Proof of Theorem ppiprm
Step | Hyp | Ref
| Expression |
1 | | fzfid 13335 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(2...𝐴) ∈
Fin) |
2 | | inss1 4205 |
. . . 4
⊢
((2...𝐴) ∩
ℙ) ⊆ (2...𝐴) |
3 | | ssfi 8732 |
. . . 4
⊢
(((2...𝐴) ∈ Fin
∧ ((2...𝐴) ∩
ℙ) ⊆ (2...𝐴))
→ ((2...𝐴) ∩
ℙ) ∈ Fin) |
4 | 1, 2, 3 | sylancl 588 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((2...𝐴) ∩ ℙ)
∈ Fin) |
5 | | zre 11979 |
. . . . . . 7
⊢ (𝐴 ∈ ℤ → 𝐴 ∈
ℝ) |
6 | 5 | adantr 483 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 ∈
ℝ) |
7 | 6 | ltp1d 11564 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 < (𝐴 + 1)) |
8 | | peano2z 12017 |
. . . . . . . 8
⊢ (𝐴 ∈ ℤ → (𝐴 + 1) ∈
ℤ) |
9 | 8 | adantr 483 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 + 1) ∈
ℤ) |
10 | 9 | zred 12081 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 + 1) ∈
ℝ) |
11 | 6, 10 | ltnled 10781 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 < (𝐴 + 1) ↔ ¬ (𝐴 + 1) ≤ 𝐴)) |
12 | 7, 11 | mpbid 234 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
¬ (𝐴 + 1) ≤ 𝐴) |
13 | | elinel1 4172 |
. . . . 5
⊢ ((𝐴 + 1) ∈ ((2...𝐴) ∩ ℙ) → (𝐴 + 1) ∈ (2...𝐴)) |
14 | | elfzle2 12905 |
. . . . 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 142 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
¬ (𝐴 + 1) ∈
((2...𝐴) ∩
ℙ)) |
17 | | ovex 7183 |
. . . 4
⊢ (𝐴 + 1) ∈ V |
18 | | hashunsng 13747 |
. . . 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 586 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(♯‘(((2...𝐴)
∩ ℙ) ∪ {(𝐴 +
1)})) = ((♯‘((2...𝐴) ∩ ℙ)) + 1)) |
21 | | ppival2 25699 |
. . . 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 12008 |
. . . . . . . 8
⊢ 2 ∈
ℤ |
24 | | zcn 11980 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ℤ → 𝐴 ∈
ℂ) |
25 | 24 | adantr 483 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 ∈
ℂ) |
26 | | ax-1cn 10589 |
. . . . . . . . . . 11
⊢ 1 ∈
ℂ |
27 | | pncan 10886 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝐴 + 1)
− 1) = 𝐴) |
28 | 25, 26, 27 | sylancl 588 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((𝐴 + 1) − 1) = 𝐴) |
29 | | prmuz2 16034 |
. . . . . . . . . . . 12
⊢ ((𝐴 + 1) ∈ ℙ →
(𝐴 + 1) ∈
(ℤ≥‘2)) |
30 | 29 | adantl 484 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 + 1) ∈
(ℤ≥‘2)) |
31 | | uz2m1nn 12317 |
. . . . . . . . . . 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 2914 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 ∈
ℕ) |
34 | | nnuz 12275 |
. . . . . . . . . 10
⊢ ℕ =
(ℤ≥‘1) |
35 | | 2m1e1 11757 |
. . . . . . . . . . 11
⊢ (2
− 1) = 1 |
36 | 35 | fveq2i 6668 |
. . . . . . . . . 10
⊢
(ℤ≥‘(2 − 1)) =
(ℤ≥‘1) |
37 | 34, 36 | eqtr4i 2847 |
. . . . . . . . 9
⊢ ℕ =
(ℤ≥‘(2 − 1)) |
38 | 33, 37 | eleqtrdi 2923 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
𝐴 ∈
(ℤ≥‘(2 − 1))) |
39 | | fzsuc2 12959 |
. . . . . . . 8
⊢ ((2
∈ ℤ ∧ 𝐴
∈ (ℤ≥‘(2 − 1))) → (2...(𝐴 + 1)) = ((2...𝐴) ∪ {(𝐴 + 1)})) |
40 | 23, 38, 39 | sylancr 589 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(2...(𝐴 + 1)) = ((2...𝐴) ∪ {(𝐴 + 1)})) |
41 | 40 | ineq1d 4188 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((2...(𝐴 + 1)) ∩
ℙ) = (((2...𝐴) ∪
{(𝐴 + 1)}) ∩
ℙ)) |
42 | | indir 4252 |
. . . . . 6
⊢
(((2...𝐴) ∪
{(𝐴 + 1)}) ∩ ℙ) =
(((2...𝐴) ∩ ℙ)
∪ ({(𝐴 + 1)} ∩
ℙ)) |
43 | 41, 42 | syl6eq 2872 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((2...(𝐴 + 1)) ∩
ℙ) = (((2...𝐴) ∩
ℙ) ∪ ({(𝐴 + 1)}
∩ ℙ))) |
44 | | simpr 487 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(𝐴 + 1) ∈
ℙ) |
45 | 44 | snssd 4736 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
{(𝐴 + 1)} ⊆
ℙ) |
46 | | df-ss 3952 |
. . . . . . 7
⊢ ({(𝐴 + 1)} ⊆ ℙ ↔
({(𝐴 + 1)} ∩ ℙ) =
{(𝐴 + 1)}) |
47 | 45, 46 | sylib 220 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
({(𝐴 + 1)} ∩ ℙ) =
{(𝐴 + 1)}) |
48 | 47 | uneq2d 4139 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(((2...𝐴) ∩ ℙ)
∪ ({(𝐴 + 1)} ∩
ℙ)) = (((2...𝐴) ∩
ℙ) ∪ {(𝐴 +
1)})) |
49 | 43, 48 | eqtrd 2856 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((2...(𝐴 + 1)) ∩
ℙ) = (((2...𝐴) ∩
ℙ) ∪ {(𝐴 +
1)})) |
50 | 49 | fveq2d 6669 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(♯‘((2...(𝐴 +
1)) ∩ ℙ)) = (♯‘(((2...𝐴) ∩ ℙ) ∪ {(𝐴 + 1)}))) |
51 | 22, 50 | eqtrd 2856 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(π‘(𝐴 + 1))
= (♯‘(((2...𝐴)
∩ ℙ) ∪ {(𝐴 +
1)}))) |
52 | | ppival2 25699 |
. . . 4
⊢ (𝐴 ∈ ℤ →
(π‘𝐴) =
(♯‘((2...𝐴)
∩ ℙ))) |
53 | 52 | adantr 483 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(π‘𝐴) =
(♯‘((2...𝐴)
∩ ℙ))) |
54 | 53 | oveq1d 7165 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
((π‘𝐴) + 1)
= ((♯‘((2...𝐴)
∩ ℙ)) + 1)) |
55 | 20, 51, 54 | 3eqtr4d 2866 |
1
⊢ ((𝐴 ∈ ℤ ∧ (𝐴 + 1) ∈ ℙ) →
(π‘(𝐴 + 1))
= ((π‘𝐴) +
1)) |