Step |
Hyp |
Ref |
Expression |
1 |
|
elfz2nn0 |
⊢ ( 𝑀 ∈ ( 0 ... 𝐿 ) ↔ ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑀 ≤ 𝐿 ) ) |
2 |
|
elfz2 |
⊢ ( 𝑁 ∈ ( 𝐿 ... 𝑋 ) ↔ ( ( 𝐿 ∈ ℤ ∧ 𝑋 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝐿 ≤ 𝑁 ∧ 𝑁 ≤ 𝑋 ) ) ) |
3 |
|
nn0re |
⊢ ( 𝑀 ∈ ℕ0 → 𝑀 ∈ ℝ ) |
4 |
|
nn0re |
⊢ ( 𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ ) |
5 |
|
zre |
⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℝ ) |
6 |
3 4 5
|
3anim123i |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ ) ) |
7 |
6
|
3expa |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ) ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ ) ) |
8 |
|
letr |
⊢ ( ( 𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( ( 𝑀 ≤ 𝐿 ∧ 𝐿 ≤ 𝑁 ) → 𝑀 ≤ 𝑁 ) ) |
9 |
7 8
|
syl |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ) ∧ 𝑁 ∈ ℤ ) → ( ( 𝑀 ≤ 𝐿 ∧ 𝐿 ≤ 𝑁 ) → 𝑀 ≤ 𝑁 ) ) |
10 |
|
simplll |
⊢ ( ( ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ) ∧ 𝑁 ∈ ℤ ) ∧ 𝑀 ≤ 𝑁 ) → 𝑀 ∈ ℕ0 ) |
11 |
|
simpr |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ) ∧ 𝑁 ∈ ℤ ) → 𝑁 ∈ ℤ ) |
12 |
11
|
adantr |
⊢ ( ( ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ) ∧ 𝑁 ∈ ℤ ) ∧ 𝑀 ≤ 𝑁 ) → 𝑁 ∈ ℤ ) |
13 |
|
elnn0z |
⊢ ( 𝑀 ∈ ℕ0 ↔ ( 𝑀 ∈ ℤ ∧ 0 ≤ 𝑀 ) ) |
14 |
|
0red |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → 0 ∈ ℝ ) |
15 |
|
zre |
⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ℝ ) |
16 |
15
|
adantr |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → 𝑀 ∈ ℝ ) |
17 |
5
|
adantl |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → 𝑁 ∈ ℝ ) |
18 |
|
letr |
⊢ ( ( 0 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( ( 0 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁 ) → 0 ≤ 𝑁 ) ) |
19 |
14 16 17 18
|
syl3anc |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 0 ≤ 𝑀 ∧ 𝑀 ≤ 𝑁 ) → 0 ≤ 𝑁 ) ) |
20 |
19
|
exp4b |
⊢ ( 𝑀 ∈ ℤ → ( 𝑁 ∈ ℤ → ( 0 ≤ 𝑀 → ( 𝑀 ≤ 𝑁 → 0 ≤ 𝑁 ) ) ) ) |
21 |
20
|
com23 |
⊢ ( 𝑀 ∈ ℤ → ( 0 ≤ 𝑀 → ( 𝑁 ∈ ℤ → ( 𝑀 ≤ 𝑁 → 0 ≤ 𝑁 ) ) ) ) |
22 |
21
|
imp |
⊢ ( ( 𝑀 ∈ ℤ ∧ 0 ≤ 𝑀 ) → ( 𝑁 ∈ ℤ → ( 𝑀 ≤ 𝑁 → 0 ≤ 𝑁 ) ) ) |
23 |
13 22
|
sylbi |
⊢ ( 𝑀 ∈ ℕ0 → ( 𝑁 ∈ ℤ → ( 𝑀 ≤ 𝑁 → 0 ≤ 𝑁 ) ) ) |
24 |
23
|
adantr |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ) → ( 𝑁 ∈ ℤ → ( 𝑀 ≤ 𝑁 → 0 ≤ 𝑁 ) ) ) |
25 |
24
|
imp |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ) ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ≤ 𝑁 → 0 ≤ 𝑁 ) ) |
26 |
25
|
imp |
⊢ ( ( ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ) ∧ 𝑁 ∈ ℤ ) ∧ 𝑀 ≤ 𝑁 ) → 0 ≤ 𝑁 ) |
27 |
|
elnn0z |
⊢ ( 𝑁 ∈ ℕ0 ↔ ( 𝑁 ∈ ℤ ∧ 0 ≤ 𝑁 ) ) |
28 |
12 26 27
|
sylanbrc |
⊢ ( ( ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ) ∧ 𝑁 ∈ ℤ ) ∧ 𝑀 ≤ 𝑁 ) → 𝑁 ∈ ℕ0 ) |
29 |
|
simpr |
⊢ ( ( ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ) ∧ 𝑁 ∈ ℤ ) ∧ 𝑀 ≤ 𝑁 ) → 𝑀 ≤ 𝑁 ) |
30 |
10 28 29
|
3jca |
⊢ ( ( ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ) ∧ 𝑁 ∈ ℤ ) ∧ 𝑀 ≤ 𝑁 ) → ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁 ) ) |
31 |
30
|
ex |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ) ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ≤ 𝑁 → ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁 ) ) ) |
32 |
9 31
|
syld |
⊢ ( ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ) ∧ 𝑁 ∈ ℤ ) → ( ( 𝑀 ≤ 𝐿 ∧ 𝐿 ≤ 𝑁 ) → ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁 ) ) ) |
33 |
32
|
exp4b |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ) → ( 𝑁 ∈ ℤ → ( 𝑀 ≤ 𝐿 → ( 𝐿 ≤ 𝑁 → ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁 ) ) ) ) ) |
34 |
33
|
com23 |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ) → ( 𝑀 ≤ 𝐿 → ( 𝑁 ∈ ℤ → ( 𝐿 ≤ 𝑁 → ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁 ) ) ) ) ) |
35 |
34
|
3impia |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑀 ≤ 𝐿 ) → ( 𝑁 ∈ ℤ → ( 𝐿 ≤ 𝑁 → ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁 ) ) ) ) |
36 |
35
|
com13 |
⊢ ( 𝐿 ≤ 𝑁 → ( 𝑁 ∈ ℤ → ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑀 ≤ 𝐿 ) → ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁 ) ) ) ) |
37 |
36
|
adantr |
⊢ ( ( 𝐿 ≤ 𝑁 ∧ 𝑁 ≤ 𝑋 ) → ( 𝑁 ∈ ℤ → ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑀 ≤ 𝐿 ) → ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁 ) ) ) ) |
38 |
37
|
com12 |
⊢ ( 𝑁 ∈ ℤ → ( ( 𝐿 ≤ 𝑁 ∧ 𝑁 ≤ 𝑋 ) → ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑀 ≤ 𝐿 ) → ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁 ) ) ) ) |
39 |
38
|
3ad2ant3 |
⊢ ( ( 𝐿 ∈ ℤ ∧ 𝑋 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝐿 ≤ 𝑁 ∧ 𝑁 ≤ 𝑋 ) → ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑀 ≤ 𝐿 ) → ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁 ) ) ) ) |
40 |
39
|
imp |
⊢ ( ( ( 𝐿 ∈ ℤ ∧ 𝑋 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( 𝐿 ≤ 𝑁 ∧ 𝑁 ≤ 𝑋 ) ) → ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑀 ≤ 𝐿 ) → ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁 ) ) ) |
41 |
2 40
|
sylbi |
⊢ ( 𝑁 ∈ ( 𝐿 ... 𝑋 ) → ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑀 ≤ 𝐿 ) → ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁 ) ) ) |
42 |
41
|
com12 |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐿 ∈ ℕ0 ∧ 𝑀 ≤ 𝐿 ) → ( 𝑁 ∈ ( 𝐿 ... 𝑋 ) → ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁 ) ) ) |
43 |
1 42
|
sylbi |
⊢ ( 𝑀 ∈ ( 0 ... 𝐿 ) → ( 𝑁 ∈ ( 𝐿 ... 𝑋 ) → ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁 ) ) ) |
44 |
43
|
imp |
⊢ ( ( 𝑀 ∈ ( 0 ... 𝐿 ) ∧ 𝑁 ∈ ( 𝐿 ... 𝑋 ) ) → ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁 ) ) |
45 |
|
elfz2nn0 |
⊢ ( 𝑀 ∈ ( 0 ... 𝑁 ) ↔ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁 ) ) |
46 |
44 45
|
sylibr |
⊢ ( ( 𝑀 ∈ ( 0 ... 𝐿 ) ∧ 𝑁 ∈ ( 𝐿 ... 𝑋 ) ) → 𝑀 ∈ ( 0 ... 𝑁 ) ) |