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Axiom ax-hgprmladder 44330
Description: There is a partition ("ladder") of primes from 7 to 8.8 x 10^30 with parts ("rungs") having lengths of at least 4 and at most N - 4, see section 1.2.2 in [Helfgott] p. 4. Temporarily provided as "axiom". (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.)
Assertion
Ref Expression
ax-hgprmladder 𝑑 ∈ (ℤ‘3)∃𝑓 ∈ (RePart‘𝑑)(((𝑓‘0) = 7 ∧ (𝑓‘1) = 13 ∧ (𝑓𝑑) = (89 · (10↑29))) ∧ ∀𝑖 ∈ (0..^𝑑)((𝑓𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝑓‘(𝑖 + 1)) − (𝑓𝑖)) < ((4 · (10↑18)) − 4) ∧ 4 < ((𝑓‘(𝑖 + 1)) − (𝑓𝑖))))

Detailed syntax breakdown of Axiom ax-hgprmladder
StepHypRef Expression
1 cc0 10526 . . . . . . 7 class 0
2 vf . . . . . . . 8 setvar 𝑓
32cv 1537 . . . . . . 7 class 𝑓
41, 3cfv 6324 . . . . . 6 class (𝑓‘0)
5 c7 11685 . . . . . 6 class 7
64, 5wceq 1538 . . . . 5 wff (𝑓‘0) = 7
7 c1 10527 . . . . . . 7 class 1
87, 3cfv 6324 . . . . . 6 class (𝑓‘1)
9 c3 11681 . . . . . . 7 class 3
107, 9cdc 12086 . . . . . 6 class 13
118, 10wceq 1538 . . . . 5 wff (𝑓‘1) = 13
12 vd . . . . . . . 8 setvar 𝑑
1312cv 1537 . . . . . . 7 class 𝑑
1413, 3cfv 6324 . . . . . 6 class (𝑓𝑑)
15 c8 11686 . . . . . . . 8 class 8
16 c9 11687 . . . . . . . 8 class 9
1715, 16cdc 12086 . . . . . . 7 class 89
187, 1cdc 12086 . . . . . . . 8 class 10
19 c2 11680 . . . . . . . . 9 class 2
2019, 16cdc 12086 . . . . . . . 8 class 29
21 cexp 13425 . . . . . . . 8 class
2218, 20, 21co 7135 . . . . . . 7 class (10↑29)
23 cmul 10531 . . . . . . 7 class ·
2417, 22, 23co 7135 . . . . . 6 class (89 · (10↑29))
2514, 24wceq 1538 . . . . 5 wff (𝑓𝑑) = (89 · (10↑29))
266, 11, 25w3a 1084 . . . 4 wff ((𝑓‘0) = 7 ∧ (𝑓‘1) = 13 ∧ (𝑓𝑑) = (89 · (10↑29)))
27 vi . . . . . . . . 9 setvar 𝑖
2827cv 1537 . . . . . . . 8 class 𝑖
2928, 3cfv 6324 . . . . . . 7 class (𝑓𝑖)
30 cprime 16005 . . . . . . . 8 class
3119csn 4525 . . . . . . . 8 class {2}
3230, 31cdif 3878 . . . . . . 7 class (ℙ ∖ {2})
3329, 32wcel 2111 . . . . . 6 wff (𝑓𝑖) ∈ (ℙ ∖ {2})
34 caddc 10529 . . . . . . . . . 10 class +
3528, 7, 34co 7135 . . . . . . . . 9 class (𝑖 + 1)
3635, 3cfv 6324 . . . . . . . 8 class (𝑓‘(𝑖 + 1))
37 cmin 10859 . . . . . . . 8 class
3836, 29, 37co 7135 . . . . . . 7 class ((𝑓‘(𝑖 + 1)) − (𝑓𝑖))
39 c4 11682 . . . . . . . . 9 class 4
407, 15cdc 12086 . . . . . . . . . 10 class 18
4118, 40, 21co 7135 . . . . . . . . 9 class (10↑18)
4239, 41, 23co 7135 . . . . . . . 8 class (4 · (10↑18))
4342, 39, 37co 7135 . . . . . . 7 class ((4 · (10↑18)) − 4)
44 clt 10664 . . . . . . 7 class <
4538, 43, 44wbr 5030 . . . . . 6 wff ((𝑓‘(𝑖 + 1)) − (𝑓𝑖)) < ((4 · (10↑18)) − 4)
4639, 38, 44wbr 5030 . . . . . 6 wff 4 < ((𝑓‘(𝑖 + 1)) − (𝑓𝑖))
4733, 45, 46w3a 1084 . . . . 5 wff ((𝑓𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝑓‘(𝑖 + 1)) − (𝑓𝑖)) < ((4 · (10↑18)) − 4) ∧ 4 < ((𝑓‘(𝑖 + 1)) − (𝑓𝑖)))
48 cfzo 13028 . . . . . 6 class ..^
491, 13, 48co 7135 . . . . 5 class (0..^𝑑)
5047, 27, 49wral 3106 . . . 4 wff 𝑖 ∈ (0..^𝑑)((𝑓𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝑓‘(𝑖 + 1)) − (𝑓𝑖)) < ((4 · (10↑18)) − 4) ∧ 4 < ((𝑓‘(𝑖 + 1)) − (𝑓𝑖)))
5126, 50wa 399 . . 3 wff (((𝑓‘0) = 7 ∧ (𝑓‘1) = 13 ∧ (𝑓𝑑) = (89 · (10↑29))) ∧ ∀𝑖 ∈ (0..^𝑑)((𝑓𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝑓‘(𝑖 + 1)) − (𝑓𝑖)) < ((4 · (10↑18)) − 4) ∧ 4 < ((𝑓‘(𝑖 + 1)) − (𝑓𝑖))))
52 ciccp 43928 . . . 4 class RePart
5313, 52cfv 6324 . . 3 class (RePart‘𝑑)
5451, 2, 53wrex 3107 . 2 wff 𝑓 ∈ (RePart‘𝑑)(((𝑓‘0) = 7 ∧ (𝑓‘1) = 13 ∧ (𝑓𝑑) = (89 · (10↑29))) ∧ ∀𝑖 ∈ (0..^𝑑)((𝑓𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝑓‘(𝑖 + 1)) − (𝑓𝑖)) < ((4 · (10↑18)) − 4) ∧ 4 < ((𝑓‘(𝑖 + 1)) − (𝑓𝑖))))
55 cuz 12231 . . 3 class
569, 55cfv 6324 . 2 class (ℤ‘3)
5754, 12, 56wrex 3107 1 wff 𝑑 ∈ (ℤ‘3)∃𝑓 ∈ (RePart‘𝑑)(((𝑓‘0) = 7 ∧ (𝑓‘1) = 13 ∧ (𝑓𝑑) = (89 · (10↑29))) ∧ ∀𝑖 ∈ (0..^𝑑)((𝑓𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝑓‘(𝑖 + 1)) − (𝑓𝑖)) < ((4 · (10↑18)) − 4) ∧ 4 < ((𝑓‘(𝑖 + 1)) − (𝑓𝑖))))
Colors of variables: wff setvar class
This axiom is referenced by:  tgblthelfgott  44331
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