Proof of Theorem ge2nprmge4
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | eluz2b2 12964 | . . 3
⊢ (𝑋 ∈
(ℤ≥‘2) ↔ (𝑋 ∈ ℕ ∧ 1 < 𝑋)) | 
| 2 |  | 4z 12653 | . . . . . . 7
⊢ 4 ∈
ℤ | 
| 3 | 2 | a1i 11 | . . . . . 6
⊢ (((𝑋 ∈ ℕ ∧ 1 <
𝑋) ∧ 𝑋 ∉ ℙ) → 4 ∈
ℤ) | 
| 4 |  | nnz 12636 | . . . . . . 7
⊢ (𝑋 ∈ ℕ → 𝑋 ∈
ℤ) | 
| 5 | 4 | ad2antrr 726 | . . . . . 6
⊢ (((𝑋 ∈ ℕ ∧ 1 <
𝑋) ∧ 𝑋 ∉ ℙ) → 𝑋 ∈ ℤ) | 
| 6 |  | 1z 12649 | . . . . . . . . . . 11
⊢ 1 ∈
ℤ | 
| 7 |  | zltp1le 12669 | . . . . . . . . . . 11
⊢ ((1
∈ ℤ ∧ 𝑋
∈ ℤ) → (1 < 𝑋 ↔ (1 + 1) ≤ 𝑋)) | 
| 8 | 6, 4, 7 | sylancr 587 | . . . . . . . . . 10
⊢ (𝑋 ∈ ℕ → (1 <
𝑋 ↔ (1 + 1) ≤ 𝑋)) | 
| 9 |  | 1p1e2 12392 | . . . . . . . . . . 11
⊢ (1 + 1) =
2 | 
| 10 | 9 | breq1i 5149 | . . . . . . . . . 10
⊢ ((1 + 1)
≤ 𝑋 ↔ 2 ≤ 𝑋) | 
| 11 | 8, 10 | bitrdi 287 | . . . . . . . . 9
⊢ (𝑋 ∈ ℕ → (1 <
𝑋 ↔ 2 ≤ 𝑋)) | 
| 12 |  | 2re 12341 | . . . . . . . . . . 11
⊢ 2 ∈
ℝ | 
| 13 |  | nnre 12274 | . . . . . . . . . . 11
⊢ (𝑋 ∈ ℕ → 𝑋 ∈
ℝ) | 
| 14 |  | leloe 11348 | . . . . . . . . . . 11
⊢ ((2
∈ ℝ ∧ 𝑋
∈ ℝ) → (2 ≤ 𝑋 ↔ (2 < 𝑋 ∨ 2 = 𝑋))) | 
| 15 | 12, 13, 14 | sylancr 587 | . . . . . . . . . 10
⊢ (𝑋 ∈ ℕ → (2 ≤
𝑋 ↔ (2 < 𝑋 ∨ 2 = 𝑋))) | 
| 16 |  | 2z 12651 | . . . . . . . . . . . . . 14
⊢ 2 ∈
ℤ | 
| 17 |  | zltp1le 12669 | . . . . . . . . . . . . . 14
⊢ ((2
∈ ℤ ∧ 𝑋
∈ ℤ) → (2 < 𝑋 ↔ (2 + 1) ≤ 𝑋)) | 
| 18 | 16, 4, 17 | sylancr 587 | . . . . . . . . . . . . 13
⊢ (𝑋 ∈ ℕ → (2 <
𝑋 ↔ (2 + 1) ≤ 𝑋)) | 
| 19 |  | 2p1e3 12409 | . . . . . . . . . . . . . 14
⊢ (2 + 1) =
3 | 
| 20 | 19 | breq1i 5149 | . . . . . . . . . . . . 13
⊢ ((2 + 1)
≤ 𝑋 ↔ 3 ≤ 𝑋) | 
| 21 | 18, 20 | bitrdi 287 | . . . . . . . . . . . 12
⊢ (𝑋 ∈ ℕ → (2 <
𝑋 ↔ 3 ≤ 𝑋)) | 
| 22 |  | 3re 12347 | . . . . . . . . . . . . . 14
⊢ 3 ∈
ℝ | 
| 23 |  | leloe 11348 | . . . . . . . . . . . . . 14
⊢ ((3
∈ ℝ ∧ 𝑋
∈ ℝ) → (3 ≤ 𝑋 ↔ (3 < 𝑋 ∨ 3 = 𝑋))) | 
| 24 | 22, 13, 23 | sylancr 587 | . . . . . . . . . . . . 13
⊢ (𝑋 ∈ ℕ → (3 ≤
𝑋 ↔ (3 < 𝑋 ∨ 3 = 𝑋))) | 
| 25 |  | df-4 12332 | . . . . . . . . . . . . . . . . 17
⊢ 4 = (3 +
1) | 
| 26 |  | 3z 12652 | . . . . . . . . . . . . . . . . . . 19
⊢ 3 ∈
ℤ | 
| 27 |  | zltp1le 12669 | . . . . . . . . . . . . . . . . . . 19
⊢ ((3
∈ ℤ ∧ 𝑋
∈ ℤ) → (3 < 𝑋 ↔ (3 + 1) ≤ 𝑋)) | 
| 28 | 26, 4, 27 | sylancr 587 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑋 ∈ ℕ → (3 <
𝑋 ↔ (3 + 1) ≤ 𝑋)) | 
| 29 | 28 | biimpa 476 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑋 ∈ ℕ ∧ 3 <
𝑋) → (3 + 1) ≤
𝑋) | 
| 30 | 25, 29 | eqbrtrid 5177 | . . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∈ ℕ ∧ 3 <
𝑋) → 4 ≤ 𝑋) | 
| 31 | 30 | a1d 25 | . . . . . . . . . . . . . . 15
⊢ ((𝑋 ∈ ℕ ∧ 3 <
𝑋) → (𝑋 ∉ ℙ → 4 ≤
𝑋)) | 
| 32 | 31 | ex 412 | . . . . . . . . . . . . . 14
⊢ (𝑋 ∈ ℕ → (3 <
𝑋 → (𝑋 ∉ ℙ → 4 ≤ 𝑋))) | 
| 33 |  | neleq1 3051 | . . . . . . . . . . . . . . . . 17
⊢ (𝑋 = 3 → (𝑋 ∉ ℙ ↔ 3 ∉
ℙ)) | 
| 34 | 33 | eqcoms 2744 | . . . . . . . . . . . . . . . 16
⊢ (3 =
𝑋 → (𝑋 ∉ ℙ ↔ 3 ∉
ℙ)) | 
| 35 |  | 3prm 16732 | . . . . . . . . . . . . . . . . 17
⊢ 3 ∈
ℙ | 
| 36 |  | pm2.24nel 3058 | . . . . . . . . . . . . . . . . 17
⊢ (3 ∈
ℙ → (3 ∉ ℙ → 4 ≤ 𝑋)) | 
| 37 | 35, 36 | mp1i 13 | . . . . . . . . . . . . . . . 16
⊢ (3 =
𝑋 → (3 ∉ ℙ
→ 4 ≤ 𝑋)) | 
| 38 | 34, 37 | sylbid 240 | . . . . . . . . . . . . . . 15
⊢ (3 =
𝑋 → (𝑋 ∉ ℙ → 4 ≤ 𝑋)) | 
| 39 | 38 | a1i 11 | . . . . . . . . . . . . . 14
⊢ (𝑋 ∈ ℕ → (3 =
𝑋 → (𝑋 ∉ ℙ → 4 ≤ 𝑋))) | 
| 40 | 32, 39 | jaod 859 | . . . . . . . . . . . . 13
⊢ (𝑋 ∈ ℕ → ((3 <
𝑋 ∨ 3 = 𝑋) → (𝑋 ∉ ℙ → 4 ≤ 𝑋))) | 
| 41 | 24, 40 | sylbid 240 | . . . . . . . . . . . 12
⊢ (𝑋 ∈ ℕ → (3 ≤
𝑋 → (𝑋 ∉ ℙ → 4 ≤ 𝑋))) | 
| 42 | 21, 41 | sylbid 240 | . . . . . . . . . . 11
⊢ (𝑋 ∈ ℕ → (2 <
𝑋 → (𝑋 ∉ ℙ → 4 ≤ 𝑋))) | 
| 43 |  | neleq1 3051 | . . . . . . . . . . . . . 14
⊢ (𝑋 = 2 → (𝑋 ∉ ℙ ↔ 2 ∉
ℙ)) | 
| 44 | 43 | eqcoms 2744 | . . . . . . . . . . . . 13
⊢ (2 =
𝑋 → (𝑋 ∉ ℙ ↔ 2 ∉
ℙ)) | 
| 45 |  | 2prm 16730 | . . . . . . . . . . . . . 14
⊢ 2 ∈
ℙ | 
| 46 |  | pm2.24nel 3058 | . . . . . . . . . . . . . 14
⊢ (2 ∈
ℙ → (2 ∉ ℙ → 4 ≤ 𝑋)) | 
| 47 | 45, 46 | mp1i 13 | . . . . . . . . . . . . 13
⊢ (2 =
𝑋 → (2 ∉ ℙ
→ 4 ≤ 𝑋)) | 
| 48 | 44, 47 | sylbid 240 | . . . . . . . . . . . 12
⊢ (2 =
𝑋 → (𝑋 ∉ ℙ → 4 ≤ 𝑋)) | 
| 49 | 48 | a1i 11 | . . . . . . . . . . 11
⊢ (𝑋 ∈ ℕ → (2 =
𝑋 → (𝑋 ∉ ℙ → 4 ≤ 𝑋))) | 
| 50 | 42, 49 | jaod 859 | . . . . . . . . . 10
⊢ (𝑋 ∈ ℕ → ((2 <
𝑋 ∨ 2 = 𝑋) → (𝑋 ∉ ℙ → 4 ≤ 𝑋))) | 
| 51 | 15, 50 | sylbid 240 | . . . . . . . . 9
⊢ (𝑋 ∈ ℕ → (2 ≤
𝑋 → (𝑋 ∉ ℙ → 4 ≤ 𝑋))) | 
| 52 | 11, 51 | sylbid 240 | . . . . . . . 8
⊢ (𝑋 ∈ ℕ → (1 <
𝑋 → (𝑋 ∉ ℙ → 4 ≤ 𝑋))) | 
| 53 | 52 | imp 406 | . . . . . . 7
⊢ ((𝑋 ∈ ℕ ∧ 1 <
𝑋) → (𝑋 ∉ ℙ → 4 ≤
𝑋)) | 
| 54 | 53 | imp 406 | . . . . . 6
⊢ (((𝑋 ∈ ℕ ∧ 1 <
𝑋) ∧ 𝑋 ∉ ℙ) → 4 ≤ 𝑋) | 
| 55 | 3, 5, 54 | 3jca 1128 | . . . . 5
⊢ (((𝑋 ∈ ℕ ∧ 1 <
𝑋) ∧ 𝑋 ∉ ℙ) → (4 ∈ ℤ
∧ 𝑋 ∈ ℤ
∧ 4 ≤ 𝑋)) | 
| 56 | 55 | ex 412 | . . . 4
⊢ ((𝑋 ∈ ℕ ∧ 1 <
𝑋) → (𝑋 ∉ ℙ → (4
∈ ℤ ∧ 𝑋
∈ ℤ ∧ 4 ≤ 𝑋))) | 
| 57 |  | eluz2 12885 | . . . 4
⊢ (𝑋 ∈
(ℤ≥‘4) ↔ (4 ∈ ℤ ∧ 𝑋 ∈ ℤ ∧ 4 ≤
𝑋)) | 
| 58 | 56, 57 | imbitrrdi 252 | . . 3
⊢ ((𝑋 ∈ ℕ ∧ 1 <
𝑋) → (𝑋 ∉ ℙ → 𝑋 ∈
(ℤ≥‘4))) | 
| 59 | 1, 58 | sylbi 217 | . 2
⊢ (𝑋 ∈
(ℤ≥‘2) → (𝑋 ∉ ℙ → 𝑋 ∈
(ℤ≥‘4))) | 
| 60 | 59 | imp 406 | 1
⊢ ((𝑋 ∈
(ℤ≥‘2) ∧ 𝑋 ∉ ℙ) → 𝑋 ∈
(ℤ≥‘4)) |