| Step | Hyp | Ref
| Expression |
| 1 | | 10nn 12749 |
. . 3
⊢ ;10 ∈ ℕ |
| 2 | | 2nn0 12543 |
. . . 4
⊢ 2 ∈
ℕ0 |
| 3 | | 7nn0 12548 |
. . . 4
⊢ 7 ∈
ℕ0 |
| 4 | 2, 3 | deccl 12748 |
. . 3
⊢ ;27 ∈
ℕ0 |
| 5 | | nnexpcl 14115 |
. . 3
⊢ ((;10 ∈ ℕ ∧ ;27 ∈ ℕ0) →
(;10↑;27) ∈ ℕ) |
| 6 | 1, 4, 5 | mp2an 692 |
. 2
⊢ (;10↑;27) ∈ ℕ |
| 7 | 6 | nnrei 12275 |
. . . 4
⊢ (;10↑;27) ∈ ℝ |
| 8 | 7 | leidi 11797 |
. . 3
⊢ (;10↑;27) ≤ (;10↑;27) |
| 9 | | simpl 482 |
. . . . . 6
⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → 𝑛 ∈ 𝑂) |
| 10 | | inss2 4238 |
. . . . . . . . . . . . . 14
⊢ (𝑂 ∩ ℙ) ⊆
ℙ |
| 11 | | prmssnn 16713 |
. . . . . . . . . . . . . 14
⊢ ℙ
⊆ ℕ |
| 12 | 10, 11 | sstri 3993 |
. . . . . . . . . . . . 13
⊢ (𝑂 ∩ ℙ) ⊆
ℕ |
| 13 | 12 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑂 ∩ ℙ) ⊆
ℕ) |
| 14 | | tgoldbachgt.o |
. . . . . . . . . . . . . . 15
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} |
| 15 | 14 | eleq2i 2833 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ 𝑂 ↔ 𝑛 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}) |
| 16 | | elrabi 3687 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} → 𝑛 ∈ ℤ) |
| 17 | 15, 16 | sylbi 217 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ 𝑂 → 𝑛 ∈ ℤ) |
| 18 | 17 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 𝑛 ∈ ℤ) |
| 19 | | 3nn0 12544 |
. . . . . . . . . . . . 13
⊢ 3 ∈
ℕ0 |
| 20 | 19 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 3 ∈
ℕ0) |
| 21 | | simpr 484 |
. . . . . . . . . . . 12
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) |
| 22 | 13, 18, 20, 21 | reprf 34627 |
. . . . . . . . . . 11
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 𝑐:(0..^3)⟶(𝑂 ∩ ℙ)) |
| 23 | | c0ex 11255 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
V |
| 24 | 23 | tpid1 4768 |
. . . . . . . . . . . . 13
⊢ 0 ∈
{0, 1, 2} |
| 25 | | fzo0to3tp 13791 |
. . . . . . . . . . . . 13
⊢ (0..^3) =
{0, 1, 2} |
| 26 | 24, 25 | eleqtrri 2840 |
. . . . . . . . . . . 12
⊢ 0 ∈
(0..^3) |
| 27 | 26 | a1i 11 |
. . . . . . . . . . 11
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 0 ∈
(0..^3)) |
| 28 | 22, 27 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ (𝑂 ∩ ℙ)) |
| 29 | 28 | elin2d 4205 |
. . . . . . . . 9
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ ℙ) |
| 30 | | 1ex 11257 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
V |
| 31 | 30 | tpid2 4770 |
. . . . . . . . . . . . 13
⊢ 1 ∈
{0, 1, 2} |
| 32 | 31, 25 | eleqtrri 2840 |
. . . . . . . . . . . 12
⊢ 1 ∈
(0..^3) |
| 33 | 32 | a1i 11 |
. . . . . . . . . . 11
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 1 ∈
(0..^3)) |
| 34 | 22, 33 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ (𝑂 ∩ ℙ)) |
| 35 | 34 | elin2d 4205 |
. . . . . . . . 9
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ ℙ) |
| 36 | | 2ex 12343 |
. . . . . . . . . . . . . 14
⊢ 2 ∈
V |
| 37 | 36 | tpid3 4773 |
. . . . . . . . . . . . 13
⊢ 2 ∈
{0, 1, 2} |
| 38 | 37, 25 | eleqtrri 2840 |
. . . . . . . . . . . 12
⊢ 2 ∈
(0..^3) |
| 39 | 38 | a1i 11 |
. . . . . . . . . . 11
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 2 ∈
(0..^3)) |
| 40 | 22, 39 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ (𝑂 ∩ ℙ)) |
| 41 | 40 | elin2d 4205 |
. . . . . . . . 9
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ ℙ) |
| 42 | 28 | elin1d 4204 |
. . . . . . . . . . 11
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ 𝑂) |
| 43 | 34 | elin1d 4204 |
. . . . . . . . . . 11
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ 𝑂) |
| 44 | 40 | elin1d 4204 |
. . . . . . . . . . 11
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ 𝑂) |
| 45 | 42, 43, 44 | 3jca 1129 |
. . . . . . . . . 10
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → ((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ (𝑐‘2) ∈ 𝑂)) |
| 46 | 25 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (0..^3) = {0, 1,
2}) |
| 47 | 46 | sumeq1d 15736 |
. . . . . . . . . . 11
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → Σ𝑖 ∈ (0..^3)(𝑐‘𝑖) = Σ𝑖 ∈ {0, 1, 2} (𝑐‘𝑖)) |
| 48 | 13, 18, 20, 21 | reprsum 34628 |
. . . . . . . . . . 11
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → Σ𝑖 ∈ (0..^3)(𝑐‘𝑖) = 𝑛) |
| 49 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑖 = 0 → (𝑐‘𝑖) = (𝑐‘0)) |
| 50 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑖 = 1 → (𝑐‘𝑖) = (𝑐‘1)) |
| 51 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑖 = 2 → (𝑐‘𝑖) = (𝑐‘2)) |
| 52 | 12, 28 | sselid 3981 |
. . . . . . . . . . . . . 14
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ ℕ) |
| 53 | 52 | nncnd 12282 |
. . . . . . . . . . . . 13
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘0) ∈ ℂ) |
| 54 | 12, 34 | sselid 3981 |
. . . . . . . . . . . . . 14
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ ℕ) |
| 55 | 54 | nncnd 12282 |
. . . . . . . . . . . . 13
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘1) ∈ ℂ) |
| 56 | 12, 40 | sselid 3981 |
. . . . . . . . . . . . . 14
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ ℕ) |
| 57 | 56 | nncnd 12282 |
. . . . . . . . . . . . 13
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (𝑐‘2) ∈ ℂ) |
| 58 | 53, 55, 57 | 3jca 1129 |
. . . . . . . . . . . 12
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → ((𝑐‘0) ∈ ℂ ∧ (𝑐‘1) ∈ ℂ ∧
(𝑐‘2) ∈
ℂ)) |
| 59 | 23 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 0 ∈
V) |
| 60 | 30 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 1 ∈
V) |
| 61 | 36 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 2 ∈
V) |
| 62 | 59, 60, 61 | 3jca 1129 |
. . . . . . . . . . . 12
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (0 ∈ V ∧ 1
∈ V ∧ 2 ∈ V)) |
| 63 | | 0ne1 12337 |
. . . . . . . . . . . . 13
⊢ 0 ≠
1 |
| 64 | 63 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 0 ≠
1) |
| 65 | | 0ne2 12473 |
. . . . . . . . . . . . 13
⊢ 0 ≠
2 |
| 66 | 65 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 0 ≠
2) |
| 67 | | 1ne2 12474 |
. . . . . . . . . . . . 13
⊢ 1 ≠
2 |
| 68 | 67 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 1 ≠
2) |
| 69 | 49, 50, 51, 58, 62, 64, 66, 68 | sumtp 15785 |
. . . . . . . . . . 11
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → Σ𝑖 ∈ {0, 1, 2} (𝑐‘𝑖) = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2))) |
| 70 | 47, 48, 69 | 3eqtr3d 2785 |
. . . . . . . . . 10
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → 𝑛 = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2))) |
| 71 | 45, 70 | jca 511 |
. . . . . . . . 9
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → (((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ (𝑐‘2) ∈ 𝑂) ∧ 𝑛 = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2)))) |
| 72 | | eleq1 2829 |
. . . . . . . . . . . 12
⊢ (𝑝 = (𝑐‘0) → (𝑝 ∈ 𝑂 ↔ (𝑐‘0) ∈ 𝑂)) |
| 73 | 72 | 3anbi1d 1442 |
. . . . . . . . . . 11
⊢ (𝑝 = (𝑐‘0) → ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ↔ ((𝑐‘0) ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂))) |
| 74 | | oveq1 7438 |
. . . . . . . . . . . . 13
⊢ (𝑝 = (𝑐‘0) → (𝑝 + 𝑞) = ((𝑐‘0) + 𝑞)) |
| 75 | 74 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢ (𝑝 = (𝑐‘0) → ((𝑝 + 𝑞) + 𝑟) = (((𝑐‘0) + 𝑞) + 𝑟)) |
| 76 | 75 | eqeq2d 2748 |
. . . . . . . . . . 11
⊢ (𝑝 = (𝑐‘0) → (𝑛 = ((𝑝 + 𝑞) + 𝑟) ↔ 𝑛 = (((𝑐‘0) + 𝑞) + 𝑟))) |
| 77 | 73, 76 | anbi12d 632 |
. . . . . . . . . 10
⊢ (𝑝 = (𝑐‘0) → (((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟)) ↔ (((𝑐‘0) ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = (((𝑐‘0) + 𝑞) + 𝑟)))) |
| 78 | | eleq1 2829 |
. . . . . . . . . . . 12
⊢ (𝑞 = (𝑐‘1) → (𝑞 ∈ 𝑂 ↔ (𝑐‘1) ∈ 𝑂)) |
| 79 | 78 | 3anbi2d 1443 |
. . . . . . . . . . 11
⊢ (𝑞 = (𝑐‘1) → (((𝑐‘0) ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ↔ ((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ 𝑟 ∈ 𝑂))) |
| 80 | | oveq2 7439 |
. . . . . . . . . . . . 13
⊢ (𝑞 = (𝑐‘1) → ((𝑐‘0) + 𝑞) = ((𝑐‘0) + (𝑐‘1))) |
| 81 | 80 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢ (𝑞 = (𝑐‘1) → (((𝑐‘0) + 𝑞) + 𝑟) = (((𝑐‘0) + (𝑐‘1)) + 𝑟)) |
| 82 | 81 | eqeq2d 2748 |
. . . . . . . . . . 11
⊢ (𝑞 = (𝑐‘1) → (𝑛 = (((𝑐‘0) + 𝑞) + 𝑟) ↔ 𝑛 = (((𝑐‘0) + (𝑐‘1)) + 𝑟))) |
| 83 | 79, 82 | anbi12d 632 |
. . . . . . . . . 10
⊢ (𝑞 = (𝑐‘1) → ((((𝑐‘0) ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = (((𝑐‘0) + 𝑞) + 𝑟)) ↔ (((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = (((𝑐‘0) + (𝑐‘1)) + 𝑟)))) |
| 84 | | eleq1 2829 |
. . . . . . . . . . . 12
⊢ (𝑟 = (𝑐‘2) → (𝑟 ∈ 𝑂 ↔ (𝑐‘2) ∈ 𝑂)) |
| 85 | 84 | 3anbi3d 1444 |
. . . . . . . . . . 11
⊢ (𝑟 = (𝑐‘2) → (((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ↔ ((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ (𝑐‘2) ∈ 𝑂))) |
| 86 | | oveq2 7439 |
. . . . . . . . . . . 12
⊢ (𝑟 = (𝑐‘2) → (((𝑐‘0) + (𝑐‘1)) + 𝑟) = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2))) |
| 87 | 86 | eqeq2d 2748 |
. . . . . . . . . . 11
⊢ (𝑟 = (𝑐‘2) → (𝑛 = (((𝑐‘0) + (𝑐‘1)) + 𝑟) ↔ 𝑛 = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2)))) |
| 88 | 85, 87 | anbi12d 632 |
. . . . . . . . . 10
⊢ (𝑟 = (𝑐‘2) → ((((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = (((𝑐‘0) + (𝑐‘1)) + 𝑟)) ↔ (((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ (𝑐‘2) ∈ 𝑂) ∧ 𝑛 = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2))))) |
| 89 | 77, 83, 88 | rspc3ev 3639 |
. . . . . . . . 9
⊢ ((((𝑐‘0) ∈ ℙ ∧
(𝑐‘1) ∈ ℙ
∧ (𝑐‘2) ∈
ℙ) ∧ (((𝑐‘0) ∈ 𝑂 ∧ (𝑐‘1) ∈ 𝑂 ∧ (𝑐‘2) ∈ 𝑂) ∧ 𝑛 = (((𝑐‘0) + (𝑐‘1)) + (𝑐‘2)))) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟))) |
| 90 | 29, 35, 41, 71, 89 | syl31anc 1375 |
. . . . . . . 8
⊢ (((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟))) |
| 91 | 90 | adantr 480 |
. . . . . . 7
⊢ ((((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) ∧ ⊤) →
∃𝑝 ∈ ℙ
∃𝑞 ∈ ℙ
∃𝑟 ∈ ℙ
((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟))) |
| 92 | 6 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (;10↑;27) ∈ ℕ) |
| 93 | 92 | nnred 12281 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (;10↑;27) ∈ ℝ) |
| 94 | 17 | zred 12722 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ 𝑂 → 𝑛 ∈ ℝ) |
| 95 | 94 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → 𝑛 ∈ ℝ) |
| 96 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (;10↑;27) < 𝑛) |
| 97 | 93, 95, 96 | ltled 11409 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (;10↑;27) ≤ 𝑛) |
| 98 | 14, 9, 97 | tgoldbachgtd 34677 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → 0 < (♯‘((𝑂 ∩
ℙ)(repr‘3)𝑛))) |
| 99 | | ovex 7464 |
. . . . . . . . . . . . . . 15
⊢ ((𝑂 ∩
ℙ)(repr‘3)𝑛)
∈ V |
| 100 | | hashneq0 14403 |
. . . . . . . . . . . . . . 15
⊢ (((𝑂 ∩
ℙ)(repr‘3)𝑛)
∈ V → (0 < (♯‘((𝑂 ∩ ℙ)(repr‘3)𝑛)) ↔ ((𝑂 ∩ ℙ)(repr‘3)𝑛) ≠
∅)) |
| 101 | 99, 100 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ (0 <
(♯‘((𝑂 ∩
ℙ)(repr‘3)𝑛))
↔ ((𝑂 ∩
ℙ)(repr‘3)𝑛)
≠ ∅) |
| 102 | 98, 101 | sylib 218 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ((𝑂 ∩ ℙ)(repr‘3)𝑛) ≠ ∅) |
| 103 | 102 | neneqd 2945 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ¬ ((𝑂 ∩ ℙ)(repr‘3)𝑛) = ∅) |
| 104 | | neq0 4352 |
. . . . . . . . . . . 12
⊢ (¬
((𝑂 ∩
ℙ)(repr‘3)𝑛) =
∅ ↔ ∃𝑐
𝑐 ∈ ((𝑂 ∩
ℙ)(repr‘3)𝑛)) |
| 105 | 103, 104 | sylib 218 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ∃𝑐 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)) |
| 106 | | tru 1544 |
. . . . . . . . . . 11
⊢
⊤ |
| 107 | 105, 106 | jctil 519 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → (⊤ ∧ ∃𝑐 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛))) |
| 108 | | 19.42v 1953 |
. . . . . . . . . 10
⊢
(∃𝑐(⊤
∧ 𝑐 ∈ ((𝑂 ∩
ℙ)(repr‘3)𝑛))
↔ (⊤ ∧ ∃𝑐 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛))) |
| 109 | 107, 108 | sylibr 234 |
. . . . . . . . 9
⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ∃𝑐(⊤ ∧ 𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛))) |
| 110 | | exancom 1861 |
. . . . . . . . 9
⊢
(∃𝑐(⊤
∧ 𝑐 ∈ ((𝑂 ∩
ℙ)(repr‘3)𝑛))
↔ ∃𝑐(𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛) ∧
⊤)) |
| 111 | 109, 110 | sylib 218 |
. . . . . . . 8
⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ∃𝑐(𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛) ∧
⊤)) |
| 112 | | df-rex 3071 |
. . . . . . . 8
⊢
(∃𝑐 ∈
((𝑂 ∩
ℙ)(repr‘3)𝑛)⊤ ↔ ∃𝑐(𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛) ∧
⊤)) |
| 113 | 111, 112 | sylibr 234 |
. . . . . . 7
⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ∃𝑐 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑛)⊤) |
| 114 | 91, 113 | r19.29a 3162 |
. . . . . 6
⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟))) |
| 115 | | tgoldbachgt.g |
. . . . . . . . 9
⊢ 𝐺 = {𝑧 ∈ 𝑂 ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))} |
| 116 | 115 | eleq2i 2833 |
. . . . . . . 8
⊢ (𝑛 ∈ 𝐺 ↔ 𝑛 ∈ {𝑧 ∈ 𝑂 ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}) |
| 117 | | eqeq1 2741 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑛 → (𝑧 = ((𝑝 + 𝑞) + 𝑟) ↔ 𝑛 = ((𝑝 + 𝑞) + 𝑟))) |
| 118 | 117 | anbi2d 630 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑛 → (((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟)) ↔ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))) |
| 119 | 118 | rexbidv 3179 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑛 → (∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟)) ↔ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))) |
| 120 | 119 | rexbidv 3179 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑛 → (∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟)) ↔ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))) |
| 121 | 120 | rexbidv 3179 |
. . . . . . . . 9
⊢ (𝑧 = 𝑛 → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟)) ↔ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))) |
| 122 | 121 | elrab3 3693 |
. . . . . . . 8
⊢ (𝑛 ∈ 𝑂 → (𝑛 ∈ {𝑧 ∈ 𝑂 ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))} ↔ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))) |
| 123 | 116, 122 | bitrid 283 |
. . . . . . 7
⊢ (𝑛 ∈ 𝑂 → (𝑛 ∈ 𝐺 ↔ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))) |
| 124 | 123 | biimpar 477 |
. . . . . 6
⊢ ((𝑛 ∈ 𝑂 ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑛 = ((𝑝 + 𝑞) + 𝑟))) → 𝑛 ∈ 𝐺) |
| 125 | 9, 114, 124 | syl2anc 584 |
. . . . 5
⊢ ((𝑛 ∈ 𝑂 ∧ (;10↑;27) < 𝑛) → 𝑛 ∈ 𝐺) |
| 126 | 125 | ex 412 |
. . . 4
⊢ (𝑛 ∈ 𝑂 → ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺)) |
| 127 | 126 | rgen 3063 |
. . 3
⊢
∀𝑛 ∈
𝑂 ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺) |
| 128 | 8, 127 | pm3.2i 470 |
. 2
⊢ ((;10↑;27) ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺)) |
| 129 | | breq1 5146 |
. . . 4
⊢ (𝑚 = (;10↑;27) → (𝑚 ≤ (;10↑;27) ↔ (;10↑;27) ≤ (;10↑;27))) |
| 130 | | breq1 5146 |
. . . . . 6
⊢ (𝑚 = (;10↑;27) → (𝑚 < 𝑛 ↔ (;10↑;27) < 𝑛)) |
| 131 | 130 | imbi1d 341 |
. . . . 5
⊢ (𝑚 = (;10↑;27) → ((𝑚 < 𝑛 → 𝑛 ∈ 𝐺) ↔ ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺))) |
| 132 | 131 | ralbidv 3178 |
. . . 4
⊢ (𝑚 = (;10↑;27) → (∀𝑛 ∈ 𝑂 (𝑚 < 𝑛 → 𝑛 ∈ 𝐺) ↔ ∀𝑛 ∈ 𝑂 ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺))) |
| 133 | 129, 132 | anbi12d 632 |
. . 3
⊢ (𝑚 = (;10↑;27) → ((𝑚 ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 (𝑚 < 𝑛 → 𝑛 ∈ 𝐺)) ↔ ((;10↑;27) ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺)))) |
| 134 | 133 | rspcev 3622 |
. 2
⊢ (((;10↑;27) ∈ ℕ ∧ ((;10↑;27) ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 ((;10↑;27) < 𝑛 → 𝑛 ∈ 𝐺))) → ∃𝑚 ∈ ℕ (𝑚 ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 (𝑚 < 𝑛 → 𝑛 ∈ 𝐺))) |
| 135 | 6, 128, 134 | mp2an 692 |
1
⊢
∃𝑚 ∈
ℕ (𝑚 ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 (𝑚 < 𝑛 → 𝑛 ∈ 𝐺)) |