Step |
Hyp |
Ref |
Expression |
1 |
|
nnsum3primesprm |
⊢ ( 𝑃 ∈ ℙ → ∃ 𝑑 ∈ ℕ ∃ 𝑓 ∈ ( ℙ ↑m ( 1 ... 𝑑 ) ) ( 𝑑 ≤ 3 ∧ 𝑃 = Σ 𝑘 ∈ ( 1 ... 𝑑 ) ( 𝑓 ‘ 𝑘 ) ) ) |
2 |
|
3lt4 |
⊢ 3 < 4 |
3 |
|
nnre |
⊢ ( 𝑑 ∈ ℕ → 𝑑 ∈ ℝ ) |
4 |
|
3re |
⊢ 3 ∈ ℝ |
5 |
4
|
a1i |
⊢ ( 𝑑 ∈ ℕ → 3 ∈ ℝ ) |
6 |
|
4re |
⊢ 4 ∈ ℝ |
7 |
6
|
a1i |
⊢ ( 𝑑 ∈ ℕ → 4 ∈ ℝ ) |
8 |
|
leltletr |
⊢ ( ( 𝑑 ∈ ℝ ∧ 3 ∈ ℝ ∧ 4 ∈ ℝ ) → ( ( 𝑑 ≤ 3 ∧ 3 < 4 ) → 𝑑 ≤ 4 ) ) |
9 |
3 5 7 8
|
syl3anc |
⊢ ( 𝑑 ∈ ℕ → ( ( 𝑑 ≤ 3 ∧ 3 < 4 ) → 𝑑 ≤ 4 ) ) |
10 |
2 9
|
mpan2i |
⊢ ( 𝑑 ∈ ℕ → ( 𝑑 ≤ 3 → 𝑑 ≤ 4 ) ) |
11 |
10
|
anim1d |
⊢ ( 𝑑 ∈ ℕ → ( ( 𝑑 ≤ 3 ∧ 𝑃 = Σ 𝑘 ∈ ( 1 ... 𝑑 ) ( 𝑓 ‘ 𝑘 ) ) → ( 𝑑 ≤ 4 ∧ 𝑃 = Σ 𝑘 ∈ ( 1 ... 𝑑 ) ( 𝑓 ‘ 𝑘 ) ) ) ) |
12 |
11
|
reximdv |
⊢ ( 𝑑 ∈ ℕ → ( ∃ 𝑓 ∈ ( ℙ ↑m ( 1 ... 𝑑 ) ) ( 𝑑 ≤ 3 ∧ 𝑃 = Σ 𝑘 ∈ ( 1 ... 𝑑 ) ( 𝑓 ‘ 𝑘 ) ) → ∃ 𝑓 ∈ ( ℙ ↑m ( 1 ... 𝑑 ) ) ( 𝑑 ≤ 4 ∧ 𝑃 = Σ 𝑘 ∈ ( 1 ... 𝑑 ) ( 𝑓 ‘ 𝑘 ) ) ) ) |
13 |
12
|
reximia |
⊢ ( ∃ 𝑑 ∈ ℕ ∃ 𝑓 ∈ ( ℙ ↑m ( 1 ... 𝑑 ) ) ( 𝑑 ≤ 3 ∧ 𝑃 = Σ 𝑘 ∈ ( 1 ... 𝑑 ) ( 𝑓 ‘ 𝑘 ) ) → ∃ 𝑑 ∈ ℕ ∃ 𝑓 ∈ ( ℙ ↑m ( 1 ... 𝑑 ) ) ( 𝑑 ≤ 4 ∧ 𝑃 = Σ 𝑘 ∈ ( 1 ... 𝑑 ) ( 𝑓 ‘ 𝑘 ) ) ) |
14 |
1 13
|
syl |
⊢ ( 𝑃 ∈ ℙ → ∃ 𝑑 ∈ ℕ ∃ 𝑓 ∈ ( ℙ ↑m ( 1 ... 𝑑 ) ) ( 𝑑 ≤ 4 ∧ 𝑃 = Σ 𝑘 ∈ ( 1 ... 𝑑 ) ( 𝑓 ‘ 𝑘 ) ) ) |