Step | Hyp | Ref
| Expression |
1 | | simp-4r 784 |
. . . 4
⊢
(((((𝑋 ≠ ∅
∧ 𝐷 ∈
(PsMet‘𝑋)) ∧
𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝐷 ∈ (PsMet‘𝑋)) |
2 | | simplr 769 |
. . . . . 6
⊢
(((((𝑋 ≠ ∅
∧ 𝐷 ∈
(PsMet‘𝑋)) ∧
𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝑎 ∈ ℝ+) |
3 | 2 | rphalfcld 12539 |
. . . . 5
⊢
(((((𝑋 ≠ ∅
∧ 𝐷 ∈
(PsMet‘𝑋)) ∧
𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (𝑎 / 2) ∈
ℝ+) |
4 | | eqidd 2740 |
. . . . 5
⊢
(((((𝑋 ≠ ∅
∧ 𝐷 ∈
(PsMet‘𝑋)) ∧
𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (◡𝐷 “ (0[,)(𝑎 / 2))) = (◡𝐷 “ (0[,)(𝑎 / 2)))) |
5 | | oveq2 7191 |
. . . . . . 7
⊢ (𝑏 = (𝑎 / 2) → (0[,)𝑏) = (0[,)(𝑎 / 2))) |
6 | 5 | imaeq2d 5913 |
. . . . . 6
⊢ (𝑏 = (𝑎 / 2) → (◡𝐷 “ (0[,)𝑏)) = (◡𝐷 “ (0[,)(𝑎 / 2)))) |
7 | 6 | rspceeqv 3544 |
. . . . 5
⊢ (((𝑎 / 2) ∈ ℝ+
∧ (◡𝐷 “ (0[,)(𝑎 / 2))) = (◡𝐷 “ (0[,)(𝑎 / 2)))) → ∃𝑏 ∈ ℝ+ (◡𝐷 “ (0[,)(𝑎 / 2))) = (◡𝐷 “ (0[,)𝑏))) |
8 | 3, 4, 7 | syl2anc 587 |
. . . 4
⊢
(((((𝑋 ≠ ∅
∧ 𝐷 ∈
(PsMet‘𝑋)) ∧
𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ∃𝑏 ∈ ℝ+ (◡𝐷 “ (0[,)(𝑎 / 2))) = (◡𝐷 “ (0[,)𝑏))) |
9 | | metust.1 |
. . . . . . 7
⊢ 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) |
10 | | oveq2 7191 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏)) |
11 | 10 | imaeq2d 5913 |
. . . . . . . . 9
⊢ (𝑎 = 𝑏 → (◡𝐷 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)𝑏))) |
12 | 11 | cbvmptv 5143 |
. . . . . . . 8
⊢ (𝑎 ∈ ℝ+
↦ (◡𝐷 “ (0[,)𝑎))) = (𝑏 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑏))) |
13 | 12 | rneqi 5790 |
. . . . . . 7
⊢ ran
(𝑎 ∈
ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑏))) |
14 | 9, 13 | eqtri 2762 |
. . . . . 6
⊢ 𝐹 = ran (𝑏 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑏))) |
15 | 14 | metustel 23316 |
. . . . 5
⊢ (𝐷 ∈ (PsMet‘𝑋) → ((◡𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹 ↔ ∃𝑏 ∈ ℝ+ (◡𝐷 “ (0[,)(𝑎 / 2))) = (◡𝐷 “ (0[,)𝑏)))) |
16 | 15 | biimpar 481 |
. . . 4
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ ∃𝑏 ∈ ℝ+
(◡𝐷 “ (0[,)(𝑎 / 2))) = (◡𝐷 “ (0[,)𝑏))) → (◡𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹) |
17 | 1, 8, 16 | syl2anc 587 |
. . 3
⊢
(((((𝑋 ≠ ∅
∧ 𝐷 ∈
(PsMet‘𝑋)) ∧
𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (◡𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹) |
18 | | relco 6087 |
. . . . 5
⊢ Rel
((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))) |
19 | 18 | a1i 11 |
. . . 4
⊢
(((((𝑋 ≠ ∅
∧ 𝐷 ∈
(PsMet‘𝑋)) ∧
𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → Rel ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) |
20 | | cossxp 6114 |
. . . . . . . . . 10
⊢ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (dom (◡𝐷 “ (0[,)(𝑎 / 2))) × ran (◡𝐷 “ (0[,)(𝑎 / 2)))) |
21 | | cnvimass 5933 |
. . . . . . . . . . . . . 14
⊢ (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ dom 𝐷 |
22 | | psmetf 23072 |
. . . . . . . . . . . . . 14
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
23 | 21, 22 | fssdm 6535 |
. . . . . . . . . . . . 13
⊢ (𝐷 ∈ (PsMet‘𝑋) → (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋)) |
24 | | dmss 5755 |
. . . . . . . . . . . . . 14
⊢ ((◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋) → dom (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ dom (𝑋 × 𝑋)) |
25 | | rnss 5792 |
. . . . . . . . . . . . . 14
⊢ ((◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋) → ran (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ ran (𝑋 × 𝑋)) |
26 | | xpss12 5550 |
. . . . . . . . . . . . . 14
⊢ ((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 587 |
. . . . . . . . . . . . 13
⊢ ((◡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 17 |
. . . . . . . . . . . 12
⊢ (𝐷 ∈ (PsMet‘𝑋) → (dom (◡𝐷 “ (0[,)(𝑎 / 2))) × ran (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋))) |
29 | 28 | adantl 485 |
. . . . . . . . . . 11
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (dom (◡𝐷 “ (0[,)(𝑎 / 2))) × ran (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋))) |
30 | | dmxp 5782 |
. . . . . . . . . . . . 13
⊢ (𝑋 ≠ ∅ → dom (𝑋 × 𝑋) = 𝑋) |
31 | | rnxp 6012 |
. . . . . . . . . . . . 13
⊢ (𝑋 ≠ ∅ → ran (𝑋 × 𝑋) = 𝑋) |
32 | 30, 31 | xpeq12d 5566 |
. . . . . . . . . . . 12
⊢ (𝑋 ≠ ∅ → (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)) = (𝑋 × 𝑋)) |
33 | 32 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)) = (𝑋 × 𝑋)) |
34 | 29, 33 | sseqtrd 3927 |
. . . . . . . . . 10
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (dom (◡𝐷 “ (0[,)(𝑎 / 2))) × ran (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (𝑋 × 𝑋)) |
35 | 20, 34 | sstrid 3898 |
. . . . . . . . 9
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (𝑋 × 𝑋)) |
36 | 35 | ad3antrrr 730 |
. . . . . . . 8
⊢
(((((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 3887 |
. . . . . . 7
⊢
((((((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 5571 |
. . . . . . 7
⊢
(〈𝑝, 𝑞〉 ∈ (𝑋 × 𝑋) ↔ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) |
39 | 37, 38 | sylib 221 |
. . . . . 6
⊢
((((((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 767 |
. . . . . . 7
⊢
(((((((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 771 |
. . . . . . 7
⊢
(((((((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 773 |
. . . . . . 7
⊢
(((((((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 769 |
. . . . . . 7
⊢
(((((((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 775 |
. . . . . . . . . . . . . . 15
⊢
(((((((((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 1143 |
. . . . . . . . . . . . . 14
⊢
(((((((((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, 1 | syl 17 |
. . . . . . . . . . . . 13
⊢
(((((((((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, 2 | syl 17 |
. . . . . . . . . . . . 13
⊢
(((((((((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 515 |
. . . . . . . . . . . 12
⊢
(((((((((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 1144 |
. . . . . . . . . . . 12
⊢
(((((((((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 1145 |
. . . . . . . . . . . 12
⊢
(((((((((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 1129 |
. . . . . . . . . . 11
⊢
(((((((((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 769 |
. . . . . . . . . . 11
⊢
(((((((((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 771 |
. . . . . . . . . . 11
⊢
(((((((((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 773 |
. . . . . . . . . . 11
⊢
(((((((((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 767 |
. . . . . . . . . . . . . . 15
⊢
(((((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 1143 |
. . . . . . . . . . . . . 14
⊢
(((((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 498 |
. . . . . . . . . . . . 13
⊢
(((((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 6519 |
. . . . . . . . . . . . 13
⊢ (𝐷 ∈ (PsMet‘𝑋) → Fun 𝐷) |
59 | 57, 58 | syl 17 |
. . . . . . . . . . . 12
⊢
(((((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 1144 |
. . . . . . . . . . . . . 14
⊢
(((((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 1145 |
. . . . . . . . . . . . . 14
⊢
(((((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 5573 |
. . . . . . . . . . . . 13
⊢
(((((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 6526 |
. . . . . . . . . . . . . 14
⊢ (𝐷 ∈ (PsMet‘𝑋) → dom 𝐷 = (𝑋 × 𝑋)) |
64 | 57, 63 | syl 17 |
. . . . . . . . . . . . 13
⊢
(((((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 2837 |
. . . . . . . . . . . 12
⊢
(((((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 10779 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
ℝ* |
67 | 66 | a1i 11 |
. . . . . . . . . . . . 13
⊢
(((((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 499 |
. . . . . . . . . . . . . 14
⊢
(((((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 12528 |
. . . . . . . . . . . . 13
⊢
(((((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 17 |
. . . . . . . . . . . . . 14
⊢
(((((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 6875 |
. . . . . . . . . . . . 13
⊢
(((((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 23078 |
. . . . . . . . . . . . . . 15
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) → 0 ≤ (𝑝𝐷𝑞)) |
73 | 57, 60, 61, 72 | syl3anc 1372 |
. . . . . . . . . . . . . 14
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+)
∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 0 ≤ (𝑝𝐷𝑞)) |
74 | | df-ov 7186 |
. . . . . . . . . . . . . 14
⊢ (𝑝𝐷𝑞) = (𝐷‘〈𝑝, 𝑞〉) |
75 | 73, 74 | breqtrdi 5081 |
. . . . . . . . . . . . 13
⊢
(((((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 2838 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+)
∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑞) ∈
ℝ*) |
77 | | 0red 10735 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((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 12527 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((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 11976 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((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 10782 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((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 7186 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑝𝐷𝑟) = (𝐷‘〈𝑝, 𝑟〉) |
82 | | simplr 769 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((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 5573 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((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 2837 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((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 771 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((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 5041 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ↔ 〈𝑝, 𝑟〉 ∈ (◡𝐷 “ (0[,)(𝑎 / 2)))) |
87 | 85, 86 | sylib 221 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((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 6843 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((Fun
𝐷 ∧ 〈𝑝, 𝑟〉 ∈ dom 𝐷) → ((𝐷‘〈𝑝, 𝑟〉) ∈ (0[,)(𝑎 / 2)) ↔ 〈𝑝, 𝑟〉 ∈ (◡𝐷 “ (0[,)(𝑎 / 2))))) |
89 | 88 | biimpar 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((Fun
𝐷 ∧ 〈𝑝, 𝑟〉 ∈ dom 𝐷) ∧ 〈𝑝, 𝑟〉 ∈ (◡𝐷 “ (0[,)(𝑎 / 2)))) → (𝐷‘〈𝑝, 𝑟〉) ∈ (0[,)(𝑎 / 2))) |
90 | 59, 84, 87, 89 | syl21anc 837 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((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 2838 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+)
∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) |
92 | | elico2 12898 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((0
∈ ℝ ∧ (𝑎 /
2) ∈ ℝ*) → ((𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2)) ↔ ((𝑝𝐷𝑟) ∈ ℝ ∧ 0 ≤ (𝑝𝐷𝑟) ∧ (𝑝𝐷𝑟) < (𝑎 / 2)))) |
93 | 92 | biimpa 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((0
∈ ℝ ∧ (𝑎 /
2) ∈ ℝ*) ∧ (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) → ((𝑝𝐷𝑟) ∈ ℝ ∧ 0 ≤ (𝑝𝐷𝑟) ∧ (𝑝𝐷𝑟) < (𝑎 / 2))) |
94 | 93 | simp1d 1143 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((0
∈ ℝ ∧ (𝑎 /
2) ∈ ℝ*) ∧ (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) → (𝑝𝐷𝑟) ∈ ℝ) |
95 | 77, 80, 91, 94 | syl21anc 837 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+)
∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑟) ∈ ℝ) |
96 | | df-ov 7186 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑟𝐷𝑞) = (𝐷‘〈𝑟, 𝑞〉) |
97 | 82, 61 | opelxpd 5573 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((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 2837 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((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 773 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((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 5041 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞 ↔ 〈𝑟, 𝑞〉 ∈ (◡𝐷 “ (0[,)(𝑎 / 2)))) |
101 | 99, 100 | sylib 221 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((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 6843 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((Fun
𝐷 ∧ 〈𝑟, 𝑞〉 ∈ dom 𝐷) → ((𝐷‘〈𝑟, 𝑞〉) ∈ (0[,)(𝑎 / 2)) ↔ 〈𝑟, 𝑞〉 ∈ (◡𝐷 “ (0[,)(𝑎 / 2))))) |
103 | 102 | biimpar 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((Fun
𝐷 ∧ 〈𝑟, 𝑞〉 ∈ dom 𝐷) ∧ 〈𝑟, 𝑞〉 ∈ (◡𝐷 “ (0[,)(𝑎 / 2)))) → (𝐷‘〈𝑟, 𝑞〉) ∈ (0[,)(𝑎 / 2))) |
104 | 59, 98, 101, 103 | syl21anc 837 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((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 2838 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+)
∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) |
106 | | elico2 12898 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((0
∈ ℝ ∧ (𝑎 /
2) ∈ ℝ*) → ((𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2)) ↔ ((𝑟𝐷𝑞) ∈ ℝ ∧ 0 ≤ (𝑟𝐷𝑞) ∧ (𝑟𝐷𝑞) < (𝑎 / 2)))) |
107 | 106 | biimpa 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((0
∈ ℝ ∧ (𝑎 /
2) ∈ ℝ*) ∧ (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) → ((𝑟𝐷𝑞) ∈ ℝ ∧ 0 ≤ (𝑟𝐷𝑞) ∧ (𝑟𝐷𝑞) < (𝑎 / 2))) |
108 | 107 | simp1d 1143 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((0
∈ ℝ ∧ (𝑎 /
2) ∈ ℝ*) ∧ (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) → (𝑟𝐷𝑞) ∈ ℝ) |
109 | 77, 80, 105, 108 | syl21anc 837 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+)
∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑟𝐷𝑞) ∈ ℝ) |
110 | 95, 109 | rexaddd 12723 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+)
∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)) = ((𝑝𝐷𝑟) + (𝑟𝐷𝑞))) |
111 | 95, 109 | readdcld 10761 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+)
∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) + (𝑟𝐷𝑞)) ∈ ℝ) |
112 | 110, 111 | eqeltrd 2834 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+)
∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)) ∈ ℝ) |
113 | 112 | rexrd 10782 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+)
∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)) ∈
ℝ*) |
114 | | psmettri 23077 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑋)) → (𝑝𝐷𝑞) ≤ ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞))) |
115 | 57, 60, 61, 82, 114 | syl13anc 1373 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+)
∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑞) ≤ ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞))) |
116 | 93 | simp3d 1145 |
. . . . . . . . . . . . . . . . . 18
⊢ (((0
∈ ℝ ∧ (𝑎 /
2) ∈ ℝ*) ∧ (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) → (𝑝𝐷𝑟) < (𝑎 / 2)) |
117 | 77, 80, 91, 116 | syl21anc 837 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+)
∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑟) < (𝑎 / 2)) |
118 | 107 | simp3d 1145 |
. . . . . . . . . . . . . . . . . 18
⊢ (((0
∈ ℝ ∧ (𝑎 /
2) ∈ ℝ*) ∧ (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) → (𝑟𝐷𝑞) < (𝑎 / 2)) |
119 | 77, 80, 105, 118 | syl21anc 837 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+)
∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑟𝐷𝑞) < (𝑎 / 2)) |
120 | 95, 109, 78, 117, 119 | lt2halvesd 11977 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+)
∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) + (𝑟𝐷𝑞)) < 𝑎) |
121 | 110, 120 | eqbrtrd 5062 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+)
∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)) < 𝑎) |
122 | 76, 113, 69, 115, 121 | xrlelttrd 12649 |
. . . . . . . . . . . . . 14
⊢
(((((𝐷 ∈
(PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+)
∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑞) < 𝑎) |
123 | 74, 122 | eqbrtrrid 5076 |
. . . . . . . . . . . . 13
⊢
(((((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 12884 |
. . . . . . . . . . . 12
⊢
(((((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 6843 |
. . . . . . . . . . . . . 14
⊢ ((Fun
𝐷 ∧ 〈𝑝, 𝑞〉 ∈ dom 𝐷) → ((𝐷‘〈𝑝, 𝑞〉) ∈ (0[,)𝑎) ↔ 〈𝑝, 𝑞〉 ∈ (◡𝐷 “ (0[,)𝑎)))) |
126 | 125 | biimpa 480 |
. . . . . . . . . . . . 13
⊢ (((Fun
𝐷 ∧ 〈𝑝, 𝑞〉 ∈ dom 𝐷) ∧ (𝐷‘〈𝑝, 𝑞〉) ∈ (0[,)𝑎)) → 〈𝑝, 𝑞〉 ∈ (◡𝐷 “ (0[,)𝑎))) |
127 | | df-br 5041 |
. . . . . . . . . . . . 13
⊢ (𝑝(◡𝐷 “ (0[,)𝑎))𝑞 ↔ 〈𝑝, 𝑞〉 ∈ (◡𝐷 “ (0[,)𝑎))) |
128 | 126, 127 | sylibr 237 |
. . . . . . . . . . . 12
⊢ (((Fun
𝐷 ∧ 〈𝑝, 𝑞〉 ∈ dom 𝐷) ∧ (𝐷‘〈𝑝, 𝑞〉) ∈ (0[,)𝑎)) → 𝑝(◡𝐷 “ (0[,)𝑎))𝑞) |
129 | 59, 65, 124, 128 | syl21anc 837 |
. . . . . . . . . . 11
⊢
(((((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 838 |
. . . . . . . . . 10
⊢
(((((((((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 499 |
. . . . . . . . . . 11
⊢
(((((((((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 5051 |
. . . . . . . . . 10
⊢
(((((((((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)) → (𝑝𝐴𝑞 ↔ 𝑝(◡𝐷 “ (0[,)𝑎))𝑞)) |
133 | 130, 132 | mpbird 260 |
. . . . . . . . 9
⊢
(((((((((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)) → 𝑝𝐴𝑞) |
134 | | simpr 488 |
. . . . . . . . . . . . 13
⊢
(((((((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 5041 |
. . . . . . . . . . . . 13
⊢ (𝑝((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))𝑞 ↔ 〈𝑝, 𝑞〉 ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) |
136 | 134, 135 | sylibr 237 |
. . . . . . . . . . . 12
⊢
(((((((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 3404 |
. . . . . . . . . . . . 13
⊢ 𝑝 ∈ V |
138 | | vex 3404 |
. . . . . . . . . . . . 13
⊢ 𝑞 ∈ V |
139 | 137, 138 | brco 5723 |
. . . . . . . . . . . 12
⊢ (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 221 |
. . . . . . . . . . 11
⊢
(((((((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 485 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋)) |
142 | 141, 25 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ ran (𝑋 × 𝑋)) |
143 | 31 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑋 × 𝑋) = 𝑋) |
144 | 142, 143 | sseqtrd 3927 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ 𝑋) |
145 | 144 | adantr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟) → ran (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ 𝑋) |
146 | | vex 3404 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑟 ∈ V |
147 | 137, 146 | brelrn 5795 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 → 𝑟 ∈ ran (◡𝐷 “ (0[,)(𝑎 / 2)))) |
148 | 147 | adantl 485 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟) → 𝑟 ∈ ran (◡𝐷 “ (0[,)(𝑎 / 2)))) |
149 | 145, 148 | sseldd 3888 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟) → 𝑟 ∈ 𝑋) |
150 | 149 | adantrr 717 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑟 ∈ 𝑋) |
151 | 150 | ex 416 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞) → 𝑟 ∈ 𝑋)) |
152 | 151 | ancrd 555 |
. . . . . . . . . . . . . . 15
⊢ ((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 1924 |
. . . . . . . . . . . . . 14
⊢ ((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 730 |
. . . . . . . . . . . . 13
⊢
(((((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 1134 |
. . . . . . . . . . . 12
⊢
((((((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 484 |
. . . . . . . . . . 11
⊢
(((((((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 15 |
. . . . . . . . . 10
⊢
(((((((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 3060 |
. . . . . . . . . 10
⊢
(∃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 237 |
. . . . . . . . 9
⊢
(((((((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 3200 |
. . . . . . . 8
⊢
(((((((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))))) → 𝑝𝐴𝑞) |
161 | | df-br 5041 |
. . . . . . . 8
⊢ (𝑝𝐴𝑞 ↔ 〈𝑝, 𝑞〉 ∈ 𝐴) |
162 | 160, 161 | sylib 221 |
. . . . . . 7
⊢
(((((((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 1374 |
. . . . . 6
⊢
(((((((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 687 |
. . . . 5
⊢
((((((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 416 |
. . . 4
⊢
(((((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 5642 |
. . 3
⊢
(((((𝑋 ≠ ∅
∧ 𝐷 ∈
(PsMet‘𝑋)) ∧
𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ 𝐴) |
167 | | id 22 |
. . . . . 6
⊢ (𝑣 = (◡𝐷 “ (0[,)(𝑎 / 2))) → 𝑣 = (◡𝐷 “ (0[,)(𝑎 / 2)))) |
168 | 167, 167 | coeq12d 5717 |
. . . . 5
⊢ (𝑣 = (◡𝐷 “ (0[,)(𝑎 / 2))) → (𝑣 ∘ 𝑣) = ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) |
169 | 168 | sseq1d 3918 |
. . . 4
⊢ (𝑣 = (◡𝐷 “ (0[,)(𝑎 / 2))) → ((𝑣 ∘ 𝑣) ⊆ 𝐴 ↔ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ 𝐴)) |
170 | 169 | rspcev 3529 |
. . 3
⊢ (((◡𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹 ∧ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ 𝐴) → ∃𝑣 ∈ 𝐹 (𝑣 ∘ 𝑣) ⊆ 𝐴) |
171 | 17, 166, 170 | syl2anc 587 |
. 2
⊢
(((((𝑋 ≠ ∅
∧ 𝐷 ∈
(PsMet‘𝑋)) ∧
𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ∃𝑣 ∈ 𝐹 (𝑣 ∘ 𝑣) ⊆ 𝐴) |
172 | 9 | metustel 23316 |
. . . 4
⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐴 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎)))) |
173 | 172 | adantl 485 |
. . 3
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐴 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎)))) |
174 | 173 | biimpa 480 |
. 2
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) → ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎))) |
175 | 171, 174 | r19.29a 3200 |
1
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) → ∃𝑣 ∈ 𝐹 (𝑣 ∘ 𝑣) ⊆ 𝐴) |