Step | Hyp | Ref
| Expression |
1 | | simp1 1054 |
. . 3
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝐷 ∈ (∞Met‘𝑋)) |
2 | | xmetresbl.1 |
. . . 4
⊢ 𝐵 = (𝑃(ball‘𝐷)𝑅) |
3 | | blssm 22033 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋) |
4 | 2, 3 | syl5eqss 3612 |
. . 3
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝐵 ⊆ 𝑋) |
5 | | xmetres2 21976 |
. . 3
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ⊆ 𝑋) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵)) |
6 | 1, 4, 5 | syl2anc 691 |
. 2
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵)) |
7 | | xmetf 21944 |
. . . . . 6
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
8 | 1, 7 | syl 17 |
. . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝐷:(𝑋 × 𝑋)⟶ℝ*) |
9 | | xpss12 5148 |
. . . . . 6
⊢ ((𝐵 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → (𝐵 × 𝐵) ⊆ (𝑋 × 𝑋)) |
10 | 4, 4, 9 | syl2anc 691 |
. . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐵 × 𝐵) ⊆ (𝑋 × 𝑋)) |
11 | 8, 10 | fssresd 5984 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ*) |
12 | | ffn 5958 |
. . . 4
⊢ ((𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ* → (𝐷 ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) |
13 | 11, 12 | syl 17 |
. . 3
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵)) |
14 | | ovres 6698 |
. . . . . 6
⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) = (𝑥𝐷𝑦)) |
15 | 14 | adantl 481 |
. . . . 5
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) = (𝑥𝐷𝑦)) |
16 | | simpl1 1057 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐷 ∈ (∞Met‘𝑋)) |
17 | | eqid 2610 |
. . . . . . . . . 10
⊢ (◡𝐷 “ ℝ) = (◡𝐷 “ ℝ) |
18 | 17 | xmeter 22048 |
. . . . . . . . 9
⊢ (𝐷 ∈ (∞Met‘𝑋) → (◡𝐷 “ ℝ) Er 𝑋) |
19 | 16, 18 | syl 17 |
. . . . . . . 8
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (◡𝐷 “ ℝ) Er 𝑋) |
20 | 17 | blssec 22050 |
. . . . . . . . . . . 12
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ [𝑃](◡𝐷 “ ℝ)) |
21 | 2, 20 | syl5eqss 3612 |
. . . . . . . . . . 11
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → 𝐵 ⊆ [𝑃](◡𝐷 “ ℝ)) |
22 | 21 | sselda 3568 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ [𝑃](◡𝐷 “ ℝ)) |
23 | 22 | adantrr 749 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ [𝑃](◡𝐷 “ ℝ)) |
24 | | simpl2 1058 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃 ∈ 𝑋) |
25 | | elecg 7672 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ [𝑃](◡𝐷 “ ℝ) ∧ 𝑃 ∈ 𝑋) → (𝑥 ∈ [𝑃](◡𝐷 “ ℝ) ↔ 𝑃(◡𝐷 “ ℝ)𝑥)) |
26 | 23, 24, 25 | syl2anc 691 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∈ [𝑃](◡𝐷 “ ℝ) ↔ 𝑃(◡𝐷 “ ℝ)𝑥)) |
27 | 23, 26 | mpbid 221 |
. . . . . . . 8
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃(◡𝐷 “ ℝ)𝑥) |
28 | 21 | sselda 3568 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ [𝑃](◡𝐷 “ ℝ)) |
29 | 28 | adantrl 748 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ [𝑃](◡𝐷 “ ℝ)) |
30 | | elecg 7672 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ [𝑃](◡𝐷 “ ℝ) ∧ 𝑃 ∈ 𝑋) → (𝑦 ∈ [𝑃](◡𝐷 “ ℝ) ↔ 𝑃(◡𝐷 “ ℝ)𝑦)) |
31 | 29, 24, 30 | syl2anc 691 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 ∈ [𝑃](◡𝐷 “ ℝ) ↔ 𝑃(◡𝐷 “ ℝ)𝑦)) |
32 | 29, 31 | mpbid 221 |
. . . . . . . 8
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑃(◡𝐷 “ ℝ)𝑦) |
33 | 19, 27, 32 | ertr3d 7647 |
. . . . . . 7
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥(◡𝐷 “ ℝ)𝑦) |
34 | 17 | xmeterval 22047 |
. . . . . . . 8
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑥(◡𝐷 “ ℝ)𝑦 ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝑥𝐷𝑦) ∈ ℝ))) |
35 | 16, 34 | syl 17 |
. . . . . . 7
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(◡𝐷 “ ℝ)𝑦 ↔ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝑥𝐷𝑦) ∈ ℝ))) |
36 | 33, 35 | mpbid 221 |
. . . . . 6
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (𝑥𝐷𝑦) ∈ ℝ)) |
37 | 36 | simp3d 1068 |
. . . . 5
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐷𝑦) ∈ ℝ) |
38 | 15, 37 | eqeltrd 2688 |
. . . 4
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ) |
39 | 38 | ralrimivva 2954 |
. . 3
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) →
∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ) |
40 | | ffnov 6662 |
. . 3
⊢ ((𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ ↔ ((𝐷 ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(𝐷 ↾ (𝐵 × 𝐵))𝑦) ∈ ℝ)) |
41 | 13, 39, 40 | sylanbrc 695 |
. 2
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ) |
42 | | ismet2 21948 |
. 2
⊢ ((𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵) ↔ ((𝐷 ↾ (𝐵 × 𝐵)) ∈ (∞Met‘𝐵) ∧ (𝐷 ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶ℝ)) |
43 | 6, 41, 42 | sylanbrc 695 |
1
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵)) |