| 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
|
ffvelcdmd |
⊢ ( ( ( ( ( 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 |
|
df-br |
⊢ ( 𝑝 ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) 𝑞 ↔ ⟨ 𝑝 , 𝑞 ⟩ ∈ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) |
| 135 |
134
|
bilanri |
⊢ ( ( ( ( ( ( ( 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 ) |
| 136 |
|
vex |
⊢ 𝑝 ∈ V |
| 137 |
|
vex |
⊢ 𝑞 ∈ V |
| 138 |
136 137
|
brco |
⊢ ( 𝑝 ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) 𝑞 ↔ ∃ 𝑟 ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) |
| 139 |
135 138
|
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 ) ) |
| 140 |
23
|
adantl |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ⊆ ( 𝑋 × 𝑋 ) ) |
| 141 |
140 25
|
syl |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ran ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ⊆ ran ( 𝑋 × 𝑋 ) ) |
| 142 |
31
|
adantr |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ran ( 𝑋 × 𝑋 ) = 𝑋 ) |
| 143 |
141 142
|
sseqtrd |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ran ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ⊆ 𝑋 ) |
| 144 |
143
|
adantr |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ) → ran ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ⊆ 𝑋 ) |
| 145 |
|
vex |
⊢ 𝑟 ∈ V |
| 146 |
136 145
|
brelrn |
⊢ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 → 𝑟 ∈ ran ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) |
| 147 |
146
|
adantl |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ) → 𝑟 ∈ ran ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) |
| 148 |
144 147
|
sseldd |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ) → 𝑟 ∈ 𝑋 ) |
| 149 |
148
|
adantrr |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) ) → 𝑟 ∈ 𝑋 ) |
| 150 |
149
|
ex |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( ( 𝑝 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑟 ∧ 𝑟 ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) 𝑞 ) → 𝑟 ∈ 𝑋 ) ) |
| 151 |
150
|
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 ) ) ) ) |
| 152 |
151
|
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 ) ) ) ) |
| 153 |
152
|
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 ) ) ) ) |
| 154 |
153
|
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 ) ) ) ) |
| 155 |
154
|
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 ) ) ) ) |
| 156 |
139 155
|
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 ) ) ) |
| 157 |
|
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 ) ) ) |
| 158 |
156 157
|
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 ) ) |
| 159 |
133 158
|
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 ) |
| 160 |
|
df-br |
⊢ ( 𝑝 𝐴 𝑞 ↔ ⟨ 𝑝 , 𝑞 ⟩ ∈ 𝐴 ) |
| 161 |
159 160
|
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 ) |
| 162 |
40 41 42 43 161
|
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 ) |
| 163 |
39 162
|
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 ) |
| 164 |
163
|
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 ) ) |
| 165 |
19 164
|
relssdv |
⊢ ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ⊆ 𝐴 ) |
| 166 |
|
id |
⊢ ( 𝑣 = ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) → 𝑣 = ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) |
| 167 |
166 166
|
coeq12d |
⊢ ( 𝑣 = ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) → ( 𝑣 ∘ 𝑣 ) = ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ) |
| 168 |
167
|
sseq1d |
⊢ ( 𝑣 = ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) → ( ( 𝑣 ∘ 𝑣 ) ⊆ 𝐴 ↔ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ⊆ 𝐴 ) ) |
| 169 |
168
|
rspcev |
⊢ ( ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∈ 𝐹 ∧ ( ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ∘ ( ◡ 𝐷 “ ( 0 [,) ( 𝑎 / 2 ) ) ) ) ⊆ 𝐴 ) → ∃ 𝑣 ∈ 𝐹 ( 𝑣 ∘ 𝑣 ) ⊆ 𝐴 ) |
| 170 |
17 165 169
|
syl2anc |
⊢ ( ( ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) ∧ 𝑎 ∈ ℝ+ ) ∧ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) → ∃ 𝑣 ∈ 𝐹 ( 𝑣 ∘ 𝑣 ) ⊆ 𝐴 ) |
| 171 |
1
|
metustel |
⊢ ( 𝐷 ∈ ( PsMet ‘ 𝑋 ) → ( 𝐴 ∈ 𝐹 ↔ ∃ 𝑎 ∈ ℝ+ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
| 172 |
171
|
adantl |
⊢ ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) → ( 𝐴 ∈ 𝐹 ↔ ∃ 𝑎 ∈ ℝ+ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) ) |
| 173 |
172
|
biimpa |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) → ∃ 𝑎 ∈ ℝ+ 𝐴 = ( ◡ 𝐷 “ ( 0 [,) 𝑎 ) ) ) |
| 174 |
170 173
|
r19.29a |
⊢ ( ( ( 𝑋 ≠ ∅ ∧ 𝐷 ∈ ( PsMet ‘ 𝑋 ) ) ∧ 𝐴 ∈ 𝐹 ) → ∃ 𝑣 ∈ 𝐹 ( 𝑣 ∘ 𝑣 ) ⊆ 𝐴 ) |