Step |
Hyp |
Ref |
Expression |
1 |
|
metust.1 |
⊢ 𝐹 = ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
2 |
|
simp-4r |
⊢ ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) |
3 |
|
simplr |
⊢ ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → 𝑎 ∈ ℝ+ ) |
4 |
3
|
rphalfcld |
⊢ ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( 𝑎 / 2 ) ∈ ℝ+ ) |
5 |
|
eqidd |
⊢ ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) = ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) |
6 |
|
oveq2 |
⊢ ( 𝑏 = ( 𝑎 / 2 ) → ( 0 [,) 𝑏 ) = ( 0 [,) ( 𝑎 / 2 ) ) ) |
7 |
6
|
imaeq2d |
⊢ ( 𝑏 = ( 𝑎 / 2 ) → ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) = ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) |
8 |
7
|
rspceeqv |
⊢ ( ( ( 𝑎 / 2 ) ∈ ℝ+ ∧ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) = ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) → ∃ 𝑏 ∈ ℝ+ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) |
9 |
4 5 8
|
syl2anc |
⊢ ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ∃ 𝑏 ∈ ℝ+ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) |
10 |
|
oveq2 |
⊢ ( 𝑎 = 𝑏 → ( 0 [,) 𝑎 ) = ( 0 [,) 𝑏 ) ) |
11 |
10
|
imaeq2d |
⊢ ( 𝑎 = 𝑏 → ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) |
12 |
11
|
cbvmptv |
⊢ ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) = ( 𝑏 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) |
13 |
12
|
rneqi |
⊢ ran ( 𝑎 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) = ran ( 𝑏 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) |
14 |
1 13
|
eqtri |
⊢ 𝐹 = ran ( 𝑏 ∈ ℝ+ ↦ ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) |
15 |
14
|
metustel |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∈ 𝐹 ↔ ∃ 𝑏 ∈ ℝ+ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) ) |
16 |
15
|
biimpar |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ∃ 𝑏 ∈ ℝ+ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) = ( ◡ 𝐷 “ ( 0 [,) 𝑏 ) ) ) → ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∈ 𝐹 ) |
17 |
2 9 16
|
syl2anc |
⊢ ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∈ 𝐹 ) |
18 |
|
relco |
⊢ Rel ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) |
19 |
18
|
a1i |
⊢ ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → Rel ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) |
20 |
|
cossxp |
⊢ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ⊆ ( dom ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) × ran ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) |
21 |
|
cnvimass |
⊢ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ⊆ dom 𝐷 |
22 |
|
psmetf |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ) |
23 |
21 22
|
fssdm |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ⊆ ( 𝑋 × 𝑋 ) ) |
24 |
|
dmss |
⊢ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ⊆ ( 𝑋 × 𝑋 ) → dom ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ⊆ dom ( 𝑋 × 𝑋 ) ) |
25 |
|
rnss |
⊢ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ⊆ ( 𝑋 × 𝑋 ) → ran ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ⊆ ran ( 𝑋 × 𝑋 ) ) |
26 |
|
xpss12 |
⊢ ( ( dom ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ⊆ dom ( 𝑋 × 𝑋 ) ∧ ran ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ⊆ ran ( 𝑋 × 𝑋 ) ) → ( dom ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) × ran ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ⊆ ( dom ( 𝑋 × 𝑋 ) × ran ( 𝑋 × 𝑋 ) ) ) |
27 |
24 25 26
|
syl2anc |
⊢ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ⊆ ( 𝑋 × 𝑋 ) → ( dom ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) × ran ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ⊆ ( dom ( 𝑋 × 𝑋 ) × ran ( 𝑋 × 𝑋 ) ) ) |
28 |
23 27
|
syl |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( dom ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) × ran ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ⊆ ( dom ( 𝑋 × 𝑋 ) × ran ( 𝑋 × 𝑋 ) ) ) |
29 |
28
|
adantl |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( dom ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) × ran ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ⊆ ( dom ( 𝑋 × 𝑋 ) × ran ( 𝑋 × 𝑋 ) ) ) |
30 |
|
dmxp |
⊢ ( 𝑋 ≠ ∅ → dom ( 𝑋 × 𝑋 ) = 𝑋 ) |
31 |
|
rnxp |
⊢ ( 𝑋 ≠ ∅ → ran ( 𝑋 × 𝑋 ) = 𝑋 ) |
32 |
30 31
|
xpeq12d |
⊢ ( 𝑋 ≠ ∅ → ( dom ( 𝑋 × 𝑋 ) × ran ( 𝑋 × 𝑋 ) ) = ( 𝑋 × 𝑋 ) ) |
33 |
32
|
adantr |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( dom ( 𝑋 × 𝑋 ) × ran ( 𝑋 × 𝑋 ) ) = ( 𝑋 × 𝑋 ) ) |
34 |
29 33
|
sseqtrd |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( dom ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) × ran ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ⊆ ( 𝑋 × 𝑋 ) ) |
35 |
20 34
|
sstrid |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ⊆ ( 𝑋 × 𝑋 ) ) |
36 |
35
|
ad3antrrr |
⊢ ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ⊆ ( 𝑋 × 𝑋 ) ) |
37 |
36
|
sselda |
⊢ ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) → 〈 𝑝 , 𝑞 〉 ∈ ( 𝑋 × 𝑋 ) ) |
38 |
|
opelxp |
⊢ ( 〈 𝑝 , 𝑞 〉 ∈ ( 𝑋 × 𝑋 ) ↔ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) |
39 |
37 38
|
sylib |
⊢ ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) → ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) |
40 |
|
simpll |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) → ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
41 |
|
simprl |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) → 𝑝 ∈ 𝑋 ) |
42 |
|
simprr |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) → 𝑞 ∈ 𝑋 ) |
43 |
|
simplr |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) → 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) |
44 |
|
simplll |
⊢ ( ( ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) |
45 |
44
|
simp1d |
⊢ ( ( ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
46 |
45 2
|
syl |
⊢ ( ( ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) |
47 |
45 3
|
syl |
⊢ ( ( ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑎 ∈ ℝ+ ) |
48 |
46 47
|
jca |
⊢ ( ( ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ) |
49 |
44
|
simp2d |
⊢ ( ( ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑝 ∈ 𝑋 ) |
50 |
44
|
simp3d |
⊢ ( ( ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑞 ∈ 𝑋 ) |
51 |
48 49 50
|
3jca |
⊢ ( ( ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) |
52 |
|
simplr |
⊢ ( ( ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑟 ∈ 𝑋 ) |
53 |
|
simprl |
⊢ ( ( ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ) |
54 |
|
simprr |
⊢ ( ( ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) |
55 |
|
simpll |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) |
56 |
55
|
simp1d |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ) |
57 |
56
|
simpld |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) |
58 |
22
|
ffund |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → Fun 𝐷 ) |
59 |
57 58
|
syl |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → Fun 𝐷 ) |
60 |
55
|
simp2d |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑝 ∈ 𝑋 ) |
61 |
55
|
simp3d |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑞 ∈ 𝑋 ) |
62 |
60 61
|
opelxpd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 〈 𝑝 , 𝑞 〉 ∈ ( 𝑋 × 𝑋 ) ) |
63 |
22
|
fdmd |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → dom 𝐷 = ( 𝑋 × 𝑋 ) ) |
64 |
57 63
|
syl |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → dom 𝐷 = ( 𝑋 × 𝑋 ) ) |
65 |
62 64
|
eleqtrrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 〈 𝑝 , 𝑞 〉 ∈ dom 𝐷 ) |
66 |
|
0xr |
⊢ 0 ∈ ℝ* |
67 |
66
|
a1i |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 0 ∈ ℝ* ) |
68 |
56
|
simprd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑎 ∈ ℝ+ ) |
69 |
68
|
rpxrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑎 ∈ ℝ* ) |
70 |
57 22
|
syl |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* ) |
71 |
70 62
|
ffvelrnd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) ∈ ℝ* ) |
72 |
|
psmetge0 |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) → 0 ≤ ( 𝑝 𝐷 𝑞 ) ) |
73 |
57 60 61 72
|
syl3anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 0 ≤ ( 𝑝 𝐷 𝑞 ) ) |
74 |
|
df-ov |
⊢ ( 𝑝 𝐷 𝑞 ) = ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) |
75 |
73 74
|
breqtrdi |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 0 ≤ ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) ) |
76 |
74 71
|
eqeltrid |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝑝 𝐷 𝑞 ) ∈ ℝ* ) |
77 |
|
0red |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 0 ∈ ℝ ) |
78 |
68
|
rpred |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑎 ∈ ℝ ) |
79 |
78
|
rehalfcld |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝑎 / 2 ) ∈ ℝ ) |
80 |
79
|
rexrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝑎 / 2 ) ∈ ℝ* ) |
81 |
|
df-ov |
⊢ ( 𝑝 𝐷 𝑟 ) = ( 𝐷 ‘ 〈 𝑝 , 𝑟 〉 ) |
82 |
|
simplr |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑟 ∈ 𝑋 ) |
83 |
60 82
|
opelxpd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 〈 𝑝 , 𝑟 〉 ∈ ( 𝑋 × 𝑋 ) ) |
84 |
83 64
|
eleqtrrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 〈 𝑝 , 𝑟 〉 ∈ dom 𝐷 ) |
85 |
|
simprl |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ) |
86 |
|
df-br |
⊢ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ↔ 〈 𝑝 , 𝑟 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) |
87 |
85 86
|
sylib |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 〈 𝑝 , 𝑟 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) |
88 |
|
fvimacnv |
⊢ ( ( Fun 𝐷 ∧ 〈 𝑝 , 𝑟 〉 ∈ dom 𝐷 ) → ( ( 𝐷 ‘ 〈 𝑝 , 𝑟 〉 ) ∈ ( 0 [,) ( 𝑎 / 2 ) ) ↔ 〈 𝑝 , 𝑟 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) |
89 |
88
|
biimpar |
⊢ ( ( ( Fun 𝐷 ∧ 〈 𝑝 , 𝑟 〉 ∈ dom 𝐷 ) ∧ 〈 𝑝 , 𝑟 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) → ( 𝐷 ‘ 〈 𝑝 , 𝑟 〉 ) ∈ ( 0 [,) ( 𝑎 / 2 ) ) ) |
90 |
59 84 87 89
|
syl21anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝐷 ‘ 〈 𝑝 , 𝑟 〉 ) ∈ ( 0 [,) ( 𝑎 / 2 ) ) ) |
91 |
81 90
|
eqeltrid |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝑝 𝐷 𝑟 ) ∈ ( 0 [,) ( 𝑎 / 2 ) ) ) |
92 |
|
elico2 |
⊢ ( ( 0 ∈ ℝ ∧ ( 𝑎 / 2 ) ∈ ℝ* ) → ( ( 𝑝 𝐷 𝑟 ) ∈ ( 0 [,) ( 𝑎 / 2 ) ) ↔ ( ( 𝑝 𝐷 𝑟 ) ∈ ℝ ∧ 0 ≤ ( 𝑝 𝐷 𝑟 ) ∧ ( 𝑝 𝐷 𝑟 ) < ( 𝑎 / 2 ) ) ) ) |
93 |
92
|
biimpa |
⊢ ( ( ( 0 ∈ ℝ ∧ ( 𝑎 / 2 ) ∈ ℝ* ) ∧ ( 𝑝 𝐷 𝑟 ) ∈ ( 0 [,) ( 𝑎 / 2 ) ) ) → ( ( 𝑝 𝐷 𝑟 ) ∈ ℝ ∧ 0 ≤ ( 𝑝 𝐷 𝑟 ) ∧ ( 𝑝 𝐷 𝑟 ) < ( 𝑎 / 2 ) ) ) |
94 |
93
|
simp1d |
⊢ ( ( ( 0 ∈ ℝ ∧ ( 𝑎 / 2 ) ∈ ℝ* ) ∧ ( 𝑝 𝐷 𝑟 ) ∈ ( 0 [,) ( 𝑎 / 2 ) ) ) → ( 𝑝 𝐷 𝑟 ) ∈ ℝ ) |
95 |
77 80 91 94
|
syl21anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝑝 𝐷 𝑟 ) ∈ ℝ ) |
96 |
|
df-ov |
⊢ ( 𝑟 𝐷 𝑞 ) = ( 𝐷 ‘ 〈 𝑟 , 𝑞 〉 ) |
97 |
82 61
|
opelxpd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 〈 𝑟 , 𝑞 〉 ∈ ( 𝑋 × 𝑋 ) ) |
98 |
97 64
|
eleqtrrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 〈 𝑟 , 𝑞 〉 ∈ dom 𝐷 ) |
99 |
|
simprr |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) |
100 |
|
df-br |
⊢ ( 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ↔ 〈 𝑟 , 𝑞 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) |
101 |
99 100
|
sylib |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 〈 𝑟 , 𝑞 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) |
102 |
|
fvimacnv |
⊢ ( ( Fun 𝐷 ∧ 〈 𝑟 , 𝑞 〉 ∈ dom 𝐷 ) → ( ( 𝐷 ‘ 〈 𝑟 , 𝑞 〉 ) ∈ ( 0 [,) ( 𝑎 / 2 ) ) ↔ 〈 𝑟 , 𝑞 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) |
103 |
102
|
biimpar |
⊢ ( ( ( Fun 𝐷 ∧ 〈 𝑟 , 𝑞 〉 ∈ dom 𝐷 ) ∧ 〈 𝑟 , 𝑞 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) → ( 𝐷 ‘ 〈 𝑟 , 𝑞 〉 ) ∈ ( 0 [,) ( 𝑎 / 2 ) ) ) |
104 |
59 98 101 103
|
syl21anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝐷 ‘ 〈 𝑟 , 𝑞 〉 ) ∈ ( 0 [,) ( 𝑎 / 2 ) ) ) |
105 |
96 104
|
eqeltrid |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝑟 𝐷 𝑞 ) ∈ ( 0 [,) ( 𝑎 / 2 ) ) ) |
106 |
|
elico2 |
⊢ ( ( 0 ∈ ℝ ∧ ( 𝑎 / 2 ) ∈ ℝ* ) → ( ( 𝑟 𝐷 𝑞 ) ∈ ( 0 [,) ( 𝑎 / 2 ) ) ↔ ( ( 𝑟 𝐷 𝑞 ) ∈ ℝ ∧ 0 ≤ ( 𝑟 𝐷 𝑞 ) ∧ ( 𝑟 𝐷 𝑞 ) < ( 𝑎 / 2 ) ) ) ) |
107 |
106
|
biimpa |
⊢ ( ( ( 0 ∈ ℝ ∧ ( 𝑎 / 2 ) ∈ ℝ* ) ∧ ( 𝑟 𝐷 𝑞 ) ∈ ( 0 [,) ( 𝑎 / 2 ) ) ) → ( ( 𝑟 𝐷 𝑞 ) ∈ ℝ ∧ 0 ≤ ( 𝑟 𝐷 𝑞 ) ∧ ( 𝑟 𝐷 𝑞 ) < ( 𝑎 / 2 ) ) ) |
108 |
107
|
simp1d |
⊢ ( ( ( 0 ∈ ℝ ∧ ( 𝑎 / 2 ) ∈ ℝ* ) ∧ ( 𝑟 𝐷 𝑞 ) ∈ ( 0 [,) ( 𝑎 / 2 ) ) ) → ( 𝑟 𝐷 𝑞 ) ∈ ℝ ) |
109 |
77 80 105 108
|
syl21anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝑟 𝐷 𝑞 ) ∈ ℝ ) |
110 |
95 109
|
rexaddd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( ( 𝑝 𝐷 𝑟 ) +𝑒 ( 𝑟 𝐷 𝑞 ) ) = ( ( 𝑝 𝐷 𝑟 ) + ( 𝑟 𝐷 𝑞 ) ) ) |
111 |
95 109
|
readdcld |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( ( 𝑝 𝐷 𝑟 ) + ( 𝑟 𝐷 𝑞 ) ) ∈ ℝ ) |
112 |
110 111
|
eqeltrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( ( 𝑝 𝐷 𝑟 ) +𝑒 ( 𝑟 𝐷 𝑞 ) ) ∈ ℝ ) |
113 |
112
|
rexrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( ( 𝑝 𝐷 𝑟 ) +𝑒 ( 𝑟 𝐷 𝑞 ) ) ∈ ℝ* ) |
114 |
|
psmettri |
⊢ ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑋 ) ) → ( 𝑝 𝐷 𝑞 ) ≤ ( ( 𝑝 𝐷 𝑟 ) +𝑒 ( 𝑟 𝐷 𝑞 ) ) ) |
115 |
57 60 61 82 114
|
syl13anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝑝 𝐷 𝑞 ) ≤ ( ( 𝑝 𝐷 𝑟 ) +𝑒 ( 𝑟 𝐷 𝑞 ) ) ) |
116 |
93
|
simp3d |
⊢ ( ( ( 0 ∈ ℝ ∧ ( 𝑎 / 2 ) ∈ ℝ* ) ∧ ( 𝑝 𝐷 𝑟 ) ∈ ( 0 [,) ( 𝑎 / 2 ) ) ) → ( 𝑝 𝐷 𝑟 ) < ( 𝑎 / 2 ) ) |
117 |
77 80 91 116
|
syl21anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝑝 𝐷 𝑟 ) < ( 𝑎 / 2 ) ) |
118 |
107
|
simp3d |
⊢ ( ( ( 0 ∈ ℝ ∧ ( 𝑎 / 2 ) ∈ ℝ* ) ∧ ( 𝑟 𝐷 𝑞 ) ∈ ( 0 [,) ( 𝑎 / 2 ) ) ) → ( 𝑟 𝐷 𝑞 ) < ( 𝑎 / 2 ) ) |
119 |
77 80 105 118
|
syl21anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝑟 𝐷 𝑞 ) < ( 𝑎 / 2 ) ) |
120 |
95 109 78 117 119
|
lt2halvesd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( ( 𝑝 𝐷 𝑟 ) + ( 𝑟 𝐷 𝑞 ) ) < 𝑎 ) |
121 |
110 120
|
eqbrtrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( ( 𝑝 𝐷 𝑟 ) +𝑒 ( 𝑟 𝐷 𝑞 ) ) < 𝑎 ) |
122 |
76 113 69 115 121
|
xrlelttrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝑝 𝐷 𝑞 ) < 𝑎 ) |
123 |
74 122
|
eqbrtrrid |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) < 𝑎 ) |
124 |
67 69 71 75 123
|
elicod |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) ∈ ( 0 [,) 𝑎 ) ) |
125 |
|
fvimacnv |
⊢ ( ( Fun 𝐷 ∧ 〈 𝑝 , 𝑞 〉 ∈ dom 𝐷 ) → ( ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) ∈ ( 0 [,) 𝑎 ) ↔ 〈 𝑝 , 𝑞 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
126 |
125
|
biimpa |
⊢ ( ( ( Fun 𝐷 ∧ 〈 𝑝 , 𝑞 〉 ∈ dom 𝐷 ) ∧ ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) ∈ ( 0 [,) 𝑎 ) ) → 〈 𝑝 , 𝑞 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
127 |
|
df-br |
⊢ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) 𝑞 ↔ 〈 𝑝 , 𝑞 〉 ∈ ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
128 |
126 127
|
sylibr |
⊢ ( ( ( Fun 𝐷 ∧ 〈 𝑝 , 𝑞 〉 ∈ dom 𝐷 ) ∧ ( 𝐷 ‘ 〈 𝑝 , 𝑞 〉 ) ∈ ( 0 [,) 𝑎 ) ) → 𝑝 ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) 𝑞 ) |
129 |
59 65 124 128
|
syl21anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑝 ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) 𝑞 ) |
130 |
51 52 53 54 129
|
syl22anc |
⊢ ( ( ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑝 ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) 𝑞 ) |
131 |
45
|
simprd |
⊢ ( ( ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
132 |
131
|
breqd |
⊢ ( ( ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → ( 𝑝 𝐴 𝑞 ↔ 𝑝 ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) 𝑞 ) ) |
133 |
130 132
|
mpbird |
⊢ ( ( ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ 𝑟 ∈ 𝑋 ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑝 𝐴 𝑞 ) |
134 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) → 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) |
135 |
|
df-br |
⊢ ( 𝑝 ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) 𝑞 ↔ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) |
136 |
134 135
|
sylibr |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) → 𝑝 ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) 𝑞 ) |
137 |
|
vex |
⊢ 𝑝 ∈ V |
138 |
|
vex |
⊢ 𝑞 ∈ V |
139 |
137 138
|
brco |
⊢ ( 𝑝 ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) 𝑞 ↔ ∃ 𝑟 ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) |
140 |
136 139
|
sylib |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) → ∃ 𝑟 ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) |
141 |
23
|
adantl |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ⊆ ( 𝑋 × 𝑋 ) ) |
142 |
141 25
|
syl |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ran ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ⊆ ran ( 𝑋 × 𝑋 ) ) |
143 |
31
|
adantr |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ran ( 𝑋 × 𝑋 ) = 𝑋 ) |
144 |
142 143
|
sseqtrd |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ran ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ⊆ 𝑋 ) |
145 |
144
|
adantr |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ) → ran ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ⊆ 𝑋 ) |
146 |
|
vex |
⊢ 𝑟 ∈ V |
147 |
137 146
|
brelrn |
⊢ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 → 𝑟 ∈ ran ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) |
148 |
147
|
adantl |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ) → 𝑟 ∈ ran ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) |
149 |
145 148
|
sseldd |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ) → 𝑟 ∈ 𝑋 ) |
150 |
149
|
adantrr |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑟 ∈ 𝑋 ) |
151 |
150
|
ex |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) → 𝑟 ∈ 𝑋 ) ) |
152 |
151
|
ancrd |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) → ( 𝑟 ∈ 𝑋 ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) ) ) |
153 |
152
|
eximdv |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( ∃ 𝑟 ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) → ∃ 𝑟 ( 𝑟 ∈ 𝑋 ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) ) ) |
154 |
153
|
ad3antrrr |
⊢ ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( ∃ 𝑟 ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) → ∃ 𝑟 ( 𝑟 ∈ 𝑋 ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) ) ) |
155 |
154
|
3ad2ant1 |
⊢ ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) → ( ∃ 𝑟 ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) → ∃ 𝑟 ( 𝑟 ∈ 𝑋 ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) ) ) |
156 |
155
|
adantr |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) → ( ∃ 𝑟 ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) → ∃ 𝑟 ( 𝑟 ∈ 𝑋 ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) ) ) |
157 |
140 156
|
mpd |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) → ∃ 𝑟 ( 𝑟 ∈ 𝑋 ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) ) |
158 |
|
df-rex |
⊢ ( ∃ 𝑟 ∈ 𝑋 ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ↔ ∃ 𝑟 ( 𝑟 ∈ 𝑋 ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) ) |
159 |
157 158
|
sylibr |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) → ∃ 𝑟 ∈ 𝑋 ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) |
160 |
133 159
|
r19.29a |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) → 𝑝 𝐴 𝑞 ) |
161 |
|
df-br |
⊢ ( 𝑝 𝐴 𝑞 ↔ 〈 𝑝 , 𝑞 〉 ∈ 𝐴 ) |
162 |
160 161
|
sylib |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) → 〈 𝑝 , 𝑞 〉 ∈ 𝐴 ) |
163 |
40 41 42 43 162
|
syl31anc |
⊢ ( ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) ∧ ( 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ) ) → 〈 𝑝 , 𝑞 〉 ∈ 𝐴 ) |
164 |
39 163
|
mpdan |
⊢ ( ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ∧ 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) → 〈 𝑝 , 𝑞 〉 ∈ 𝐴 ) |
165 |
164
|
ex |
⊢ ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( 〈 𝑝 , 𝑞 〉 ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) → 〈 𝑝 , 𝑞 〉 ∈ 𝐴 ) ) |
166 |
19 165
|
relssdv |
⊢ ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ⊆ 𝐴 ) |
167 |
|
id |
⊢ ( 𝑣 = ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) → 𝑣 = ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) |
168 |
167 167
|
coeq12d |
⊢ ( 𝑣 = ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) → ( 𝑣 ∘ 𝑣 ) = ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) |
169 |
168
|
sseq1d |
⊢ ( 𝑣 = ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) → ( ( 𝑣 ∘ 𝑣 ) ⊆ 𝐴 ↔ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ⊆ 𝐴 ) ) |
170 |
169
|
rspcev |
⊢ ( ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∈ 𝐹 ∧ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ⊆ 𝐴 ) → ∃ 𝑣 ∈ 𝐹 ( 𝑣 ∘ 𝑣 ) ⊆ 𝐴 ) |
171 |
17 166 170
|
syl2anc |
⊢ ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ∃ 𝑣 ∈ 𝐹 ( 𝑣 ∘ 𝑣 ) ⊆ 𝐴 ) |
172 |
1
|
metustel |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( 𝐴 ∈ 𝐹 ↔ ∃ 𝑎 ∈ ℝ+ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
173 |
172
|
adantl |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( 𝐴 ∈ 𝐹 ↔ ∃ 𝑎 ∈ ℝ+ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
174 |
173
|
biimpa |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) → ∃ 𝑎 ∈ ℝ+ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
175 |
171 174
|
r19.29a |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) → ∃ 𝑣 ∈ 𝐹 ( 𝑣 ∘ 𝑣 ) ⊆ 𝐴 ) |