Proof of Theorem ppiprm
Step | Hyp | Ref
| Expression |
1 | | fzfid 12634 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(2...𝐴) ∈
Fin) |
2 | | inss1 3795 |
. . . 4
⊢
((2...𝐴) ∩
ℙ) ⊆ (2...𝐴) |
3 | | ssfi 8065 |
. . . 4
⊢
(((2...𝐴) ∈ Fin
∧ ((2...𝐴) ∩
ℙ) ⊆ (2...𝐴))
→ ((2...𝐴) ∩
ℙ) ∈ Fin) |
4 | 1, 2, 3 | sylancl 693 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((2...𝐴) ∩ ℙ)
∈ Fin) |
5 | | zre 11258 |
. . . . . . 7
⊢ (𝐴 ∈ ℤ → 𝐴 ∈
ℝ) |
6 | 5 | adantr 480 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 ∈
ℝ) |
7 | 6 | ltp1d 10833 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 < (𝐴 + 1)) |
8 | | peano2z 11295 |
. . . . . . . 8
⊢ (𝐴 ∈ ℤ → (𝐴 + 1) ∈
ℤ) |
9 | 8 | adantr 480 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 + 1) ∈
ℤ) |
10 | 9 | zred 11358 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 + 1) ∈
ℝ) |
11 | 6, 10 | ltnled 10063 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 < (𝐴 + 1) ↔ ¬ (𝐴 + 1) ≤ 𝐴)) |
12 | 7, 11 | mpbid 221 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
¬ (𝐴 + 1) ≤ 𝐴) |
13 | 2 | sseli 3564 |
. . . . 5
⊢ ((𝐴 + 1) ∈ ((2...𝐴) ∩ ℙ) → (𝐴 + 1) ∈ (2...𝐴)) |
14 | | elfzle2 12216 |
. . . . 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 134 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
¬ (𝐴 + 1) ∈
((2...𝐴) ∩
ℙ)) |
17 | | ovex 6577 |
. . . 4
⊢ (𝐴 + 1) ∈ V |
18 | | hashunsng 13042 |
. . . 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 691 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(#‘(((2...𝐴) ∩
ℙ) ∪ {(𝐴 + 1)}))
= ((#‘((2...𝐴) ∩
ℙ)) + 1)) |
21 | | ppival2 24654 |
. . . 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 11286 |
. . . . . . . 8
⊢ 2 ∈
ℤ |
24 | | zcn 11259 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ℤ → 𝐴 ∈
ℂ) |
25 | 24 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 ∈
ℂ) |
26 | | ax-1cn 9873 |
. . . . . . . . . . 11
⊢ 1 ∈
ℂ |
27 | | pncan 10166 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝐴 + 1)
− 1) = 𝐴) |
28 | 25, 26, 27 | sylancl 693 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((𝐴 + 1) − 1) = 𝐴) |
29 | | prmuz2 15246 |
. . . . . . . . . . . 12
⊢ ((𝐴 + 1) ∈ ℙ →
(𝐴 + 1) ∈
(ℤ≥‘2)) |
30 | 29 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 + 1) ∈
(ℤ≥‘2)) |
31 | | uz2m1nn 11639 |
. . . . . . . . . . 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 2689 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 ∈
ℕ) |
34 | | nnuz 11599 |
. . . . . . . . . 10
⊢ ℕ =
(ℤ≥‘1) |
35 | | 2m1e1 11012 |
. . . . . . . . . . 11
⊢ (2
− 1) = 1 |
36 | 35 | fveq2i 6106 |
. . . . . . . . . 10
⊢
(ℤ≥‘(2 − 1)) =
(ℤ≥‘1) |
37 | 34, 36 | eqtr4i 2635 |
. . . . . . . . 9
⊢ ℕ =
(ℤ≥‘(2 − 1)) |
38 | 33, 37 | syl6eleq 2698 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 ∈
(ℤ≥‘(2 − 1))) |
39 | | fzsuc2 12268 |
. . . . . . . 8
⊢ ((2
∈ ℤ ∧ 𝐴
∈ (ℤ≥‘(2 − 1))) → (2...(𝐴 + 1)) = ((2...𝐴) ∪ {(𝐴 + 1)})) |
40 | 23, 38, 39 | sylancr 694 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(2...(𝐴 + 1)) = ((2...𝐴) ∪ {(𝐴 + 1)})) |
41 | 40 | ineq1d 3775 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((2...(𝐴 + 1)) ∩
ℙ) = (((2...𝐴) ∪
{(𝐴 + 1)}) ∩
ℙ)) |
42 | | indir 3834 |
. . . . . 6
⊢
(((2...𝐴) ∪
{(𝐴 + 1)}) ∩ ℙ) =
(((2...𝐴) ∩ ℙ)
∪ ({(𝐴 + 1)} ∩
ℙ)) |
43 | 41, 42 | syl6eq 2660 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((2...(𝐴 + 1)) ∩
ℙ) = (((2...𝐴) ∩
ℙ) ∪ ({(𝐴 + 1)}
∩ ℙ))) |
44 | | simpr 476 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 + 1) ∈
ℙ) |
45 | 44 | snssd 4281 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
{(𝐴 + 1)} ⊆
ℙ) |
46 | | df-ss 3554 |
. . . . . . 7
⊢ ({(𝐴 + 1)} ⊆ ℙ ↔
({(𝐴 + 1)} ∩ ℙ) =
{(𝐴 + 1)}) |
47 | 45, 46 | sylib 207 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
({(𝐴 + 1)} ∩ ℙ) =
{(𝐴 + 1)}) |
48 | 47 | uneq2d 3729 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(((2...𝐴) ∩ ℙ)
∪ ({(𝐴 + 1)} ∩
ℙ)) = (((2...𝐴) ∩
ℙ) ∪ {(𝐴 +
1)})) |
49 | 43, 48 | eqtrd 2644 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((2...(𝐴 + 1)) ∩
ℙ) = (((2...𝐴) ∩
ℙ) ∪ {(𝐴 +
1)})) |
50 | 49 | fveq2d 6107 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(#‘((2...(𝐴 + 1))
∩ ℙ)) = (#‘(((2...𝐴) ∩ ℙ) ∪ {(𝐴 + 1)}))) |
51 | 22, 50 | eqtrd 2644 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(π‘(𝐴 + 1))
= (#‘(((2...𝐴) ∩
ℙ) ∪ {(𝐴 +
1)}))) |
52 | | ppival2 24654 |
. . . 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 6564 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((π‘𝐴) + 1)
= ((#‘((2...𝐴) ∩
ℙ)) + 1)) |
55 | 20, 51, 54 | 3eqtr4d 2654 |
1
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(π‘(𝐴 + 1))
= ((π‘𝐴) +
1)) |