Step |
Hyp |
Ref |
Expression |
1 |
|
metust.1 |
⊢ 𝐹 = ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
2 |
|
simpll |
⊢ ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) → 𝑎 ∈ ℝ+ ) |
3 |
2
|
rpred |
⊢ ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) → 𝑎 ∈ ℝ ) |
4 |
|
simplr |
⊢ ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) → 𝑏 ∈ ℝ+ ) |
5 |
4
|
rpred |
⊢ ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) → 𝑏 ∈ ℝ ) |
6 |
|
simpllr |
⊢ ( ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) ∧ 𝑎 ≤ 𝑏 ) → 𝑏 ∈ ℝ+ ) |
7 |
6
|
rpred |
⊢ ( ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) ∧ 𝑎 ≤ 𝑏 ) → 𝑏 ∈ ℝ ) |
8 |
|
0xr |
⊢ 0 ∈ ℝ* |
9 |
8
|
a1i |
⊢ ( ( 𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏 ) → 0 ∈ ℝ* ) |
10 |
|
simpl |
⊢ ( ( 𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏 ) → 𝑏 ∈ ℝ ) |
11 |
10
|
rexrd |
⊢ ( ( 𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏 ) → 𝑏 ∈ ℝ* ) |
12 |
|
0le0 |
⊢ 0 ≤ 0 |
13 |
12
|
a1i |
⊢ ( ( 𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏 ) → 0 ≤ 0 ) |
14 |
|
simpr |
⊢ ( ( 𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏 ) → 𝑎 ≤ 𝑏 ) |
15 |
|
icossico |
⊢ ( ( ( 0 ∈ ℝ* ∧ 𝑏 ∈ ℝ* ) ∧ ( 0 ≤ 0 ∧ 𝑎 ≤ 𝑏 ) ) → ( 0 [,) 𝑎 ) ⊆ ( 0 [,) 𝑏 ) ) |
16 |
9 11 13 14 15
|
syl22anc |
⊢ ( ( 𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏 ) → ( 0 [,) 𝑎 ) ⊆ ( 0 [,) 𝑏 ) ) |
17 |
|
imass2 |
⊢ ( ( 0 [,) 𝑎 ) ⊆ ( 0 [,) 𝑏 ) → ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) |
18 |
16 17
|
syl |
⊢ ( ( 𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏 ) → ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) |
19 |
7 18
|
sylancom |
⊢ ( ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) ∧ 𝑎 ≤ 𝑏 ) → ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) |
20 |
|
simplrl |
⊢ ( ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) ∧ 𝑎 ≤ 𝑏 ) → 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
21 |
|
simplrr |
⊢ ( ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) ∧ 𝑎 ≤ 𝑏 ) → 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) |
22 |
19 20 21
|
3sstr4d |
⊢ ( ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) ∧ 𝑎 ≤ 𝑏 ) → 𝐴 ⊆ 𝐵 ) |
23 |
22
|
orcd |
⊢ ( ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) ∧ 𝑎 ≤ 𝑏 ) → ( 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴 ) ) |
24 |
|
simplll |
⊢ ( ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) ∧ 𝑏 ≤ 𝑎 ) → 𝑎 ∈ ℝ+ ) |
25 |
24
|
rpred |
⊢ ( ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) ∧ 𝑏 ≤ 𝑎 ) → 𝑎 ∈ ℝ ) |
26 |
8
|
a1i |
⊢ ( ( 𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎 ) → 0 ∈ ℝ* ) |
27 |
|
simpl |
⊢ ( ( 𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎 ) → 𝑎 ∈ ℝ ) |
28 |
27
|
rexrd |
⊢ ( ( 𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎 ) → 𝑎 ∈ ℝ* ) |
29 |
12
|
a1i |
⊢ ( ( 𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎 ) → 0 ≤ 0 ) |
30 |
|
simpr |
⊢ ( ( 𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎 ) → 𝑏 ≤ 𝑎 ) |
31 |
|
icossico |
⊢ ( ( ( 0 ∈ ℝ* ∧ 𝑎 ∈ ℝ* ) ∧ ( 0 ≤ 0 ∧ 𝑏 ≤ 𝑎 ) ) → ( 0 [,) 𝑏 ) ⊆ ( 0 [,) 𝑎 ) ) |
32 |
26 28 29 30 31
|
syl22anc |
⊢ ( ( 𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎 ) → ( 0 [,) 𝑏 ) ⊆ ( 0 [,) 𝑎 ) ) |
33 |
|
imass2 |
⊢ ( ( 0 [,) 𝑏 ) ⊆ ( 0 [,) 𝑎 ) → ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
34 |
32 33
|
syl |
⊢ ( ( 𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎 ) → ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
35 |
25 34
|
sylancom |
⊢ ( ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) ∧ 𝑏 ≤ 𝑎 ) → ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ⊆ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
36 |
|
simplrr |
⊢ ( ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) ∧ 𝑏 ≤ 𝑎 ) → 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) |
37 |
|
simplrl |
⊢ ( ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) ∧ 𝑏 ≤ 𝑎 ) → 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
38 |
35 36 37
|
3sstr4d |
⊢ ( ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) ∧ 𝑏 ≤ 𝑎 ) → 𝐵 ⊆ 𝐴 ) |
39 |
38
|
olcd |
⊢ ( ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) ∧ 𝑏 ≤ 𝑎 ) → ( 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴 ) ) |
40 |
3 5 23 39
|
lecasei |
⊢ ( ( ( 𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) → ( 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴 ) ) |
41 |
40
|
adantlll |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑏 ∈ ℝ+ ) ∧ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) → ( 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴 ) ) |
42 |
1
|
metustel |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( 𝐴 ∈ 𝐹 ↔ ∃ 𝑎 ∈ ℝ+ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
43 |
42
|
biimpa |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ) → ∃ 𝑎 ∈ ℝ+ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
44 |
43
|
3adant3 |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹 ) → ∃ 𝑎 ∈ ℝ+ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
45 |
|
oveq2 |
⊢ ( 𝑎 = 𝑏 → ( 0 [,) 𝑎 ) = ( 0 [,) 𝑏 ) ) |
46 |
45
|
imaeq2d |
⊢ ( 𝑎 = 𝑏 → ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) |
47 |
46
|
cbvmptv |
⊢ ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) = ( 𝑏 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) |
48 |
47
|
rneqi |
⊢ ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) = ran ( 𝑏 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) |
49 |
1 48
|
eqtri |
⊢ 𝐹 = ran ( 𝑏 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) |
50 |
49
|
metustel |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( 𝐵 ∈ 𝐹 ↔ ∃ 𝑏 ∈ ℝ+ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) |
51 |
50
|
biimpa |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐵 ∈ 𝐹 ) → ∃ 𝑏 ∈ ℝ+ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) |
52 |
51
|
3adant2 |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹 ) → ∃ 𝑏 ∈ ℝ+ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) |
53 |
|
reeanv |
⊢ ( ∃ 𝑎 ∈ ℝ+ ∃ 𝑏 ∈ ℝ+ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ↔ ( ∃ 𝑎 ∈ ℝ+ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ ∃ 𝑏 ∈ ℝ+ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) |
54 |
44 52 53
|
sylanbrc |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹 ) → ∃ 𝑎 ∈ ℝ+ ∃ 𝑏 ∈ ℝ+ ( 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ∧ 𝐵 = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) |
55 |
41 54
|
r19.29vva |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹 ) → ( 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴 ) ) |