Step |
Hyp |
Ref |
Expression |
1 |
|
ppisval |
⊢ ( 𝐴 ∈ ℝ → ( ( 0 [,] 𝐴 ) ∩ ℙ ) = ( ( 2 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) |
2 |
1
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 2 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( 0 [,] 𝐴 ) ∩ ℙ ) = ( ( 2 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) |
3 |
|
fzss1 |
⊢ ( 2 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 2 ... ( ⌊ ‘ 𝐴 ) ) ⊆ ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) ) |
4 |
3
|
adantl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 2 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( 2 ... ( ⌊ ‘ 𝐴 ) ) ⊆ ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) ) |
5 |
4
|
ssrind |
⊢ ( ( 𝐴 ∈ ℝ ∧ 2 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( 2 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ⊆ ( ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) |
6 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 2 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) → 𝑥 ∈ ( ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) |
7 |
|
elin |
⊢ ( 𝑥 ∈ ( ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ↔ ( 𝑥 ∈ ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) ∧ 𝑥 ∈ ℙ ) ) |
8 |
6 7
|
sylib |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 2 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) → ( 𝑥 ∈ ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) ∧ 𝑥 ∈ ℙ ) ) |
9 |
8
|
simprd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 2 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) → 𝑥 ∈ ℙ ) |
10 |
|
prmuz2 |
⊢ ( 𝑥 ∈ ℙ → 𝑥 ∈ ( ℤ≥ ‘ 2 ) ) |
11 |
9 10
|
syl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 2 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) → 𝑥 ∈ ( ℤ≥ ‘ 2 ) ) |
12 |
8
|
simpld |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 2 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) → 𝑥 ∈ ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) ) |
13 |
|
elfzuz3 |
⊢ ( 𝑥 ∈ ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) → ( ⌊ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 𝑥 ) ) |
14 |
12 13
|
syl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 2 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) → ( ⌊ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 𝑥 ) ) |
15 |
|
elfzuzb |
⊢ ( 𝑥 ∈ ( 2 ... ( ⌊ ‘ 𝐴 ) ) ↔ ( 𝑥 ∈ ( ℤ≥ ‘ 2 ) ∧ ( ⌊ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 𝑥 ) ) ) |
16 |
11 14 15
|
sylanbrc |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 2 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) → 𝑥 ∈ ( 2 ... ( ⌊ ‘ 𝐴 ) ) ) |
17 |
16 9
|
elind |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 2 ∈ ( ℤ≥ ‘ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) → 𝑥 ∈ ( ( 2 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) |
18 |
5 17
|
eqelssd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 2 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( 2 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) = ( ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) |
19 |
2 18
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 2 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( ( 0 [,] 𝐴 ) ∩ ℙ ) = ( ( 𝑀 ... ( ⌊ ‘ 𝐴 ) ) ∩ ℙ ) ) |