Step |
Hyp |
Ref |
Expression |
1 |
|
metdscn.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ inf ( ran ( 𝑦 ∈ 𝑆 ↦ ( 𝑥 𝐷 𝑦 ) ) , ℝ* , < ) ) |
2 |
|
metdscn.j |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
3 |
|
metdscn.c |
⊢ 𝐶 = ( dist ‘ ℝ*𝑠 ) |
4 |
|
metdscn.k |
⊢ 𝐾 = ( MetOpen ‘ 𝐶 ) |
5 |
|
metdscnlem.1 |
⊢ ( 𝜑 → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
6 |
|
metdscnlem.2 |
⊢ ( 𝜑 → 𝑆 ⊆ 𝑋 ) |
7 |
|
metdscnlem.3 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑋 ) |
8 |
|
metdscnlem.4 |
⊢ ( 𝜑 → 𝐵 ∈ 𝑋 ) |
9 |
|
metdscnlem.5 |
⊢ ( 𝜑 → 𝑅 ∈ ℝ+ ) |
10 |
|
metdscnlem.6 |
⊢ ( 𝜑 → ( 𝐴 𝐷 𝐵 ) < 𝑅 ) |
11 |
1
|
metdsf |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) |
12 |
5 6 11
|
syl2anc |
⊢ ( 𝜑 → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) |
13 |
12 7
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
14 |
|
eliccxr |
⊢ ( ( 𝐹 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) → ( 𝐹 ‘ 𝐴 ) ∈ ℝ* ) |
15 |
13 14
|
syl |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ∈ ℝ* ) |
16 |
12 8
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ ( 0 [,] +∞ ) ) |
17 |
|
eliccxr |
⊢ ( ( 𝐹 ‘ 𝐵 ) ∈ ( 0 [,] +∞ ) → ( 𝐹 ‘ 𝐵 ) ∈ ℝ* ) |
18 |
16 17
|
syl |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ ℝ* ) |
19 |
18
|
xnegcld |
⊢ ( 𝜑 → -𝑒 ( 𝐹 ‘ 𝐵 ) ∈ ℝ* ) |
20 |
15 19
|
xaddcld |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐴 ) +𝑒 -𝑒 ( 𝐹 ‘ 𝐵 ) ) ∈ ℝ* ) |
21 |
|
xmetcl |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) ∈ ℝ* ) |
22 |
5 7 8 21
|
syl3anc |
⊢ ( 𝜑 → ( 𝐴 𝐷 𝐵 ) ∈ ℝ* ) |
23 |
9
|
rpxrd |
⊢ ( 𝜑 → 𝑅 ∈ ℝ* ) |
24 |
1
|
metdstri |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐹 ‘ 𝐴 ) ≤ ( ( 𝐴 𝐷 𝐵 ) +𝑒 ( 𝐹 ‘ 𝐵 ) ) ) |
25 |
5 6 7 8 24
|
syl22anc |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ≤ ( ( 𝐴 𝐷 𝐵 ) +𝑒 ( 𝐹 ‘ 𝐵 ) ) ) |
26 |
|
elxrge0 |
⊢ ( ( 𝐹 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ↔ ( ( 𝐹 ‘ 𝐴 ) ∈ ℝ* ∧ 0 ≤ ( 𝐹 ‘ 𝐴 ) ) ) |
27 |
26
|
simprbi |
⊢ ( ( 𝐹 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) → 0 ≤ ( 𝐹 ‘ 𝐴 ) ) |
28 |
13 27
|
syl |
⊢ ( 𝜑 → 0 ≤ ( 𝐹 ‘ 𝐴 ) ) |
29 |
|
elxrge0 |
⊢ ( ( 𝐹 ‘ 𝐵 ) ∈ ( 0 [,] +∞ ) ↔ ( ( 𝐹 ‘ 𝐵 ) ∈ ℝ* ∧ 0 ≤ ( 𝐹 ‘ 𝐵 ) ) ) |
30 |
29
|
simprbi |
⊢ ( ( 𝐹 ‘ 𝐵 ) ∈ ( 0 [,] +∞ ) → 0 ≤ ( 𝐹 ‘ 𝐵 ) ) |
31 |
16 30
|
syl |
⊢ ( 𝜑 → 0 ≤ ( 𝐹 ‘ 𝐵 ) ) |
32 |
|
ge0nemnf |
⊢ ( ( ( 𝐹 ‘ 𝐵 ) ∈ ℝ* ∧ 0 ≤ ( 𝐹 ‘ 𝐵 ) ) → ( 𝐹 ‘ 𝐵 ) ≠ -∞ ) |
33 |
18 31 32
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ≠ -∞ ) |
34 |
|
xmetge0 |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → 0 ≤ ( 𝐴 𝐷 𝐵 ) ) |
35 |
5 7 8 34
|
syl3anc |
⊢ ( 𝜑 → 0 ≤ ( 𝐴 𝐷 𝐵 ) ) |
36 |
|
xlesubadd |
⊢ ( ( ( ( 𝐹 ‘ 𝐴 ) ∈ ℝ* ∧ ( 𝐹 ‘ 𝐵 ) ∈ ℝ* ∧ ( 𝐴 𝐷 𝐵 ) ∈ ℝ* ) ∧ ( 0 ≤ ( 𝐹 ‘ 𝐴 ) ∧ ( 𝐹 ‘ 𝐵 ) ≠ -∞ ∧ 0 ≤ ( 𝐴 𝐷 𝐵 ) ) ) → ( ( ( 𝐹 ‘ 𝐴 ) +𝑒 -𝑒 ( 𝐹 ‘ 𝐵 ) ) ≤ ( 𝐴 𝐷 𝐵 ) ↔ ( 𝐹 ‘ 𝐴 ) ≤ ( ( 𝐴 𝐷 𝐵 ) +𝑒 ( 𝐹 ‘ 𝐵 ) ) ) ) |
37 |
15 18 22 28 33 35 36
|
syl33anc |
⊢ ( 𝜑 → ( ( ( 𝐹 ‘ 𝐴 ) +𝑒 -𝑒 ( 𝐹 ‘ 𝐵 ) ) ≤ ( 𝐴 𝐷 𝐵 ) ↔ ( 𝐹 ‘ 𝐴 ) ≤ ( ( 𝐴 𝐷 𝐵 ) +𝑒 ( 𝐹 ‘ 𝐵 ) ) ) ) |
38 |
25 37
|
mpbird |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐴 ) +𝑒 -𝑒 ( 𝐹 ‘ 𝐵 ) ) ≤ ( 𝐴 𝐷 𝐵 ) ) |
39 |
20 22 23 38 10
|
xrlelttrd |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝐴 ) +𝑒 -𝑒 ( 𝐹 ‘ 𝐵 ) ) < 𝑅 ) |