Proof of Theorem ppiprm
| Step | Hyp | Ref
| Expression |
| 1 | | fzfid 14014 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(2...𝐴) ∈
Fin) |
| 2 | | inss1 4237 |
. . . 4
⊢
((2...𝐴) ∩
ℙ) ⊆ (2...𝐴) |
| 3 | | ssfi 9213 |
. . . 4
⊢
(((2...𝐴) ∈ Fin
∧ ((2...𝐴) ∩
ℙ) ⊆ (2...𝐴))
→ ((2...𝐴) ∩
ℙ) ∈ Fin) |
| 4 | 1, 2, 3 | sylancl 586 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((2...𝐴) ∩ ℙ)
∈ Fin) |
| 5 | | zre 12617 |
. . . . . . 7
⊢ (𝐴 ∈ ℤ → 𝐴 ∈
ℝ) |
| 6 | 5 | adantr 480 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 ∈
ℝ) |
| 7 | 6 | ltp1d 12198 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 < (𝐴 + 1)) |
| 8 | | peano2z 12658 |
. . . . . . . 8
⊢ (𝐴 ∈ ℤ → (𝐴 + 1) ∈
ℤ) |
| 9 | 8 | adantr 480 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 + 1) ∈
ℤ) |
| 10 | 9 | zred 12722 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 + 1) ∈
ℝ) |
| 11 | 6, 10 | ltnled 11408 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 < (𝐴 + 1) ↔ ¬ (𝐴 + 1) ≤ 𝐴)) |
| 12 | 7, 11 | mpbid 232 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
¬ (𝐴 + 1) ≤ 𝐴) |
| 13 | | elinel1 4201 |
. . . . 5
⊢ ((𝐴 + 1) ∈ ((2...𝐴) ∩ ℙ) → (𝐴 + 1) ∈ (2...𝐴)) |
| 14 | | elfzle2 13568 |
. . . . 5
⊢ ((𝐴 + 1) ∈ (2...𝐴) → (𝐴 + 1) ≤ 𝐴) |
| 15 | 13, 14 | syl 17 |
. . . 4
⊢ ((𝐴 + 1) ∈ ((2...𝐴) ∩ ℙ) → (𝐴 + 1) ≤ 𝐴) |
| 16 | 12, 15 | nsyl 140 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
¬ (𝐴 + 1) ∈
((2...𝐴) ∩
ℙ)) |
| 17 | | ovex 7464 |
. . . 4
⊢ (𝐴 + 1) ∈ V |
| 18 | | hashunsng 14431 |
. . . 4
⊢ ((𝐴 + 1) ∈ V →
((((2...𝐴) ∩ ℙ)
∈ Fin ∧ ¬ (𝐴 +
1) ∈ ((2...𝐴) ∩
ℙ)) → (♯‘(((2...𝐴) ∩ ℙ) ∪ {(𝐴 + 1)})) = ((♯‘((2...𝐴) ∩ ℙ)) +
1))) |
| 19 | 17, 18 | ax-mp 5 |
. . 3
⊢
((((2...𝐴) ∩
ℙ) ∈ Fin ∧ ¬ (𝐴 + 1) ∈ ((2...𝐴) ∩ ℙ)) →
(♯‘(((2...𝐴)
∩ ℙ) ∪ {(𝐴 +
1)})) = ((♯‘((2...𝐴) ∩ ℙ)) + 1)) |
| 20 | 4, 16, 19 | syl2anc 584 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(♯‘(((2...𝐴)
∩ ℙ) ∪ {(𝐴 +
1)})) = ((♯‘((2...𝐴) ∩ ℙ)) + 1)) |
| 21 | | ppival2 27171 |
. . . 4
⊢ ((𝐴 + 1) ∈ ℤ →
(π‘(𝐴 + 1))
= (♯‘((2...(𝐴 +
1)) ∩ ℙ))) |
| 22 | 9, 21 | syl 17 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(π‘(𝐴 + 1))
= (♯‘((2...(𝐴 +
1)) ∩ ℙ))) |
| 23 | | 2z 12649 |
. . . . . . . 8
⊢ 2 ∈
ℤ |
| 24 | | zcn 12618 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ℤ → 𝐴 ∈
ℂ) |
| 25 | 24 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 ∈
ℂ) |
| 26 | | ax-1cn 11213 |
. . . . . . . . . . 11
⊢ 1 ∈
ℂ |
| 27 | | pncan 11514 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝐴 + 1)
− 1) = 𝐴) |
| 28 | 25, 26, 27 | sylancl 586 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((𝐴 + 1) − 1) = 𝐴) |
| 29 | | prmuz2 16733 |
. . . . . . . . . . . 12
⊢ ((𝐴 + 1) ∈ ℙ →
(𝐴 + 1) ∈
(ℤ≥‘2)) |
| 30 | 29 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 + 1) ∈
(ℤ≥‘2)) |
| 31 | | uz2m1nn 12965 |
. . . . . . . . . . 11
⊢ ((𝐴 + 1) ∈
(ℤ≥‘2) → ((𝐴 + 1) − 1) ∈
ℕ) |
| 32 | 30, 31 | syl 17 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((𝐴 + 1) − 1) ∈
ℕ) |
| 33 | 28, 32 | eqeltrrd 2842 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 ∈
ℕ) |
| 34 | | nnuz 12921 |
. . . . . . . . . 10
⊢ ℕ =
(ℤ≥‘1) |
| 35 | | 2m1e1 12392 |
. . . . . . . . . . 11
⊢ (2
− 1) = 1 |
| 36 | 35 | fveq2i 6909 |
. . . . . . . . . 10
⊢
(ℤ≥‘(2 − 1)) =
(ℤ≥‘1) |
| 37 | 34, 36 | eqtr4i 2768 |
. . . . . . . . 9
⊢ ℕ =
(ℤ≥‘(2 − 1)) |
| 38 | 33, 37 | eleqtrdi 2851 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 ∈
(ℤ≥‘(2 − 1))) |
| 39 | | fzsuc2 13622 |
. . . . . . . 8
⊢ ((2
∈ ℤ ∧ 𝐴
∈ (ℤ≥‘(2 − 1))) → (2...(𝐴 + 1)) = ((2...𝐴) ∪ {(𝐴 + 1)})) |
| 40 | 23, 38, 39 | sylancr 587 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(2...(𝐴 + 1)) = ((2...𝐴) ∪ {(𝐴 + 1)})) |
| 41 | 40 | ineq1d 4219 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((2...(𝐴 + 1)) ∩
ℙ) = (((2...𝐴) ∪
{(𝐴 + 1)}) ∩
ℙ)) |
| 42 | | indir 4286 |
. . . . . 6
⊢
(((2...𝐴) ∪
{(𝐴 + 1)}) ∩ ℙ) =
(((2...𝐴) ∩ ℙ)
∪ ({(𝐴 + 1)} ∩
ℙ)) |
| 43 | 41, 42 | eqtrdi 2793 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((2...(𝐴 + 1)) ∩
ℙ) = (((2...𝐴) ∩
ℙ) ∪ ({(𝐴 + 1)}
∩ ℙ))) |
| 44 | | simpr 484 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 + 1) ∈
ℙ) |
| 45 | 44 | snssd 4809 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
{(𝐴 + 1)} ⊆
ℙ) |
| 46 | | dfss2 3969 |
. . . . . . 7
⊢ ({(𝐴 + 1)} ⊆ ℙ ↔
({(𝐴 + 1)} ∩ ℙ) =
{(𝐴 + 1)}) |
| 47 | 45, 46 | sylib 218 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
({(𝐴 + 1)} ∩ ℙ) =
{(𝐴 + 1)}) |
| 48 | 47 | uneq2d 4168 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(((2...𝐴) ∩ ℙ)
∪ ({(𝐴 + 1)} ∩
ℙ)) = (((2...𝐴) ∩
ℙ) ∪ {(𝐴 +
1)})) |
| 49 | 43, 48 | eqtrd 2777 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((2...(𝐴 + 1)) ∩
ℙ) = (((2...𝐴) ∩
ℙ) ∪ {(𝐴 +
1)})) |
| 50 | 49 | fveq2d 6910 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(♯‘((2...(𝐴 +
1)) ∩ ℙ)) = (♯‘(((2...𝐴) ∩ ℙ) ∪ {(𝐴 + 1)}))) |
| 51 | 22, 50 | eqtrd 2777 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(π‘(𝐴 + 1))
= (♯‘(((2...𝐴)
∩ ℙ) ∪ {(𝐴 +
1)}))) |
| 52 | | ppival2 27171 |
. . . 4
⊢ (𝐴 ∈ ℤ →
(π‘𝐴) =
(♯‘((2...𝐴)
∩ ℙ))) |
| 53 | 52 | adantr 480 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(π‘𝐴) =
(♯‘((2...𝐴)
∩ ℙ))) |
| 54 | 53 | oveq1d 7446 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((π‘𝐴) + 1)
= ((♯‘((2...𝐴)
∩ ℙ)) + 1)) |
| 55 | 20, 51, 54 | 3eqtr4d 2787 |
1
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(π‘(𝐴 + 1))
= ((π‘𝐴) +
1)) |