Step | Hyp | Ref
| Expression |
1 | | simp-4r 774 |
. . . 4
⊢
(((((𝑋 ≠ ∅
∧ 𝐷 ∈
(PsMet‘𝑋)) ∧
𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝐷 ∈ (PsMet‘𝑋)) |
2 | | simplr 759 |
. . . . . 6
⊢
(((((𝑋 ≠ ∅
∧ 𝐷 ∈
(PsMet‘𝑋)) ∧
𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝑎 ∈ ℝ+) |
3 | 2 | rphalfcld 12197 |
. . . . 5
⊢
(((((𝑋 ≠ ∅
∧ 𝐷 ∈
(PsMet‘𝑋)) ∧
𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (𝑎 / 2) ∈
ℝ+) |
4 | | eqidd 2779 |
. . . . 5
⊢
(((((𝑋 ≠ ∅
∧ 𝐷 ∈
(PsMet‘𝑋)) ∧
𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (◡𝐷 “ (0[,)(𝑎 / 2))) = (◡𝐷 “ (0[,)(𝑎 / 2)))) |
5 | | oveq2 6932 |
. . . . . . 7
⊢ (𝑏 = (𝑎 / 2) → (0[,)𝑏) = (0[,)(𝑎 / 2))) |
6 | 5 | imaeq2d 5722 |
. . . . . 6
⊢ (𝑏 = (𝑎 / 2) → (◡𝐷 “ (0[,)𝑏)) = (◡𝐷 “ (0[,)(𝑎 / 2)))) |
7 | 6 | rspceeqv 3529 |
. . . . 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 579 |
. . . 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 6932 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏)) |
11 | 10 | imaeq2d 5722 |
. . . . . . . . 9
⊢ (𝑎 = 𝑏 → (◡𝐷 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)𝑏))) |
12 | 11 | cbvmptv 4987 |
. . . . . . . 8
⊢ (𝑎 ∈ ℝ+
↦ (◡𝐷 “ (0[,)𝑎))) = (𝑏 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑏))) |
13 | 12 | rneqi 5599 |
. . . . . . 7
⊢ ran
(𝑎 ∈
ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑏))) |
14 | 9, 13 | eqtri 2802 |
. . . . . 6
⊢ 𝐹 = ran (𝑏 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑏))) |
15 | 14 | metustel 22767 |
. . . . 5
⊢ (𝐷 ∈ (PsMet‘𝑋) → ((◡𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹 ↔ ∃𝑏 ∈ ℝ+ (◡𝐷 “ (0[,)(𝑎 / 2))) = (◡𝐷 “ (0[,)𝑏)))) |
16 | 15 | biimpar 471 |
. . . 4
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ ∃𝑏 ∈ ℝ+
(◡𝐷 “ (0[,)(𝑎 / 2))) = (◡𝐷 “ (0[,)𝑏))) → (◡𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹) |
17 | 1, 8, 16 | syl2anc 579 |
. . 3
⊢
(((((𝑋 ≠ ∅
∧ 𝐷 ∈
(PsMet‘𝑋)) ∧
𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (◡𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹) |
18 | | relco 5889 |
. . . . 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 5914 |
. . . . . . . . . 10
⊢ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (dom (◡𝐷 “ (0[,)(𝑎 / 2))) × ran (◡𝐷 “ (0[,)(𝑎 / 2)))) |
21 | | cnvimass 5741 |
. . . . . . . . . . . . . 14
⊢ (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ dom 𝐷 |
22 | | psmetf 22523 |
. . . . . . . . . . . . . 14
⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
23 | 21, 22 | fssdm 6309 |
. . . . . . . . . . . . 13
⊢ (𝐷 ∈ (PsMet‘𝑋) → (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋)) |
24 | | dmss 5570 |
. . . . . . . . . . . . . 14
⊢ ((◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋) → dom (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ dom (𝑋 × 𝑋)) |
25 | | rnss 5601 |
. . . . . . . . . . . . . 14
⊢ ((◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋) → ran (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ ran (𝑋 × 𝑋)) |
26 | | xpss12 5372 |
. . . . . . . . . . . . . 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 579 |
. . . . . . . . . . . . 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 475 |
. . . . . . . . . . 11
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (dom (◡𝐷 “ (0[,)(𝑎 / 2))) × ran (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋))) |
30 | | dmxp 5591 |
. . . . . . . . . . . . 13
⊢ (𝑋 ≠ ∅ → dom (𝑋 × 𝑋) = 𝑋) |
31 | | rnxp 5820 |
. . . . . . . . . . . . 13
⊢ (𝑋 ≠ ∅ → ran (𝑋 × 𝑋) = 𝑋) |
32 | 30, 31 | xpeq12d 5388 |
. . . . . . . . . . . 12
⊢ (𝑋 ≠ ∅ → (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)) = (𝑋 × 𝑋)) |
33 | 32 | adantr 474 |
. . . . . . . . . . 11
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)) = (𝑋 × 𝑋)) |
34 | 29, 33 | sseqtrd 3860 |
. . . . . . . . . 10
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (dom (◡𝐷 “ (0[,)(𝑎 / 2))) × ran (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (𝑋 × 𝑋)) |
35 | 20, 34 | syl5ss 3832 |
. . . . . . . . 9
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (𝑋 × 𝑋)) |
36 | 35 | ad3antrrr 720 |
. . . . . . . 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 3821 |
. . . . . . 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 5393 |
. . . . . . 7
⊢
(〈𝑝, 𝑞〉 ∈ (𝑋 × 𝑋) ↔ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) |
39 | 37, 38 | sylib 210 |
. . . . . 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 757 |
. . . . . . 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 761 |
. . . . . . 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 763 |
. . . . . . 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 759 |
. . . . . . 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 765 |
. . . . . . . . . . . . . . 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 1133 |
. . . . . . . . . . . . . 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 507 |
. . . . . . . . . . . 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 1134 |
. . . . . . . . . . . 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 1135 |
. . . . . . . . . . . 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 1119 |
. . . . . . . . . . 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 759 |
. . . . . . . . . . 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 761 |
. . . . . . . . . . 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 763 |
. . . . . . . . . . 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 757 |
. . . . . . . . . . . . . . 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 1133 |
. . . . . . . . . . . . . 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 490 |
. . . . . . . . . . . . 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 6297 |
. . . . . . . . . . . . 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 1134 |
. . . . . . . . . . . . . 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 1135 |
. . . . . . . . . . . . . 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, 38 | sylanbrc 578 |
. . . . . . . . . . . . 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 6302 |
. . . . . . . . . . . . . 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 2862 |
. . . . . . . . . . . 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 10425 |
. . . . . . . . . . . . . 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 491 |
. . . . . . . . . . . . . 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 12186 |
. . . . . . . . . . . . 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 6626 |
. . . . . . . . . . . . 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 22529 |
. . . . . . . . . . . . . . 15
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) → 0 ≤ (𝑝𝐷𝑞)) |
73 | 57, 60, 61, 72 | syl3anc 1439 |
. . . . . . . . . . . . . 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 6927 |
. . . . . . . . . . . . . 14
⊢ (𝑝𝐷𝑞) = (𝐷‘〈𝑝, 𝑞〉) |
75 | 73, 74 | syl6breq 4929 |
. . . . . . . . . . . . 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 | syl5eqel 2863 |
. . . . . . . . . . . . . . 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 10382 |
. . . . . . . . . . . . . . . . . . 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 12185 |
. . . . . . . . . . . . . . . . . . . . 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 11633 |
. . . . . . . . . . . . . . . . . . . 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 10428 |
. . . . . . . . . . . . . . . . . . 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 6927 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑝𝐷𝑟) = (𝐷‘〈𝑝, 𝑟〉) |
82 | | simplr 759 |
. . . . . . . . . . . . . . . . . . . . . . 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 | | opelxp 5393 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(〈𝑝, 𝑟〉 ∈ (𝑋 × 𝑋) ↔ (𝑝 ∈ 𝑋 ∧ 𝑟 ∈ 𝑋)) |
84 | 60, 82, 83 | sylanbrc 578 |
. . . . . . . . . . . . . . . . . . . . . 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)) |
85 | 84, 64 | eleqtrrd 2862 |
. . . . . . . . . . . . . . . . . . . . 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) |
86 | | simprl 761 |
. . . . . . . . . . . . . . . . . . . . . 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) |
87 | | df-br 4889 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ↔ 〈𝑝, 𝑟〉 ∈ (◡𝐷 “ (0[,)(𝑎 / 2)))) |
88 | 86, 87 | sylib 210 |
. . . . . . . . . . . . . . . . . . . . 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)))) |
89 | | fvimacnv 6597 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((Fun
𝐷 ∧ 〈𝑝, 𝑟〉 ∈ dom 𝐷) → ((𝐷‘〈𝑝, 𝑟〉) ∈ (0[,)(𝑎 / 2)) ↔ 〈𝑝, 𝑟〉 ∈ (◡𝐷 “ (0[,)(𝑎 / 2))))) |
90 | 89 | biimpar 471 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((Fun
𝐷 ∧ 〈𝑝, 𝑟〉 ∈ dom 𝐷) ∧ 〈𝑝, 𝑟〉 ∈ (◡𝐷 “ (0[,)(𝑎 / 2)))) → (𝐷‘〈𝑝, 𝑟〉) ∈ (0[,)(𝑎 / 2))) |
91 | 59, 85, 88, 90 | syl21anc 828 |
. . . . . . . . . . . . . . . . . . . 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))) |
92 | 81, 91 | syl5eqel 2863 |
. . . . . . . . . . . . . . . . . . 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))) |
93 | | elico2 12553 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((0
∈ ℝ ∧ (𝑎 /
2) ∈ ℝ*) → ((𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2)) ↔ ((𝑝𝐷𝑟) ∈ ℝ ∧ 0 ≤ (𝑝𝐷𝑟) ∧ (𝑝𝐷𝑟) < (𝑎 / 2)))) |
94 | 93 | biimpa 470 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((0
∈ ℝ ∧ (𝑎 /
2) ∈ ℝ*) ∧ (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) → ((𝑝𝐷𝑟) ∈ ℝ ∧ 0 ≤ (𝑝𝐷𝑟) ∧ (𝑝𝐷𝑟) < (𝑎 / 2))) |
95 | 94 | simp1d 1133 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((0
∈ ℝ ∧ (𝑎 /
2) ∈ ℝ*) ∧ (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) → (𝑝𝐷𝑟) ∈ ℝ) |
96 | 77, 80, 92, 95 | syl21anc 828 |
. . . . . . . . . . . . . . . . . 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) |
97 | | df-ov 6927 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑟𝐷𝑞) = (𝐷‘〈𝑟, 𝑞〉) |
98 | | opelxp 5393 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(〈𝑟, 𝑞〉 ∈ (𝑋 × 𝑋) ↔ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) |
99 | 82, 61, 98 | sylanbrc 578 |
. . . . . . . . . . . . . . . . . . . . . 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)) |
100 | 99, 64 | eleqtrrd 2862 |
. . . . . . . . . . . . . . . . . . . . 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) |
101 | | simprr 763 |
. . . . . . . . . . . . . . . . . . . . . 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) |
102 | | df-br 4889 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞 ↔ 〈𝑟, 𝑞〉 ∈ (◡𝐷 “ (0[,)(𝑎 / 2)))) |
103 | 101, 102 | sylib 210 |
. . . . . . . . . . . . . . . . . . . . 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)))) |
104 | | fvimacnv 6597 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((Fun
𝐷 ∧ 〈𝑟, 𝑞〉 ∈ dom 𝐷) → ((𝐷‘〈𝑟, 𝑞〉) ∈ (0[,)(𝑎 / 2)) ↔ 〈𝑟, 𝑞〉 ∈ (◡𝐷 “ (0[,)(𝑎 / 2))))) |
105 | 104 | biimpar 471 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((Fun
𝐷 ∧ 〈𝑟, 𝑞〉 ∈ dom 𝐷) ∧ 〈𝑟, 𝑞〉 ∈ (◡𝐷 “ (0[,)(𝑎 / 2)))) → (𝐷‘〈𝑟, 𝑞〉) ∈ (0[,)(𝑎 / 2))) |
106 | 59, 100, 103, 105 | syl21anc 828 |
. . . . . . . . . . . . . . . . . . . 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))) |
107 | 97, 106 | syl5eqel 2863 |
. . . . . . . . . . . . . . . . . . 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))) |
108 | | elico2 12553 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((0
∈ ℝ ∧ (𝑎 /
2) ∈ ℝ*) → ((𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2)) ↔ ((𝑟𝐷𝑞) ∈ ℝ ∧ 0 ≤ (𝑟𝐷𝑞) ∧ (𝑟𝐷𝑞) < (𝑎 / 2)))) |
109 | 108 | biimpa 470 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((0
∈ ℝ ∧ (𝑎 /
2) ∈ ℝ*) ∧ (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) → ((𝑟𝐷𝑞) ∈ ℝ ∧ 0 ≤ (𝑟𝐷𝑞) ∧ (𝑟𝐷𝑞) < (𝑎 / 2))) |
110 | 109 | simp1d 1133 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((0
∈ ℝ ∧ (𝑎 /
2) ∈ ℝ*) ∧ (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) → (𝑟𝐷𝑞) ∈ ℝ) |
111 | 77, 80, 107, 110 | syl21anc 828 |
. . . . . . . . . . . . . . . . . 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) |
112 | | rexadd 12379 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑝𝐷𝑟) ∈ ℝ ∧ (𝑟𝐷𝑞) ∈ ℝ) → ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)) = ((𝑝𝐷𝑟) + (𝑟𝐷𝑞))) |
113 | 96, 111, 112 | syl2anc 579 |
. . . . . . . . . . . . . . . . 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))) |
114 | 96, 111 | readdcld 10408 |
. . . . . . . . . . . . . . . . 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) |
115 | 113, 114 | eqeltrd 2859 |
. . . . . . . . . . . . . . . 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) |
116 | 115 | rexrd 10428 |
. . . . . . . . . . . . . . 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*) |
117 | | psmettri 22528 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑋)) → (𝑝𝐷𝑞) ≤ ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞))) |
118 | 57, 60, 61, 82, 117 | syl13anc 1440 |
. . . . . . . . . . . . . . 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))) |
119 | 94 | simp3d 1135 |
. . . . . . . . . . . . . . . . . 18
⊢ (((0
∈ ℝ ∧ (𝑎 /
2) ∈ ℝ*) ∧ (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) → (𝑝𝐷𝑟) < (𝑎 / 2)) |
120 | 77, 80, 92, 119 | syl21anc 828 |
. . . . . . . . . . . . . . . . 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)) |
121 | 109 | simp3d 1135 |
. . . . . . . . . . . . . . . . . 18
⊢ (((0
∈ ℝ ∧ (𝑎 /
2) ∈ ℝ*) ∧ (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) → (𝑟𝐷𝑞) < (𝑎 / 2)) |
122 | 77, 80, 107, 121 | syl21anc 828 |
. . . . . . . . . . . . . . . . 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)) |
123 | 96, 111, 78, 120, 122 | lt2halvesd 11634 |
. . . . . . . . . . . . . . . 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) |
124 | 113, 123 | eqbrtrd 4910 |
. . . . . . . . . . . . . . 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) |
125 | 76, 116, 69, 118, 124 | xrlelttrd 12307 |
. . . . . . . . . . . . . 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) |
126 | 74, 125 | syl5eqbrr 4924 |
. . . . . . . . . . . . 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) |
127 | | elico1 12534 |
. . . . . . . . . . . . . 14
⊢ ((0
∈ ℝ* ∧ 𝑎 ∈ ℝ*) → ((𝐷‘〈𝑝, 𝑞〉) ∈ (0[,)𝑎) ↔ ((𝐷‘〈𝑝, 𝑞〉) ∈ ℝ* ∧ 0
≤ (𝐷‘〈𝑝, 𝑞〉) ∧ (𝐷‘〈𝑝, 𝑞〉) < 𝑎))) |
128 | 127 | biimpar 471 |
. . . . . . . . . . . . 13
⊢ (((0
∈ ℝ* ∧ 𝑎 ∈ ℝ*) ∧ ((𝐷‘〈𝑝, 𝑞〉) ∈ ℝ* ∧ 0
≤ (𝐷‘〈𝑝, 𝑞〉) ∧ (𝐷‘〈𝑝, 𝑞〉) < 𝑎)) → (𝐷‘〈𝑝, 𝑞〉) ∈ (0[,)𝑎)) |
129 | 67, 69, 71, 75, 126, 128 | syl23anc 1445 |
. . . . . . . . . . . 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)) |
130 | | fvimacnv 6597 |
. . . . . . . . . . . . . 14
⊢ ((Fun
𝐷 ∧ 〈𝑝, 𝑞〉 ∈ dom 𝐷) → ((𝐷‘〈𝑝, 𝑞〉) ∈ (0[,)𝑎) ↔ 〈𝑝, 𝑞〉 ∈ (◡𝐷 “ (0[,)𝑎)))) |
131 | 130 | biimpa 470 |
. . . . . . . . . . . . 13
⊢ (((Fun
𝐷 ∧ 〈𝑝, 𝑞〉 ∈ dom 𝐷) ∧ (𝐷‘〈𝑝, 𝑞〉) ∈ (0[,)𝑎)) → 〈𝑝, 𝑞〉 ∈ (◡𝐷 “ (0[,)𝑎))) |
132 | | df-br 4889 |
. . . . . . . . . . . . 13
⊢ (𝑝(◡𝐷 “ (0[,)𝑎))𝑞 ↔ 〈𝑝, 𝑞〉 ∈ (◡𝐷 “ (0[,)𝑎))) |
133 | 131, 132 | sylibr 226 |
. . . . . . . . . . . 12
⊢ (((Fun
𝐷 ∧ 〈𝑝, 𝑞〉 ∈ dom 𝐷) ∧ (𝐷‘〈𝑝, 𝑞〉) ∈ (0[,)𝑎)) → 𝑝(◡𝐷 “ (0[,)𝑎))𝑞) |
134 | 59, 65, 129, 133 | syl21anc 828 |
. . . . . . . . . . 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) |
135 | 51, 52, 53, 54, 134 | syl22anc 829 |
. . . . . . . . . 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) |
136 | 45 | simprd 491 |
. . . . . . . . . . 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))) |
137 | 136 | breqd 4899 |
. . . . . . . . . 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)) |
138 | 135, 137 | mpbird 249 |
. . . . . . . . 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) |
139 | | simpr 479 |
. . . . . . . . . . . . 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))))) |
140 | | df-br 4889 |
. . . . . . . . . . . . 13
⊢ (𝑝((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))𝑞 ↔ 〈𝑝, 𝑞〉 ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) |
141 | 139, 140 | sylibr 226 |
. . . . . . . . . . . 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) |
142 | | vex 3401 |
. . . . . . . . . . . . 13
⊢ 𝑝 ∈ V |
143 | | vex 3401 |
. . . . . . . . . . . . 13
⊢ 𝑞 ∈ V |
144 | 142, 143 | brco 5540 |
. . . . . . . . . . . 12
⊢ (𝑝((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))𝑞 ↔ ∃𝑟(𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) |
145 | 141, 144 | sylib 210 |
. . . . . . . . . . 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)) |
146 | 23 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋)) |
147 | 146, 25 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ ran (𝑋 × 𝑋)) |
148 | 31 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑋 × 𝑋) = 𝑋) |
149 | 147, 148 | sseqtrd 3860 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ 𝑋) |
150 | 149 | adantr 474 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟) → ran (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ 𝑋) |
151 | | vex 3401 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑟 ∈ V |
152 | 142, 151 | brelrn 5604 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 → 𝑟 ∈ ran (◡𝐷 “ (0[,)(𝑎 / 2)))) |
153 | 152 | adantl 475 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟) → 𝑟 ∈ ran (◡𝐷 “ (0[,)(𝑎 / 2)))) |
154 | 150, 153 | sseldd 3822 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟) → 𝑟 ∈ 𝑋) |
155 | 154 | adantrr 707 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑟 ∈ 𝑋) |
156 | 155 | ex 403 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞) → 𝑟 ∈ 𝑋)) |
157 | 156 | ancrd 547 |
. . . . . . . . . . . . . . 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)))) |
158 | 157 | eximdv 1960 |
. . . . . . . . . . . . . 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)))) |
159 | 158 | ad3antrrr 720 |
. . . . . . . . . . . . 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)))) |
160 | 159 | 3ad2ant1 1124 |
. . . . . . . . . . . 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)))) |
161 | 160 | adantr 474 |
. . . . . . . . . . 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)))) |
162 | 145, 161 | 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))) |
163 | | df-rex 3096 |
. . . . . . . . . 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))) |
164 | 162, 163 | sylibr 226 |
. . . . . . . . 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)) |
165 | 138, 164 | r19.29a 3264 |
. . . . . . . 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) |
166 | | df-br 4889 |
. . . . . . . 8
⊢ (𝑝𝐴𝑞 ↔ 〈𝑝, 𝑞〉 ∈ 𝐴) |
167 | 165, 166 | sylib 210 |
. . . . . . 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) |
168 | 40, 41, 42, 43, 167 | syl31anc 1441 |
. . . . . 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) |
169 | 39, 168 | mpdan 677 |
. . . . 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) |
170 | 169 | ex 403 |
. . . 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)) |
171 | 19, 170 | relssdv 5461 |
. . 3
⊢
(((((𝑋 ≠ ∅
∧ 𝐷 ∈
(PsMet‘𝑋)) ∧
𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ 𝐴) |
172 | | id 22 |
. . . . . 6
⊢ (𝑣 = (◡𝐷 “ (0[,)(𝑎 / 2))) → 𝑣 = (◡𝐷 “ (0[,)(𝑎 / 2)))) |
173 | 172, 172 | coeq12d 5534 |
. . . . 5
⊢ (𝑣 = (◡𝐷 “ (0[,)(𝑎 / 2))) → (𝑣 ∘ 𝑣) = ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) |
174 | 173 | sseq1d 3851 |
. . . 4
⊢ (𝑣 = (◡𝐷 “ (0[,)(𝑎 / 2))) → ((𝑣 ∘ 𝑣) ⊆ 𝐴 ↔ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ 𝐴)) |
175 | 174 | rspcev 3511 |
. . 3
⊢ (((◡𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹 ∧ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ 𝐴) → ∃𝑣 ∈ 𝐹 (𝑣 ∘ 𝑣) ⊆ 𝐴) |
176 | 17, 171, 175 | syl2anc 579 |
. 2
⊢
(((((𝑋 ≠ ∅
∧ 𝐷 ∈
(PsMet‘𝑋)) ∧
𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ∃𝑣 ∈ 𝐹 (𝑣 ∘ 𝑣) ⊆ 𝐴) |
177 | 9 | metustel 22767 |
. . . 4
⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐴 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎)))) |
178 | 177 | adantl 475 |
. . 3
⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐴 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎)))) |
179 | 178 | biimpa 470 |
. 2
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) → ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎))) |
180 | 176, 179 | r19.29a 3264 |
1
⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) → ∃𝑣 ∈ 𝐹 (𝑣 ∘ 𝑣) ⊆ 𝐴) |