Step |
Hyp |
Ref |
Expression |
1 |
|
elfz2nn0 |
⊢ ( 𝑁 ∈ ( 0 ... 𝑅 ) ↔ ( 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅 ) ) |
2 |
|
elfz2 |
⊢ ( 𝑀 ∈ ( 𝑁 ... 𝑅 ) ↔ ( ( 𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ∧ ( 𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅 ) ) ) |
3 |
|
simplr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ) ∧ 𝑁 ≤ 𝑀 ) → 𝑀 ∈ ℤ ) |
4 |
|
0red |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ) → 0 ∈ ℝ ) |
5 |
|
nn0re |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ ) |
6 |
5
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ) → 𝑁 ∈ ℝ ) |
7 |
|
zre |
⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ℝ ) |
8 |
7
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ) → 𝑀 ∈ ℝ ) |
9 |
4 6 8
|
3jca |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ) → ( 0 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ) ) |
10 |
9
|
adantr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ) ∧ 𝑁 ≤ 𝑀 ) → ( 0 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ) ) |
11 |
|
nn0ge0 |
⊢ ( 𝑁 ∈ ℕ0 → 0 ≤ 𝑁 ) |
12 |
11
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ) → 0 ≤ 𝑁 ) |
13 |
12
|
anim1i |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ) ∧ 𝑁 ≤ 𝑀 ) → ( 0 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀 ) ) |
14 |
|
letr |
⊢ ( ( 0 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ) → ( ( 0 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀 ) → 0 ≤ 𝑀 ) ) |
15 |
10 13 14
|
sylc |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ) ∧ 𝑁 ≤ 𝑀 ) → 0 ≤ 𝑀 ) |
16 |
|
elnn0z |
⊢ ( 𝑀 ∈ ℕ0 ↔ ( 𝑀 ∈ ℤ ∧ 0 ≤ 𝑀 ) ) |
17 |
3 15 16
|
sylanbrc |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ) ∧ 𝑁 ≤ 𝑀 ) → 𝑀 ∈ ℕ0 ) |
18 |
17
|
exp31 |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑀 ∈ ℤ → ( 𝑁 ≤ 𝑀 → 𝑀 ∈ ℕ0 ) ) ) |
19 |
18
|
com23 |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 ≤ 𝑀 → ( 𝑀 ∈ ℤ → 𝑀 ∈ ℕ0 ) ) ) |
20 |
19
|
3ad2ant1 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅 ) → ( 𝑁 ≤ 𝑀 → ( 𝑀 ∈ ℤ → 𝑀 ∈ ℕ0 ) ) ) |
21 |
20
|
com13 |
⊢ ( 𝑀 ∈ ℤ → ( 𝑁 ≤ 𝑀 → ( ( 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅 ) → 𝑀 ∈ ℕ0 ) ) ) |
22 |
21
|
adantrd |
⊢ ( 𝑀 ∈ ℤ → ( ( 𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅 ) → ( ( 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅 ) → 𝑀 ∈ ℕ0 ) ) ) |
23 |
22
|
3ad2ant3 |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ ) → ( ( 𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅 ) → ( ( 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅 ) → 𝑀 ∈ ℕ0 ) ) ) |
24 |
23
|
imp |
⊢ ( ( ( 𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ∧ ( 𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅 ) ) → ( ( 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅 ) → 𝑀 ∈ ℕ0 ) ) |
25 |
24
|
imp |
⊢ ( ( ( ( 𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ∧ ( 𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅 ) ) ∧ ( 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅 ) ) → 𝑀 ∈ ℕ0 ) |
26 |
|
simpr2 |
⊢ ( ( ( ( 𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ∧ ( 𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅 ) ) ∧ ( 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅 ) ) → 𝑅 ∈ ℕ0 ) |
27 |
|
simplrr |
⊢ ( ( ( ( 𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ∧ ( 𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅 ) ) ∧ ( 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅 ) ) → 𝑀 ≤ 𝑅 ) |
28 |
25 26 27
|
3jca |
⊢ ( ( ( ( 𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ∧ ( 𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅 ) ) ∧ ( 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅 ) ) → ( 𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅 ) ) |
29 |
28
|
ex |
⊢ ( ( ( 𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ∧ ( 𝑁 ≤ 𝑀 ∧ 𝑀 ≤ 𝑅 ) ) → ( ( 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅 ) → ( 𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅 ) ) ) |
30 |
2 29
|
sylbi |
⊢ ( 𝑀 ∈ ( 𝑁 ... 𝑅 ) → ( ( 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅 ) → ( 𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅 ) ) ) |
31 |
30
|
com12 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑁 ≤ 𝑅 ) → ( 𝑀 ∈ ( 𝑁 ... 𝑅 ) → ( 𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅 ) ) ) |
32 |
1 31
|
sylbi |
⊢ ( 𝑁 ∈ ( 0 ... 𝑅 ) → ( 𝑀 ∈ ( 𝑁 ... 𝑅 ) → ( 𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅 ) ) ) |
33 |
32
|
imp |
⊢ ( ( 𝑁 ∈ ( 0 ... 𝑅 ) ∧ 𝑀 ∈ ( 𝑁 ... 𝑅 ) ) → ( 𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅 ) ) |
34 |
|
elfz2nn0 |
⊢ ( 𝑀 ∈ ( 0 ... 𝑅 ) ↔ ( 𝑀 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ≤ 𝑅 ) ) |
35 |
33 34
|
sylibr |
⊢ ( ( 𝑁 ∈ ( 0 ... 𝑅 ) ∧ 𝑀 ∈ ( 𝑁 ... 𝑅 ) ) → 𝑀 ∈ ( 0 ... 𝑅 ) ) |