Proof of Theorem sbgoldbo
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | nfra1 3283 | . 2
⊢
Ⅎ𝑛∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) | 
| 2 |  | 3z 12652 | . . . . 5
⊢ 3 ∈
ℤ | 
| 3 |  | 6nn 12356 | . . . . . 6
⊢ 6 ∈
ℕ | 
| 4 | 3 | nnzi 12643 | . . . . 5
⊢ 6 ∈
ℤ | 
| 5 |  | 3re 12347 | . . . . . 6
⊢ 3 ∈
ℝ | 
| 6 |  | 6re 12357 | . . . . . 6
⊢ 6 ∈
ℝ | 
| 7 |  | 3lt6 12450 | . . . . . 6
⊢ 3 <
6 | 
| 8 | 5, 6, 7 | ltleii 11385 | . . . . 5
⊢ 3 ≤
6 | 
| 9 |  | eluz2 12885 | . . . . 5
⊢ (6 ∈
(ℤ≥‘3) ↔ (3 ∈ ℤ ∧ 6 ∈
ℤ ∧ 3 ≤ 6)) | 
| 10 | 2, 4, 8, 9 | mpbir3an 1341 | . . . 4
⊢ 6 ∈
(ℤ≥‘3) | 
| 11 |  | uzsplit 13637 | . . . . 5
⊢ (6 ∈
(ℤ≥‘3) → (ℤ≥‘3) =
((3...(6 − 1)) ∪ (ℤ≥‘6))) | 
| 12 | 11 | eleq2d 2826 | . . . 4
⊢ (6 ∈
(ℤ≥‘3) → (𝑛 ∈ (ℤ≥‘3)
↔ 𝑛 ∈ ((3...(6
− 1)) ∪ (ℤ≥‘6)))) | 
| 13 | 10, 12 | ax-mp 5 | . . 3
⊢ (𝑛 ∈
(ℤ≥‘3) ↔ 𝑛 ∈ ((3...(6 − 1)) ∪
(ℤ≥‘6))) | 
| 14 |  | elun 4152 | . . . . 5
⊢ (𝑛 ∈ ((3...(6 − 1))
∪ (ℤ≥‘6)) ↔ (𝑛 ∈ (3...(6 − 1)) ∨ 𝑛 ∈
(ℤ≥‘6))) | 
| 15 |  | 6m1e5 12398 | . . . . . . . . . 10
⊢ (6
− 1) = 5 | 
| 16 | 15 | oveq2i 7443 | . . . . . . . . 9
⊢ (3...(6
− 1)) = (3...5) | 
| 17 |  | 5nn 12353 | . . . . . . . . . . . 12
⊢ 5 ∈
ℕ | 
| 18 | 17 | nnzi 12643 | . . . . . . . . . . 11
⊢ 5 ∈
ℤ | 
| 19 |  | 5re 12354 | . . . . . . . . . . . 12
⊢ 5 ∈
ℝ | 
| 20 |  | 3lt5 12445 | . . . . . . . . . . . 12
⊢ 3 <
5 | 
| 21 | 5, 19, 20 | ltleii 11385 | . . . . . . . . . . 11
⊢ 3 ≤
5 | 
| 22 |  | eluz2 12885 | . . . . . . . . . . 11
⊢ (5 ∈
(ℤ≥‘3) ↔ (3 ∈ ℤ ∧ 5 ∈
ℤ ∧ 3 ≤ 5)) | 
| 23 | 2, 18, 21, 22 | mpbir3an 1341 | . . . . . . . . . 10
⊢ 5 ∈
(ℤ≥‘3) | 
| 24 |  | fzopredsuc 47340 | . . . . . . . . . 10
⊢ (5 ∈
(ℤ≥‘3) → (3...5) = (({3} ∪ ((3 + 1)..^5))
∪ {5})) | 
| 25 | 23, 24 | ax-mp 5 | . . . . . . . . 9
⊢ (3...5) =
(({3} ∪ ((3 + 1)..^5)) ∪ {5}) | 
| 26 | 16, 25 | eqtri 2764 | . . . . . . . 8
⊢ (3...(6
− 1)) = (({3} ∪ ((3 + 1)..^5)) ∪ {5}) | 
| 27 | 26 | eleq2i 2832 | . . . . . . 7
⊢ (𝑛 ∈ (3...(6 − 1))
↔ 𝑛 ∈ (({3} ∪
((3 + 1)..^5)) ∪ {5})) | 
| 28 |  | elun 4152 | . . . . . . . . 9
⊢ (𝑛 ∈ (({3} ∪ ((3 +
1)..^5)) ∪ {5}) ↔ (𝑛 ∈ ({3} ∪ ((3 + 1)..^5)) ∨ 𝑛 ∈ {5})) | 
| 29 |  | elun 4152 | . . . . . . . . . . 11
⊢ (𝑛 ∈ ({3} ∪ ((3 +
1)..^5)) ↔ (𝑛 ∈
{3} ∨ 𝑛 ∈ ((3 +
1)..^5))) | 
| 30 |  | elsni 4642 | . . . . . . . . . . . . 13
⊢ (𝑛 ∈ {3} → 𝑛 = 3) | 
| 31 |  | 1ex 11258 | . . . . . . . . . . . . . . . . . . 19
⊢ 1 ∈
V | 
| 32 | 31 | snid 4661 | . . . . . . . . . . . . . . . . . 18
⊢ 1 ∈
{1} | 
| 33 | 32 | orci 865 | . . . . . . . . . . . . . . . . 17
⊢ (1 ∈
{1} ∨ 1 ∈ ℙ) | 
| 34 |  | elun 4152 | . . . . . . . . . . . . . . . . 17
⊢ (1 ∈
({1} ∪ ℙ) ↔ (1 ∈ {1} ∨ 1 ∈
ℙ)) | 
| 35 | 33, 34 | mpbir 231 | . . . . . . . . . . . . . . . 16
⊢ 1 ∈
({1} ∪ ℙ) | 
| 36 |  | sbgoldbo.p | . . . . . . . . . . . . . . . 16
⊢ 𝑃 = ({1} ∪
ℙ) | 
| 37 | 35, 36 | eleqtrri 2839 | . . . . . . . . . . . . . . 15
⊢ 1 ∈
𝑃 | 
| 38 | 37 | a1i 11 | . . . . . . . . . . . . . 14
⊢ (𝑛 = 3 → 1 ∈ 𝑃) | 
| 39 |  | simpl 482 | . . . . . . . . . . . . . . . 16
⊢ ((𝑛 = 3 ∧ 𝑝 = 1) → 𝑛 = 3) | 
| 40 |  | oveq1 7439 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑝 = 1 → (𝑝 + 𝑞) = (1 + 𝑞)) | 
| 41 | 40 | oveq1d 7447 | . . . . . . . . . . . . . . . . 17
⊢ (𝑝 = 1 → ((𝑝 + 𝑞) + 𝑟) = ((1 + 𝑞) + 𝑟)) | 
| 42 | 41 | adantl 481 | . . . . . . . . . . . . . . . 16
⊢ ((𝑛 = 3 ∧ 𝑝 = 1) → ((𝑝 + 𝑞) + 𝑟) = ((1 + 𝑞) + 𝑟)) | 
| 43 | 39, 42 | eqeq12d 2752 | . . . . . . . . . . . . . . 15
⊢ ((𝑛 = 3 ∧ 𝑝 = 1) → (𝑛 = ((𝑝 + 𝑞) + 𝑟) ↔ 3 = ((1 + 𝑞) + 𝑟))) | 
| 44 | 43 | 2rexbidv 3221 | . . . . . . . . . . . . . 14
⊢ ((𝑛 = 3 ∧ 𝑝 = 1) → (∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟) ↔ ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 3 = ((1 + 𝑞) + 𝑟))) | 
| 45 |  | oveq2 7440 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑞 = 1 → (1 + 𝑞) = (1 + 1)) | 
| 46 | 45 | oveq1d 7447 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑞 = 1 → ((1 + 𝑞) + 𝑟) = ((1 + 1) + 𝑟)) | 
| 47 | 46 | eqeq2d 2747 | . . . . . . . . . . . . . . . . 17
⊢ (𝑞 = 1 → (3 = ((1 + 𝑞) + 𝑟) ↔ 3 = ((1 + 1) + 𝑟))) | 
| 48 | 47 | rexbidv 3178 | . . . . . . . . . . . . . . . 16
⊢ (𝑞 = 1 → (∃𝑟 ∈ 𝑃 3 = ((1 + 𝑞) + 𝑟) ↔ ∃𝑟 ∈ 𝑃 3 = ((1 + 1) + 𝑟))) | 
| 49 | 48 | adantl 481 | . . . . . . . . . . . . . . 15
⊢ ((𝑛 = 3 ∧ 𝑞 = 1) → (∃𝑟 ∈ 𝑃 3 = ((1 + 𝑞) + 𝑟) ↔ ∃𝑟 ∈ 𝑃 3 = ((1 + 1) + 𝑟))) | 
| 50 |  | df-3 12331 | . . . . . . . . . . . . . . . . . . 19
⊢ 3 = (2 +
1) | 
| 51 |  | df-2 12330 | . . . . . . . . . . . . . . . . . . . 20
⊢ 2 = (1 +
1) | 
| 52 | 51 | oveq1i 7442 | . . . . . . . . . . . . . . . . . . 19
⊢ (2 + 1) =
((1 + 1) + 1) | 
| 53 | 50, 52 | eqtri 2764 | . . . . . . . . . . . . . . . . . 18
⊢ 3 = ((1 +
1) + 1) | 
| 54 |  | oveq2 7440 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑟 = 1 → ((1 + 1) + 𝑟) = ((1 + 1) +
1)) | 
| 55 | 53, 54 | eqtr4id 2795 | . . . . . . . . . . . . . . . . 17
⊢ (𝑟 = 1 → 3 = ((1 + 1) + 𝑟)) | 
| 56 | 55 | adantl 481 | . . . . . . . . . . . . . . . 16
⊢ ((𝑛 = 3 ∧ 𝑟 = 1) → 3 = ((1 + 1) + 𝑟)) | 
| 57 | 38, 56 | rspcedeq2vd 3629 | . . . . . . . . . . . . . . 15
⊢ (𝑛 = 3 → ∃𝑟 ∈ 𝑃 3 = ((1 + 1) + 𝑟)) | 
| 58 | 38, 49, 57 | rspcedvd 3623 | . . . . . . . . . . . . . 14
⊢ (𝑛 = 3 → ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 3 = ((1 + 𝑞) + 𝑟)) | 
| 59 | 38, 44, 58 | rspcedvd 3623 | . . . . . . . . . . . . 13
⊢ (𝑛 = 3 → ∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 60 | 30, 59 | syl 17 | . . . . . . . . . . . 12
⊢ (𝑛 ∈ {3} → ∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 61 |  | 3p1e4 12412 | . . . . . . . . . . . . . . . . 17
⊢ (3 + 1) =
4 | 
| 62 |  | df-5 12333 | . . . . . . . . . . . . . . . . 17
⊢ 5 = (4 +
1) | 
| 63 | 61, 62 | oveq12i 7444 | . . . . . . . . . . . . . . . 16
⊢ ((3 +
1)..^5) = (4..^(4 + 1)) | 
| 64 |  | 4z 12653 | . . . . . . . . . . . . . . . . 17
⊢ 4 ∈
ℤ | 
| 65 |  | fzval3 13774 | . . . . . . . . . . . . . . . . 17
⊢ (4 ∈
ℤ → (4...4) = (4..^(4 + 1))) | 
| 66 | 64, 65 | ax-mp 5 | . . . . . . . . . . . . . . . 16
⊢ (4...4) =
(4..^(4 + 1)) | 
| 67 | 63, 66 | eqtr4i 2767 | . . . . . . . . . . . . . . 15
⊢ ((3 +
1)..^5) = (4...4) | 
| 68 | 67 | eleq2i 2832 | . . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ((3 + 1)..^5) ↔
𝑛 ∈
(4...4)) | 
| 69 |  | fzsn 13607 | . . . . . . . . . . . . . . . 16
⊢ (4 ∈
ℤ → (4...4) = {4}) | 
| 70 | 64, 69 | ax-mp 5 | . . . . . . . . . . . . . . 15
⊢ (4...4) =
{4} | 
| 71 | 70 | eleq2i 2832 | . . . . . . . . . . . . . 14
⊢ (𝑛 ∈ (4...4) ↔ 𝑛 ∈ {4}) | 
| 72 | 68, 71 | bitri 275 | . . . . . . . . . . . . 13
⊢ (𝑛 ∈ ((3 + 1)..^5) ↔
𝑛 ∈
{4}) | 
| 73 |  | elsni 4642 | . . . . . . . . . . . . . 14
⊢ (𝑛 ∈ {4} → 𝑛 = 4) | 
| 74 |  | 2prm 16730 | . . . . . . . . . . . . . . . . . . 19
⊢ 2 ∈
ℙ | 
| 75 | 74 | olci 866 | . . . . . . . . . . . . . . . . . 18
⊢ (2 ∈
{1} ∨ 2 ∈ ℙ) | 
| 76 |  | elun 4152 | . . . . . . . . . . . . . . . . . 18
⊢ (2 ∈
({1} ∪ ℙ) ↔ (2 ∈ {1} ∨ 2 ∈
ℙ)) | 
| 77 | 75, 76 | mpbir 231 | . . . . . . . . . . . . . . . . 17
⊢ 2 ∈
({1} ∪ ℙ) | 
| 78 | 77, 36 | eleqtrri 2839 | . . . . . . . . . . . . . . . 16
⊢ 2 ∈
𝑃 | 
| 79 | 78 | a1i 11 | . . . . . . . . . . . . . . 15
⊢ (𝑛 = 4 → 2 ∈ 𝑃) | 
| 80 |  | oveq1 7439 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑝 = 2 → (𝑝 + 𝑞) = (2 + 𝑞)) | 
| 81 | 80 | oveq1d 7447 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑝 = 2 → ((𝑝 + 𝑞) + 𝑟) = ((2 + 𝑞) + 𝑟)) | 
| 82 | 81 | eqeq2d 2747 | . . . . . . . . . . . . . . . . 17
⊢ (𝑝 = 2 → (𝑛 = ((𝑝 + 𝑞) + 𝑟) ↔ 𝑛 = ((2 + 𝑞) + 𝑟))) | 
| 83 | 82 | 2rexbidv 3221 | . . . . . . . . . . . . . . . 16
⊢ (𝑝 = 2 → (∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟) ↔ ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((2 + 𝑞) + 𝑟))) | 
| 84 | 83 | adantl 481 | . . . . . . . . . . . . . . 15
⊢ ((𝑛 = 4 ∧ 𝑝 = 2) → (∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟) ↔ ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((2 + 𝑞) + 𝑟))) | 
| 85 | 37 | a1i 11 | . . . . . . . . . . . . . . . 16
⊢ (𝑛 = 4 → 1 ∈ 𝑃) | 
| 86 |  | oveq2 7440 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑞 = 1 → (2 + 𝑞) = (2 + 1)) | 
| 87 | 86 | oveq1d 7447 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑞 = 1 → ((2 + 𝑞) + 𝑟) = ((2 + 1) + 𝑟)) | 
| 88 | 87 | eqeq2d 2747 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑞 = 1 → (𝑛 = ((2 + 𝑞) + 𝑟) ↔ 𝑛 = ((2 + 1) + 𝑟))) | 
| 89 | 88 | rexbidv 3178 | . . . . . . . . . . . . . . . . 17
⊢ (𝑞 = 1 → (∃𝑟 ∈ 𝑃 𝑛 = ((2 + 𝑞) + 𝑟) ↔ ∃𝑟 ∈ 𝑃 𝑛 = ((2 + 1) + 𝑟))) | 
| 90 | 89 | adantl 481 | . . . . . . . . . . . . . . . 16
⊢ ((𝑛 = 4 ∧ 𝑞 = 1) → (∃𝑟 ∈ 𝑃 𝑛 = ((2 + 𝑞) + 𝑟) ↔ ∃𝑟 ∈ 𝑃 𝑛 = ((2 + 1) + 𝑟))) | 
| 91 |  | simpl 482 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 = 4 ∧ 𝑟 = 1) → 𝑛 = 4) | 
| 92 |  | df-4 12332 | . . . . . . . . . . . . . . . . . . . . 21
⊢ 4 = (3 +
1) | 
| 93 | 50 | oveq1i 7442 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (3 + 1) =
((2 + 1) + 1) | 
| 94 | 92, 93 | eqtri 2764 | . . . . . . . . . . . . . . . . . . . 20
⊢ 4 = ((2 +
1) + 1) | 
| 95 | 94 | a1i 11 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝑛 = 4 ∧ 𝑟 = 1) → 4 = ((2 + 1) +
1)) | 
| 96 |  | oveq2 7440 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑟 = 1 → ((2 + 1) + 𝑟) = ((2 + 1) +
1)) | 
| 97 | 96 | eqcomd 2742 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑟 = 1 → ((2 + 1) + 1) = ((2
+ 1) + 𝑟)) | 
| 98 | 97 | adantl 481 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝑛 = 4 ∧ 𝑟 = 1) → ((2 + 1) + 1) = ((2 + 1) + 𝑟)) | 
| 99 | 95, 98 | eqtrd 2776 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 = 4 ∧ 𝑟 = 1) → 4 = ((2 + 1) + 𝑟)) | 
| 100 | 91, 99 | eqtrd 2776 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑛 = 4 ∧ 𝑟 = 1) → 𝑛 = ((2 + 1) + 𝑟)) | 
| 101 | 85, 100 | rspcedeq2vd 3629 | . . . . . . . . . . . . . . . 16
⊢ (𝑛 = 4 → ∃𝑟 ∈ 𝑃 𝑛 = ((2 + 1) + 𝑟)) | 
| 102 | 85, 90, 101 | rspcedvd 3623 | . . . . . . . . . . . . . . 15
⊢ (𝑛 = 4 → ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((2 + 𝑞) + 𝑟)) | 
| 103 | 79, 84, 102 | rspcedvd 3623 | . . . . . . . . . . . . . 14
⊢ (𝑛 = 4 → ∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 104 | 73, 103 | syl 17 | . . . . . . . . . . . . 13
⊢ (𝑛 ∈ {4} → ∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 105 | 72, 104 | sylbi 217 | . . . . . . . . . . . 12
⊢ (𝑛 ∈ ((3 + 1)..^5) →
∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 106 | 60, 105 | jaoi 857 | . . . . . . . . . . 11
⊢ ((𝑛 ∈ {3} ∨ 𝑛 ∈ ((3 + 1)..^5)) →
∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 107 | 29, 106 | sylbi 217 | . . . . . . . . . 10
⊢ (𝑛 ∈ ({3} ∪ ((3 +
1)..^5)) → ∃𝑝
∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 108 |  | elsni 4642 | . . . . . . . . . . 11
⊢ (𝑛 ∈ {5} → 𝑛 = 5) | 
| 109 |  | 3prm 16732 | . . . . . . . . . . . . . . . 16
⊢ 3 ∈
ℙ | 
| 110 | 109 | olci 866 | . . . . . . . . . . . . . . 15
⊢ (3 ∈
{1} ∨ 3 ∈ ℙ) | 
| 111 |  | elun 4152 | . . . . . . . . . . . . . . 15
⊢ (3 ∈
({1} ∪ ℙ) ↔ (3 ∈ {1} ∨ 3 ∈
ℙ)) | 
| 112 | 110, 111 | mpbir 231 | . . . . . . . . . . . . . 14
⊢ 3 ∈
({1} ∪ ℙ) | 
| 113 | 112, 36 | eleqtrri 2839 | . . . . . . . . . . . . 13
⊢ 3 ∈
𝑃 | 
| 114 | 113 | a1i 11 | . . . . . . . . . . . 12
⊢ (𝑛 = 5 → 3 ∈ 𝑃) | 
| 115 |  | oveq1 7439 | . . . . . . . . . . . . . . . 16
⊢ (𝑝 = 3 → (𝑝 + 𝑞) = (3 + 𝑞)) | 
| 116 | 115 | oveq1d 7447 | . . . . . . . . . . . . . . 15
⊢ (𝑝 = 3 → ((𝑝 + 𝑞) + 𝑟) = ((3 + 𝑞) + 𝑟)) | 
| 117 | 116 | eqeq2d 2747 | . . . . . . . . . . . . . 14
⊢ (𝑝 = 3 → (𝑛 = ((𝑝 + 𝑞) + 𝑟) ↔ 𝑛 = ((3 + 𝑞) + 𝑟))) | 
| 118 | 117 | 2rexbidv 3221 | . . . . . . . . . . . . 13
⊢ (𝑝 = 3 → (∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟) ↔ ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((3 + 𝑞) + 𝑟))) | 
| 119 | 118 | adantl 481 | . . . . . . . . . . . 12
⊢ ((𝑛 = 5 ∧ 𝑝 = 3) → (∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟) ↔ ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((3 + 𝑞) + 𝑟))) | 
| 120 | 37 | a1i 11 | . . . . . . . . . . . . 13
⊢ (𝑛 = 5 → 1 ∈ 𝑃) | 
| 121 |  | oveq2 7440 | . . . . . . . . . . . . . . . . 17
⊢ (𝑞 = 1 → (3 + 𝑞) = (3 + 1)) | 
| 122 | 121 | oveq1d 7447 | . . . . . . . . . . . . . . . 16
⊢ (𝑞 = 1 → ((3 + 𝑞) + 𝑟) = ((3 + 1) + 𝑟)) | 
| 123 | 122 | eqeq2d 2747 | . . . . . . . . . . . . . . 15
⊢ (𝑞 = 1 → (𝑛 = ((3 + 𝑞) + 𝑟) ↔ 𝑛 = ((3 + 1) + 𝑟))) | 
| 124 | 123 | rexbidv 3178 | . . . . . . . . . . . . . 14
⊢ (𝑞 = 1 → (∃𝑟 ∈ 𝑃 𝑛 = ((3 + 𝑞) + 𝑟) ↔ ∃𝑟 ∈ 𝑃 𝑛 = ((3 + 1) + 𝑟))) | 
| 125 | 124 | adantl 481 | . . . . . . . . . . . . 13
⊢ ((𝑛 = 5 ∧ 𝑞 = 1) → (∃𝑟 ∈ 𝑃 𝑛 = ((3 + 𝑞) + 𝑟) ↔ ∃𝑟 ∈ 𝑃 𝑛 = ((3 + 1) + 𝑟))) | 
| 126 |  | simpl 482 | . . . . . . . . . . . . . . 15
⊢ ((𝑛 = 5 ∧ 𝑟 = 1) → 𝑛 = 5) | 
| 127 | 92 | oveq1i 7442 | . . . . . . . . . . . . . . . . . 18
⊢ (4 + 1) =
((3 + 1) + 1) | 
| 128 | 62, 127 | eqtri 2764 | . . . . . . . . . . . . . . . . 17
⊢ 5 = ((3 +
1) + 1) | 
| 129 |  | oveq2 7440 | . . . . . . . . . . . . . . . . 17
⊢ (𝑟 = 1 → ((3 + 1) + 𝑟) = ((3 + 1) +
1)) | 
| 130 | 128, 129 | eqtr4id 2795 | . . . . . . . . . . . . . . . 16
⊢ (𝑟 = 1 → 5 = ((3 + 1) + 𝑟)) | 
| 131 | 130 | adantl 481 | . . . . . . . . . . . . . . 15
⊢ ((𝑛 = 5 ∧ 𝑟 = 1) → 5 = ((3 + 1) + 𝑟)) | 
| 132 | 126, 131 | eqtrd 2776 | . . . . . . . . . . . . . 14
⊢ ((𝑛 = 5 ∧ 𝑟 = 1) → 𝑛 = ((3 + 1) + 𝑟)) | 
| 133 | 120, 132 | rspcedeq2vd 3629 | . . . . . . . . . . . . 13
⊢ (𝑛 = 5 → ∃𝑟 ∈ 𝑃 𝑛 = ((3 + 1) + 𝑟)) | 
| 134 | 120, 125,
133 | rspcedvd 3623 | . . . . . . . . . . . 12
⊢ (𝑛 = 5 → ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((3 + 𝑞) + 𝑟)) | 
| 135 | 114, 119,
134 | rspcedvd 3623 | . . . . . . . . . . 11
⊢ (𝑛 = 5 → ∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 136 | 108, 135 | syl 17 | . . . . . . . . . 10
⊢ (𝑛 ∈ {5} → ∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 137 | 107, 136 | jaoi 857 | . . . . . . . . 9
⊢ ((𝑛 ∈ ({3} ∪ ((3 +
1)..^5)) ∨ 𝑛 ∈ {5})
→ ∃𝑝 ∈
𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 138 | 28, 137 | sylbi 217 | . . . . . . . 8
⊢ (𝑛 ∈ (({3} ∪ ((3 +
1)..^5)) ∪ {5}) → ∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 139 | 138 | a1d 25 | . . . . . . 7
⊢ (𝑛 ∈ (({3} ∪ ((3 +
1)..^5)) ∪ {5}) → (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟))) | 
| 140 | 27, 139 | sylbi 217 | . . . . . 6
⊢ (𝑛 ∈ (3...(6 − 1))
→ (∀𝑛 ∈
Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) →
∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟))) | 
| 141 |  | sbgoldbm 47776 | . . . . . . . 8
⊢
(∀𝑛 ∈
Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) →
∀𝑛 ∈
(ℤ≥‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 142 |  | rspa 3247 | . . . . . . . . . 10
⊢
((∀𝑛 ∈
(ℤ≥‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟) ∧ 𝑛 ∈ (ℤ≥‘6))
→ ∃𝑝 ∈
ℙ ∃𝑞 ∈
ℙ ∃𝑟 ∈
ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 143 |  | ssun2 4178 | . . . . . . . . . . . . 13
⊢ ℙ
⊆ ({1} ∪ ℙ) | 
| 144 | 143, 36 | sseqtrri 4032 | . . . . . . . . . . . 12
⊢ ℙ
⊆ 𝑃 | 
| 145 |  | rexss 4058 | . . . . . . . . . . . 12
⊢ (ℙ
⊆ 𝑃 →
(∃𝑝 ∈ ℙ
∃𝑞 ∈ ℙ
∃𝑟 ∈ ℙ
𝑛 = ((𝑝 + 𝑞) + 𝑟) ↔ ∃𝑝 ∈ 𝑃 (𝑝 ∈ ℙ ∧ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))) | 
| 146 | 144, 145 | ax-mp 5 | . . . . . . . . . . 11
⊢
(∃𝑝 ∈
ℙ ∃𝑞 ∈
ℙ ∃𝑟 ∈
ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟) ↔ ∃𝑝 ∈ 𝑃 (𝑝 ∈ ℙ ∧ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))) | 
| 147 |  | rexss 4058 | . . . . . . . . . . . . . . 15
⊢ (ℙ
⊆ 𝑃 →
(∃𝑞 ∈ ℙ
∃𝑟 ∈ ℙ
𝑛 = ((𝑝 + 𝑞) + 𝑟) ↔ ∃𝑞 ∈ 𝑃 (𝑞 ∈ ℙ ∧ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))) | 
| 148 | 144, 147 | ax-mp 5 | . . . . . . . . . . . . . 14
⊢
(∃𝑞 ∈
ℙ ∃𝑟 ∈
ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟) ↔ ∃𝑞 ∈ 𝑃 (𝑞 ∈ ℙ ∧ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))) | 
| 149 |  | rexss 4058 | . . . . . . . . . . . . . . . . . 18
⊢ (ℙ
⊆ 𝑃 →
(∃𝑟 ∈ ℙ
𝑛 = ((𝑝 + 𝑞) + 𝑟) ↔ ∃𝑟 ∈ 𝑃 (𝑟 ∈ ℙ ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))) | 
| 150 | 144, 149 | ax-mp 5 | . . . . . . . . . . . . . . . . 17
⊢
(∃𝑟 ∈
ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟) ↔ ∃𝑟 ∈ 𝑃 (𝑟 ∈ ℙ ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟))) | 
| 151 |  | simpr 484 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝑟 ∈ ℙ ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟)) → 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 152 | 151 | reximi 3083 | . . . . . . . . . . . . . . . . 17
⊢
(∃𝑟 ∈
𝑃 (𝑟 ∈ ℙ ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟)) → ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 153 | 150, 152 | sylbi 217 | . . . . . . . . . . . . . . . 16
⊢
(∃𝑟 ∈
ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟) → ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 154 | 153 | adantl 481 | . . . . . . . . . . . . . . 15
⊢ ((𝑞 ∈ ℙ ∧
∃𝑟 ∈ ℙ
𝑛 = ((𝑝 + 𝑞) + 𝑟)) → ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 155 | 154 | reximi 3083 | . . . . . . . . . . . . . 14
⊢
(∃𝑞 ∈
𝑃 (𝑞 ∈ ℙ ∧ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)) → ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 156 | 148, 155 | sylbi 217 | . . . . . . . . . . . . 13
⊢
(∃𝑞 ∈
ℙ ∃𝑟 ∈
ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟) → ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 157 | 156 | adantl 481 | . . . . . . . . . . . 12
⊢ ((𝑝 ∈ ℙ ∧
∃𝑞 ∈ ℙ
∃𝑟 ∈ ℙ
𝑛 = ((𝑝 + 𝑞) + 𝑟)) → ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 158 | 157 | reximi 3083 | . . . . . . . . . . 11
⊢
(∃𝑝 ∈
𝑃 (𝑝 ∈ ℙ ∧ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)) → ∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 159 | 146, 158 | sylbi 217 | . . . . . . . . . 10
⊢
(∃𝑝 ∈
ℙ ∃𝑞 ∈
ℙ ∃𝑟 ∈
ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟) → ∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 160 | 142, 159 | syl 17 | . . . . . . . . 9
⊢
((∀𝑛 ∈
(ℤ≥‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟) ∧ 𝑛 ∈ (ℤ≥‘6))
→ ∃𝑝 ∈
𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | 
| 161 | 160 | ex 412 | . . . . . . . 8
⊢
(∀𝑛 ∈
(ℤ≥‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟) → (𝑛 ∈ (ℤ≥‘6)
→ ∃𝑝 ∈
𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟))) | 
| 162 | 141, 161 | syl 17 | . . . . . . 7
⊢
(∀𝑛 ∈
Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) →
(𝑛 ∈
(ℤ≥‘6) → ∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟))) | 
| 163 | 162 | com12 32 | . . . . . 6
⊢ (𝑛 ∈
(ℤ≥‘6) → (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟))) | 
| 164 | 140, 163 | jaoi 857 | . . . . 5
⊢ ((𝑛 ∈ (3...(6 − 1)) ∨
𝑛 ∈
(ℤ≥‘6)) → (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟))) | 
| 165 | 14, 164 | sylbi 217 | . . . 4
⊢ (𝑛 ∈ ((3...(6 − 1))
∪ (ℤ≥‘6)) → (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟))) | 
| 166 | 165 | com12 32 | . . 3
⊢
(∀𝑛 ∈
Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) →
(𝑛 ∈ ((3...(6 −
1)) ∪ (ℤ≥‘6)) → ∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟))) | 
| 167 | 13, 166 | biimtrid 242 | . 2
⊢
(∀𝑛 ∈
Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) →
(𝑛 ∈
(ℤ≥‘3) → ∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟))) | 
| 168 | 1, 167 | ralrimi 3256 | 1
⊢
(∀𝑛 ∈
Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) →
∀𝑛 ∈
(ℤ≥‘3)∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) |