Proof of Theorem 31prm
| Step | Hyp | Ref
| Expression |
| 1 | | 2z 12649 |
. . 3
⊢ 2 ∈
ℤ |
| 2 | | 3nn0 12544 |
. . . . 5
⊢ 3 ∈
ℕ0 |
| 3 | | 1nn0 12542 |
. . . . 5
⊢ 1 ∈
ℕ0 |
| 4 | 2, 3 | deccl 12748 |
. . . 4
⊢ ;31 ∈
ℕ0 |
| 5 | 4 | nn0zi 12642 |
. . 3
⊢ ;31 ∈ ℤ |
| 6 | | 3nn 12345 |
. . . 4
⊢ 3 ∈
ℕ |
| 7 | | 2nn0 12543 |
. . . 4
⊢ 2 ∈
ℕ0 |
| 8 | | 2re 12340 |
. . . . 5
⊢ 2 ∈
ℝ |
| 9 | | 9re 12365 |
. . . . 5
⊢ 9 ∈
ℝ |
| 10 | | 2lt9 12471 |
. . . . 5
⊢ 2 <
9 |
| 11 | 8, 9, 10 | ltleii 11384 |
. . . 4
⊢ 2 ≤
9 |
| 12 | 6, 3, 7, 11 | declei 12769 |
. . 3
⊢ 2 ≤
;31 |
| 13 | | eluz2 12884 |
. . 3
⊢ (;31 ∈
(ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31)) |
| 14 | 1, 5, 12, 13 | mpbir3an 1342 |
. 2
⊢ ;31 ∈
(ℤ≥‘2) |
| 15 | | elun 4153 |
. . . . . 6
⊢ (𝑛 ∈ (({2, 3} ∩ ℙ)
∪ ({4, 5} ∩ ℙ)) ↔ (𝑛 ∈ ({2, 3} ∩ ℙ) ∨ 𝑛 ∈ ({4, 5} ∩
ℙ))) |
| 16 | | elin 3967 |
. . . . . . . 8
⊢ (𝑛 ∈ ({2, 3} ∩ ℙ)
↔ (𝑛 ∈ {2, 3}
∧ 𝑛 ∈
ℙ)) |
| 17 | | vex 3484 |
. . . . . . . . . . 11
⊢ 𝑛 ∈ V |
| 18 | 17 | elpr 4650 |
. . . . . . . . . 10
⊢ (𝑛 ∈ {2, 3} ↔ (𝑛 = 2 ∨ 𝑛 = 3)) |
| 19 | | 0nn0 12541 |
. . . . . . . . . . . . 13
⊢ 0 ∈
ℕ0 |
| 20 | | 2cn 12341 |
. . . . . . . . . . . . . 14
⊢ 2 ∈
ℂ |
| 21 | 20 | mul02i 11450 |
. . . . . . . . . . . . 13
⊢ (0
· 2) = 0 |
| 22 | | 1e0p1 12775 |
. . . . . . . . . . . . 13
⊢ 1 = (0 +
1) |
| 23 | 2, 19, 21, 22 | dec2dvds 17101 |
. . . . . . . . . . . 12
⊢ ¬ 2
∥ ;31 |
| 24 | | breq1 5146 |
. . . . . . . . . . . 12
⊢ (𝑛 = 2 → (𝑛 ∥ ;31 ↔ 2 ∥ ;31)) |
| 25 | 23, 24 | mtbiri 327 |
. . . . . . . . . . 11
⊢ (𝑛 = 2 → ¬ 𝑛 ∥ ;31) |
| 26 | | 3ndvds4 47582 |
. . . . . . . . . . . . 13
⊢ ¬ 3
∥ 4 |
| 27 | 2, 3 | 3dvdsdec 16369 |
. . . . . . . . . . . . . 14
⊢ (3
∥ ;31 ↔ 3 ∥ (3 +
1)) |
| 28 | | 3p1e4 12411 |
. . . . . . . . . . . . . . 15
⊢ (3 + 1) =
4 |
| 29 | 28 | breq2i 5151 |
. . . . . . . . . . . . . 14
⊢ (3
∥ (3 + 1) ↔ 3 ∥ 4) |
| 30 | 27, 29 | bitri 275 |
. . . . . . . . . . . . 13
⊢ (3
∥ ;31 ↔ 3 ∥
4) |
| 31 | 26, 30 | mtbir 323 |
. . . . . . . . . . . 12
⊢ ¬ 3
∥ ;31 |
| 32 | | breq1 5146 |
. . . . . . . . . . . 12
⊢ (𝑛 = 3 → (𝑛 ∥ ;31 ↔ 3 ∥ ;31)) |
| 33 | 31, 32 | mtbiri 327 |
. . . . . . . . . . 11
⊢ (𝑛 = 3 → ¬ 𝑛 ∥ ;31) |
| 34 | 25, 33 | jaoi 858 |
. . . . . . . . . 10
⊢ ((𝑛 = 2 ∨ 𝑛 = 3) → ¬ 𝑛 ∥ ;31) |
| 35 | 18, 34 | sylbi 217 |
. . . . . . . . 9
⊢ (𝑛 ∈ {2, 3} → ¬
𝑛 ∥ ;31) |
| 36 | 35 | adantr 480 |
. . . . . . . 8
⊢ ((𝑛 ∈ {2, 3} ∧ 𝑛 ∈ ℙ) → ¬
𝑛 ∥ ;31) |
| 37 | 16, 36 | sylbi 217 |
. . . . . . 7
⊢ (𝑛 ∈ ({2, 3} ∩ ℙ)
→ ¬ 𝑛 ∥
;31) |
| 38 | | elin 3967 |
. . . . . . . 8
⊢ (𝑛 ∈ ({4, 5} ∩ ℙ)
↔ (𝑛 ∈ {4, 5}
∧ 𝑛 ∈
ℙ)) |
| 39 | 17 | elpr 4650 |
. . . . . . . . . 10
⊢ (𝑛 ∈ {4, 5} ↔ (𝑛 = 4 ∨ 𝑛 = 5)) |
| 40 | | eleq1 2829 |
. . . . . . . . . . . 12
⊢ (𝑛 = 4 → (𝑛 ∈ ℙ ↔ 4 ∈
ℙ)) |
| 41 | | 4nprm 16732 |
. . . . . . . . . . . . 13
⊢ ¬ 4
∈ ℙ |
| 42 | 41 | pm2.21i 119 |
. . . . . . . . . . . 12
⊢ (4 ∈
ℙ → ¬ 𝑛
∥ ;31) |
| 43 | 40, 42 | biimtrdi 253 |
. . . . . . . . . . 11
⊢ (𝑛 = 4 → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31)) |
| 44 | | 1nn 12277 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℕ |
| 45 | | 1lt5 12446 |
. . . . . . . . . . . . . 14
⊢ 1 <
5 |
| 46 | 2, 44, 45 | dec5dvds 17102 |
. . . . . . . . . . . . 13
⊢ ¬ 5
∥ ;31 |
| 47 | | breq1 5146 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 5 → (𝑛 ∥ ;31 ↔ 5 ∥ ;31)) |
| 48 | 46, 47 | mtbiri 327 |
. . . . . . . . . . . 12
⊢ (𝑛 = 5 → ¬ 𝑛 ∥ ;31) |
| 49 | 48 | a1d 25 |
. . . . . . . . . . 11
⊢ (𝑛 = 5 → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31)) |
| 50 | 43, 49 | jaoi 858 |
. . . . . . . . . 10
⊢ ((𝑛 = 4 ∨ 𝑛 = 5) → (𝑛 ∈ ℙ → ¬ 𝑛 ∥ ;31)) |
| 51 | 39, 50 | sylbi 217 |
. . . . . . . . 9
⊢ (𝑛 ∈ {4, 5} → (𝑛 ∈ ℙ → ¬
𝑛 ∥ ;31)) |
| 52 | 51 | imp 406 |
. . . . . . . 8
⊢ ((𝑛 ∈ {4, 5} ∧ 𝑛 ∈ ℙ) → ¬
𝑛 ∥ ;31) |
| 53 | 38, 52 | sylbi 217 |
. . . . . . 7
⊢ (𝑛 ∈ ({4, 5} ∩ ℙ)
→ ¬ 𝑛 ∥
;31) |
| 54 | 37, 53 | jaoi 858 |
. . . . . 6
⊢ ((𝑛 ∈ ({2, 3} ∩ ℙ)
∨ 𝑛 ∈ ({4, 5} ∩
ℙ)) → ¬ 𝑛
∥ ;31) |
| 55 | 15, 54 | sylbi 217 |
. . . . 5
⊢ (𝑛 ∈ (({2, 3} ∩ ℙ)
∪ ({4, 5} ∩ ℙ)) → ¬ 𝑛 ∥ ;31) |
| 56 | | indir 4286 |
. . . . 5
⊢ (({2, 3}
∪ {4, 5}) ∩ ℙ) = (({2, 3} ∩ ℙ) ∪ ({4, 5} ∩
ℙ)) |
| 57 | 55, 56 | eleq2s 2859 |
. . . 4
⊢ (𝑛 ∈ (({2, 3} ∪ {4, 5})
∩ ℙ) → ¬ 𝑛 ∥ ;31) |
| 58 | | 5nn0 12546 |
. . . . . . . . 9
⊢ 5 ∈
ℕ0 |
| 59 | | 5re 12353 |
. . . . . . . . . 10
⊢ 5 ∈
ℝ |
| 60 | | 5lt9 12468 |
. . . . . . . . . 10
⊢ 5 <
9 |
| 61 | 59, 9, 60 | ltleii 11384 |
. . . . . . . . 9
⊢ 5 ≤
9 |
| 62 | | 2lt3 12438 |
. . . . . . . . 9
⊢ 2 <
3 |
| 63 | 7, 2, 58, 3, 61, 62 | decleh 12768 |
. . . . . . . 8
⊢ ;25 ≤ ;31 |
| 64 | | 6nn 12355 |
. . . . . . . . 9
⊢ 6 ∈
ℕ |
| 65 | | 1lt6 12451 |
. . . . . . . . 9
⊢ 1 <
6 |
| 66 | 2, 3, 64, 65 | declt 12761 |
. . . . . . . 8
⊢ ;31 < ;36 |
| 67 | 4 | nn0rei 12537 |
. . . . . . . . . 10
⊢ ;31 ∈ ℝ |
| 68 | | 0re 11263 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℝ |
| 69 | | 9pos 12379 |
. . . . . . . . . . . 12
⊢ 0 <
9 |
| 70 | 68, 9, 69 | ltleii 11384 |
. . . . . . . . . . 11
⊢ 0 ≤
9 |
| 71 | 6, 3, 19, 70 | declei 12769 |
. . . . . . . . . 10
⊢ 0 ≤
;31 |
| 72 | 67, 71 | pm3.2i 470 |
. . . . . . . . 9
⊢ (;31 ∈ ℝ ∧ 0 ≤ ;31) |
| 73 | | flsqrt5 47581 |
. . . . . . . . . 10
⊢ ((;31 ∈ ℝ ∧ 0 ≤ ;31) → ((;25 ≤ ;31 ∧ ;31 < ;36) ↔ (⌊‘(√‘;31)) = 5)) |
| 74 | 73 | bicomd 223 |
. . . . . . . . 9
⊢ ((;31 ∈ ℝ ∧ 0 ≤ ;31) →
((⌊‘(√‘;31)) = 5 ↔ (;25 ≤ ;31 ∧ ;31 < ;36))) |
| 75 | 72, 74 | ax-mp 5 |
. . . . . . . 8
⊢
((⌊‘(√‘;31)) = 5 ↔ (;25 ≤ ;31 ∧ ;31 < ;36)) |
| 76 | 63, 66, 75 | mpbir2an 711 |
. . . . . . 7
⊢
(⌊‘(√‘;31)) = 5 |
| 77 | 76 | oveq2i 7442 |
. . . . . 6
⊢
(2...(⌊‘(√‘;31))) = (2...5) |
| 78 | | 5nn 12352 |
. . . . . . . . . 10
⊢ 5 ∈
ℕ |
| 79 | 78 | nnzi 12641 |
. . . . . . . . 9
⊢ 5 ∈
ℤ |
| 80 | | 3z 12650 |
. . . . . . . . 9
⊢ 3 ∈
ℤ |
| 81 | 1, 79, 80 | 3pm3.2i 1340 |
. . . . . . . 8
⊢ (2 ∈
ℤ ∧ 5 ∈ ℤ ∧ 3 ∈ ℤ) |
| 82 | | 3re 12346 |
. . . . . . . . . 10
⊢ 3 ∈
ℝ |
| 83 | 8, 82, 62 | ltleii 11384 |
. . . . . . . . 9
⊢ 2 ≤
3 |
| 84 | | 3lt5 12444 |
. . . . . . . . . 10
⊢ 3 <
5 |
| 85 | 82, 59, 84 | ltleii 11384 |
. . . . . . . . 9
⊢ 3 ≤
5 |
| 86 | 83, 85 | pm3.2i 470 |
. . . . . . . 8
⊢ (2 ≤ 3
∧ 3 ≤ 5) |
| 87 | | elfz2 13554 |
. . . . . . . 8
⊢ (3 ∈
(2...5) ↔ ((2 ∈ ℤ ∧ 5 ∈ ℤ ∧ 3 ∈
ℤ) ∧ (2 ≤ 3 ∧ 3 ≤ 5))) |
| 88 | 81, 86, 87 | mpbir2an 711 |
. . . . . . 7
⊢ 3 ∈
(2...5) |
| 89 | | fzsplit 13590 |
. . . . . . 7
⊢ (3 ∈
(2...5) → (2...5) = ((2...3) ∪ ((3 + 1)...5))) |
| 90 | 88, 89 | ax-mp 5 |
. . . . . 6
⊢ (2...5) =
((2...3) ∪ ((3 + 1)...5)) |
| 91 | | df-3 12330 |
. . . . . . . . 9
⊢ 3 = (2 +
1) |
| 92 | 91 | oveq2i 7442 |
. . . . . . . 8
⊢ (2...3) =
(2...(2 + 1)) |
| 93 | | fzpr 13619 |
. . . . . . . . 9
⊢ (2 ∈
ℤ → (2...(2 + 1)) = {2, (2 + 1)}) |
| 94 | 1, 93 | ax-mp 5 |
. . . . . . . 8
⊢ (2...(2 +
1)) = {2, (2 + 1)} |
| 95 | | 2p1e3 12408 |
. . . . . . . . 9
⊢ (2 + 1) =
3 |
| 96 | 95 | preq2i 4737 |
. . . . . . . 8
⊢ {2, (2 +
1)} = {2, 3} |
| 97 | 92, 94, 96 | 3eqtri 2769 |
. . . . . . 7
⊢ (2...3) =
{2, 3} |
| 98 | 28 | oveq1i 7441 |
. . . . . . . 8
⊢ ((3 +
1)...5) = (4...5) |
| 99 | | df-5 12332 |
. . . . . . . . 9
⊢ 5 = (4 +
1) |
| 100 | 99 | oveq2i 7442 |
. . . . . . . 8
⊢ (4...5) =
(4...(4 + 1)) |
| 101 | | 4z 12651 |
. . . . . . . . . 10
⊢ 4 ∈
ℤ |
| 102 | | fzpr 13619 |
. . . . . . . . . 10
⊢ (4 ∈
ℤ → (4...(4 + 1)) = {4, (4 + 1)}) |
| 103 | 101, 102 | ax-mp 5 |
. . . . . . . . 9
⊢ (4...(4 +
1)) = {4, (4 + 1)} |
| 104 | | 4p1e5 12412 |
. . . . . . . . . 10
⊢ (4 + 1) =
5 |
| 105 | 104 | preq2i 4737 |
. . . . . . . . 9
⊢ {4, (4 +
1)} = {4, 5} |
| 106 | 103, 105 | eqtri 2765 |
. . . . . . . 8
⊢ (4...(4 +
1)) = {4, 5} |
| 107 | 98, 100, 106 | 3eqtri 2769 |
. . . . . . 7
⊢ ((3 +
1)...5) = {4, 5} |
| 108 | 97, 107 | uneq12i 4166 |
. . . . . 6
⊢ ((2...3)
∪ ((3 + 1)...5)) = ({2, 3} ∪ {4, 5}) |
| 109 | 77, 90, 108 | 3eqtri 2769 |
. . . . 5
⊢
(2...(⌊‘(√‘;31))) = ({2, 3} ∪ {4, 5}) |
| 110 | 109 | ineq1i 4216 |
. . . 4
⊢
((2...(⌊‘(√‘;31))) ∩ ℙ) = (({2, 3} ∪ {4, 5}) ∩
ℙ) |
| 111 | 57, 110 | eleq2s 2859 |
. . 3
⊢ (𝑛 ∈
((2...(⌊‘(√‘;31))) ∩ ℙ) → ¬ 𝑛 ∥ ;31) |
| 112 | 111 | rgen 3063 |
. 2
⊢
∀𝑛 ∈
((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31 |
| 113 | | isprm7 16745 |
. 2
⊢ (;31 ∈ ℙ ↔ (;31 ∈
(ℤ≥‘2) ∧ ∀𝑛 ∈
((2...(⌊‘(√‘;31))) ∩ ℙ) ¬ 𝑛 ∥ ;31)) |
| 114 | 14, 112, 113 | mpbir2an 711 |
1
⊢ ;31 ∈ ℙ |