 16 | co 6906 | . . . 4 class (1._4_2_62) |
18 | csqrt 14351 | . . . . 5 class √ | |
19 | 2, 18 | cfv 6124 | . . . 4 class (√‘𝑥) |
20 | cmul 10258 | . . . 4 class · | |
21 | 17, 19, 20 | co 6906 | . . 3 class ((1._4_2_62) · (√‘𝑥)) |
22 | clt 10392 | . . 3 class < | |
23 | 8, 21, 22 | wbr 4874 | . 2 wff ((ψ‘𝑥) − (θ‘𝑥)) < ((1._4_2_62) · (√‘𝑥)) |
24 | crp 12113 | . 2 class ℝ+ | |
25 | 23, 1, 24 | wral 3118 | 1 wff ∀𝑥 ∈ ℝ+ ((ψ‘𝑥) − (θ‘𝑥)) < ((1._4_2_62) · (√‘𝑥)) |
Colors of variables: wff setvar class |
This axiom is referenced by: hgt750lemd 31276 |
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