| Step | Hyp | Ref
| Expression |
| 1 | | peano2z 12658 |
. . . . . 6
⊢ (𝐴 ∈ ℤ → (𝐴 + 1) ∈
ℤ) |
| 2 | 1 | adantr 480 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 + 1) ∈
ℤ) |
| 3 | | zre 12617 |
. . . . 5
⊢ ((𝐴 + 1) ∈ ℤ →
(𝐴 + 1) ∈
ℝ) |
| 4 | 2, 3 | syl 17 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 + 1) ∈
ℝ) |
| 5 | | chtval 27153 |
. . . 4
⊢ ((𝐴 + 1) ∈ ℝ →
(θ‘(𝐴 + 1)) =
Σ𝑝 ∈
((0[,](𝐴 + 1)) ∩
ℙ)(log‘𝑝)) |
| 6 | 4, 5 | syl 17 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(θ‘(𝐴 + 1)) =
Σ𝑝 ∈
((0[,](𝐴 + 1)) ∩
ℙ)(log‘𝑝)) |
| 7 | | ppisval 27147 |
. . . . . 6
⊢ ((𝐴 + 1) ∈ ℝ →
((0[,](𝐴 + 1)) ∩
ℙ) = ((2...(⌊‘(𝐴 + 1))) ∩ ℙ)) |
| 8 | 4, 7 | syl 17 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((0[,](𝐴 + 1)) ∩
ℙ) = ((2...(⌊‘(𝐴 + 1))) ∩ ℙ)) |
| 9 | | flid 13848 |
. . . . . . . 8
⊢ ((𝐴 + 1) ∈ ℤ →
(⌊‘(𝐴 + 1)) =
(𝐴 + 1)) |
| 10 | 2, 9 | syl 17 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(⌊‘(𝐴 + 1)) =
(𝐴 + 1)) |
| 11 | 10 | oveq2d 7447 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(2...(⌊‘(𝐴 +
1))) = (2...(𝐴 +
1))) |
| 12 | 11 | ineq1d 4219 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((2...(⌊‘(𝐴 +
1))) ∩ ℙ) = ((2...(𝐴 + 1)) ∩ ℙ)) |
| 13 | 8, 12 | eqtrd 2777 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((0[,](𝐴 + 1)) ∩
ℙ) = ((2...(𝐴 + 1))
∩ ℙ)) |
| 14 | 13 | sumeq1d 15736 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
Σ𝑝 ∈
((0[,](𝐴 + 1)) ∩
ℙ)(log‘𝑝) =
Σ𝑝 ∈
((2...(𝐴 + 1)) ∩
ℙ)(log‘𝑝)) |
| 15 | 6, 14 | eqtrd 2777 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(θ‘(𝐴 + 1)) =
Σ𝑝 ∈
((2...(𝐴 + 1)) ∩
ℙ)(log‘𝑝)) |
| 16 | | zre 12617 |
. . . . . . . 8
⊢ (𝐴 ∈ ℤ → 𝐴 ∈
ℝ) |
| 17 | 16 | adantr 480 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 ∈
ℝ) |
| 18 | 17 | ltp1d 12198 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 < (𝐴 + 1)) |
| 19 | 17, 4 | ltnled 11408 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 < (𝐴 + 1) ↔ ¬ (𝐴 + 1) ≤ 𝐴)) |
| 20 | 18, 19 | mpbid 232 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
¬ (𝐴 + 1) ≤ 𝐴) |
| 21 | | elinel1 4201 |
. . . . . 6
⊢ ((𝐴 + 1) ∈ ((2...𝐴) ∩ ℙ) → (𝐴 + 1) ∈ (2...𝐴)) |
| 22 | | elfzle2 13568 |
. . . . . 6
⊢ ((𝐴 + 1) ∈ (2...𝐴) → (𝐴 + 1) ≤ 𝐴) |
| 23 | 21, 22 | syl 17 |
. . . . 5
⊢ ((𝐴 + 1) ∈ ((2...𝐴) ∩ ℙ) → (𝐴 + 1) ≤ 𝐴) |
| 24 | 20, 23 | nsyl 140 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
¬ (𝐴 + 1) ∈
((2...𝐴) ∩
ℙ)) |
| 25 | | disjsn 4711 |
. . . 4
⊢
((((2...𝐴) ∩
ℙ) ∩ {(𝐴 + 1)}) =
∅ ↔ ¬ (𝐴 +
1) ∈ ((2...𝐴) ∩
ℙ)) |
| 26 | 24, 25 | sylibr 234 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(((2...𝐴) ∩ ℙ)
∩ {(𝐴 + 1)}) =
∅) |
| 27 | | 2z 12649 |
. . . . . . 7
⊢ 2 ∈
ℤ |
| 28 | | zcn 12618 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ ℤ → 𝐴 ∈
ℂ) |
| 29 | 28 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 ∈
ℂ) |
| 30 | | ax-1cn 11213 |
. . . . . . . . . 10
⊢ 1 ∈
ℂ |
| 31 | | pncan 11514 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝐴 + 1)
− 1) = 𝐴) |
| 32 | 29, 30, 31 | sylancl 586 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((𝐴 + 1) − 1) = 𝐴) |
| 33 | | prmuz2 16733 |
. . . . . . . . . . 11
⊢ ((𝐴 + 1) ∈ ℙ →
(𝐴 + 1) ∈
(ℤ≥‘2)) |
| 34 | 33 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 + 1) ∈
(ℤ≥‘2)) |
| 35 | | uz2m1nn 12965 |
. . . . . . . . . 10
⊢ ((𝐴 + 1) ∈
(ℤ≥‘2) → ((𝐴 + 1) − 1) ∈
ℕ) |
| 36 | 34, 35 | syl 17 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((𝐴 + 1) − 1) ∈
ℕ) |
| 37 | 32, 36 | eqeltrrd 2842 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 ∈
ℕ) |
| 38 | | nnuz 12921 |
. . . . . . . . 9
⊢ ℕ =
(ℤ≥‘1) |
| 39 | | 2m1e1 12392 |
. . . . . . . . . 10
⊢ (2
− 1) = 1 |
| 40 | 39 | fveq2i 6909 |
. . . . . . . . 9
⊢
(ℤ≥‘(2 − 1)) =
(ℤ≥‘1) |
| 41 | 38, 40 | eqtr4i 2768 |
. . . . . . . 8
⊢ ℕ =
(ℤ≥‘(2 − 1)) |
| 42 | 37, 41 | eleqtrdi 2851 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 ∈
(ℤ≥‘(2 − 1))) |
| 43 | | fzsuc2 13622 |
. . . . . . 7
⊢ ((2
∈ ℤ ∧ 𝐴
∈ (ℤ≥‘(2 − 1))) → (2...(𝐴 + 1)) = ((2...𝐴) ∪ {(𝐴 + 1)})) |
| 44 | 27, 42, 43 | sylancr 587 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(2...(𝐴 + 1)) = ((2...𝐴) ∪ {(𝐴 + 1)})) |
| 45 | 44 | ineq1d 4219 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((2...(𝐴 + 1)) ∩
ℙ) = (((2...𝐴) ∪
{(𝐴 + 1)}) ∩
ℙ)) |
| 46 | | indir 4286 |
. . . . 5
⊢
(((2...𝐴) ∪
{(𝐴 + 1)}) ∩ ℙ) =
(((2...𝐴) ∩ ℙ)
∪ ({(𝐴 + 1)} ∩
ℙ)) |
| 47 | 45, 46 | eqtrdi 2793 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((2...(𝐴 + 1)) ∩
ℙ) = (((2...𝐴) ∩
ℙ) ∪ ({(𝐴 + 1)}
∩ ℙ))) |
| 48 | | simpr 484 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 + 1) ∈
ℙ) |
| 49 | 48 | snssd 4809 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
{(𝐴 + 1)} ⊆
ℙ) |
| 50 | | dfss2 3969 |
. . . . . 6
⊢ ({(𝐴 + 1)} ⊆ ℙ ↔
({(𝐴 + 1)} ∩ ℙ) =
{(𝐴 + 1)}) |
| 51 | 49, 50 | sylib 218 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
({(𝐴 + 1)} ∩ ℙ) =
{(𝐴 + 1)}) |
| 52 | 51 | uneq2d 4168 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(((2...𝐴) ∩ ℙ)
∪ ({(𝐴 + 1)} ∩
ℙ)) = (((2...𝐴) ∩
ℙ) ∪ {(𝐴 +
1)})) |
| 53 | 47, 52 | eqtrd 2777 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((2...(𝐴 + 1)) ∩
ℙ) = (((2...𝐴) ∩
ℙ) ∪ {(𝐴 +
1)})) |
| 54 | | fzfid 14014 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(2...(𝐴 + 1)) ∈
Fin) |
| 55 | | inss1 4237 |
. . . 4
⊢
((2...(𝐴 + 1)) ∩
ℙ) ⊆ (2...(𝐴 +
1)) |
| 56 | | ssfi 9213 |
. . . 4
⊢
(((2...(𝐴 + 1))
∈ Fin ∧ ((2...(𝐴 +
1)) ∩ ℙ) ⊆ (2...(𝐴 + 1))) → ((2...(𝐴 + 1)) ∩ ℙ) ∈
Fin) |
| 57 | 54, 55, 56 | sylancl 586 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((2...(𝐴 + 1)) ∩
ℙ) ∈ Fin) |
| 58 | | simpr 484 |
. . . . . . . 8
⊢ (((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) ∧
𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)) →
𝑝 ∈ ((2...(𝐴 + 1)) ∩
ℙ)) |
| 59 | 58 | elin2d 4205 |
. . . . . . 7
⊢ (((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) ∧
𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)) →
𝑝 ∈
ℙ) |
| 60 | | prmnn 16711 |
. . . . . . 7
⊢ (𝑝 ∈ ℙ → 𝑝 ∈
ℕ) |
| 61 | 59, 60 | syl 17 |
. . . . . 6
⊢ (((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) ∧
𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)) →
𝑝 ∈
ℕ) |
| 62 | 61 | nnrpd 13075 |
. . . . 5
⊢ (((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) ∧
𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)) →
𝑝 ∈
ℝ+) |
| 63 | 62 | relogcld 26665 |
. . . 4
⊢ (((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) ∧
𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)) →
(log‘𝑝) ∈
ℝ) |
| 64 | 63 | recnd 11289 |
. . 3
⊢ (((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) ∧
𝑝 ∈ ((2...(𝐴 + 1)) ∩ ℙ)) →
(log‘𝑝) ∈
ℂ) |
| 65 | 26, 53, 57, 64 | fsumsplit 15777 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
Σ𝑝 ∈
((2...(𝐴 + 1)) ∩
ℙ)(log‘𝑝) =
(Σ𝑝 ∈
((2...𝐴) ∩
ℙ)(log‘𝑝) +
Σ𝑝 ∈ {(𝐴 + 1)} (log‘𝑝))) |
| 66 | | chtval 27153 |
. . . . 5
⊢ (𝐴 ∈ ℝ →
(θ‘𝐴) =
Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)) |
| 67 | 17, 66 | syl 17 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(θ‘𝐴) =
Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝)) |
| 68 | | ppisval 27147 |
. . . . . . 7
⊢ (𝐴 ∈ ℝ →
((0[,]𝐴) ∩ ℙ) =
((2...(⌊‘𝐴))
∩ ℙ)) |
| 69 | 17, 68 | syl 17 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((0[,]𝐴) ∩ ℙ) =
((2...(⌊‘𝐴))
∩ ℙ)) |
| 70 | | flid 13848 |
. . . . . . . . 9
⊢ (𝐴 ∈ ℤ →
(⌊‘𝐴) = 𝐴) |
| 71 | 70 | adantr 480 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(⌊‘𝐴) = 𝐴) |
| 72 | 71 | oveq2d 7447 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(2...(⌊‘𝐴)) =
(2...𝐴)) |
| 73 | 72 | ineq1d 4219 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((2...(⌊‘𝐴))
∩ ℙ) = ((2...𝐴)
∩ ℙ)) |
| 74 | 69, 73 | eqtrd 2777 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((0[,]𝐴) ∩ ℙ) =
((2...𝐴) ∩
ℙ)) |
| 75 | 74 | sumeq1d 15736 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)(log‘𝑝) = Σ𝑝 ∈ ((2...𝐴) ∩ ℙ)(log‘𝑝)) |
| 76 | 67, 75 | eqtr2d 2778 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
Σ𝑝 ∈ ((2...𝐴) ∩ ℙ)(log‘𝑝) = (θ‘𝐴)) |
| 77 | | prmnn 16711 |
. . . . 5
⊢ ((𝐴 + 1) ∈ ℙ →
(𝐴 + 1) ∈
ℕ) |
| 78 | 77 | adantl 481 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 + 1) ∈
ℕ) |
| 79 | 78 | nnrpd 13075 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 + 1) ∈
ℝ+) |
| 80 | 79 | relogcld 26665 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(log‘(𝐴 + 1)) ∈
ℝ) |
| 81 | 80 | recnd 11289 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(log‘(𝐴 + 1)) ∈
ℂ) |
| 82 | | fveq2 6906 |
. . . . 5
⊢ (𝑝 = (𝐴 + 1) → (log‘𝑝) = (log‘(𝐴 + 1))) |
| 83 | 82 | sumsn 15782 |
. . . 4
⊢ (((𝐴 + 1) ∈ ℕ ∧
(log‘(𝐴 + 1)) ∈
ℂ) → Σ𝑝
∈ {(𝐴 + 1)}
(log‘𝑝) =
(log‘(𝐴 +
1))) |
| 84 | 78, 81, 83 | syl2anc 584 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
Σ𝑝 ∈ {(𝐴 + 1)} (log‘𝑝) = (log‘(𝐴 + 1))) |
| 85 | 76, 84 | oveq12d 7449 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(Σ𝑝 ∈
((2...𝐴) ∩
ℙ)(log‘𝑝) +
Σ𝑝 ∈ {(𝐴 + 1)} (log‘𝑝)) = ((θ‘𝐴) + (log‘(𝐴 + 1)))) |
| 86 | 15, 65, 85 | 3eqtrd 2781 |
1
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(θ‘(𝐴 + 1)) =
((θ‘𝐴) +
(log‘(𝐴 +
1)))) |