Step | Hyp | Ref
| Expression |
1 | | reex 9906 |
. . . . 5
⊢ ℝ
∈ V |
2 | | elssuni 4403 |
. . . . . 6
⊢ (𝐴 ∈ (topGen‘ran (,))
→ 𝐴 ⊆ ∪ (topGen‘ran (,))) |
3 | | uniretop 22376 |
. . . . . 6
⊢ ℝ =
∪ (topGen‘ran (,)) |
4 | 2, 3 | syl6sseqr 3615 |
. . . . 5
⊢ (𝐴 ∈ (topGen‘ran (,))
→ 𝐴 ⊆
ℝ) |
5 | | ssdomg 7887 |
. . . . 5
⊢ (ℝ
∈ V → (𝐴 ⊆
ℝ → 𝐴 ≼
ℝ)) |
6 | 1, 4, 5 | mpsyl 66 |
. . . 4
⊢ (𝐴 ∈ (topGen‘ran (,))
→ 𝐴 ≼
ℝ) |
7 | | rpnnen 14795 |
. . . 4
⊢ ℝ
≈ 𝒫 ℕ |
8 | | domentr 7901 |
. . . 4
⊢ ((𝐴 ≼ ℝ ∧ ℝ
≈ 𝒫 ℕ) → 𝐴 ≼ 𝒫 ℕ) |
9 | 6, 7, 8 | sylancl 693 |
. . 3
⊢ (𝐴 ∈ (topGen‘ran (,))
→ 𝐴 ≼ 𝒫
ℕ) |
10 | 9 | adantr 480 |
. 2
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅)
→ 𝐴 ≼ 𝒫
ℕ) |
11 | | n0 3890 |
. . . 4
⊢ (𝐴 ≠ ∅ ↔
∃𝑥 𝑥 ∈ 𝐴) |
12 | 4 | sselda 3568 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ) |
13 | | rpnnen2 14794 |
. . . . . . . . . . . . 13
⊢ 𝒫
ℕ ≼ (0[,]1) |
14 | | rphalfcl 11734 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ ℝ+
→ (𝑦 / 2) ∈
ℝ+) |
15 | 14 | rpred 11748 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ ℝ+
→ (𝑦 / 2) ∈
ℝ) |
16 | | resubcl 10224 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ (𝑦 / 2) ∈ ℝ) →
(𝑥 − (𝑦 / 2)) ∈
ℝ) |
17 | 15, 16 | sylan2 490 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 − (𝑦 / 2)) ∈
ℝ) |
18 | | readdcl 9898 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ (𝑦 / 2) ∈ ℝ) →
(𝑥 + (𝑦 / 2)) ∈ ℝ) |
19 | 15, 18 | sylan2 490 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 + (𝑦 / 2)) ∈
ℝ) |
20 | | simpl 472 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ 𝑥 ∈
ℝ) |
21 | | ltsubrp 11742 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ ∧ (𝑦 / 2) ∈
ℝ+) → (𝑥 − (𝑦 / 2)) < 𝑥) |
22 | 14, 21 | sylan2 490 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 − (𝑦 / 2)) < 𝑥) |
23 | | ltaddrp 11743 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ ∧ (𝑦 / 2) ∈
ℝ+) → 𝑥 < (𝑥 + (𝑦 / 2))) |
24 | 14, 23 | sylan2 490 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ 𝑥 < (𝑥 + (𝑦 / 2))) |
25 | 17, 20, 19, 22, 24 | lttrd 10077 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 − (𝑦 / 2)) < (𝑥 + (𝑦 / 2))) |
26 | | iccen 12188 |
. . . . . . . . . . . . . 14
⊢ (((𝑥 − (𝑦 / 2)) ∈ ℝ ∧ (𝑥 + (𝑦 / 2)) ∈ ℝ ∧ (𝑥 − (𝑦 / 2)) < (𝑥 + (𝑦 / 2))) → (0[,]1) ≈ ((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2)))) |
27 | 17, 19, 25, 26 | syl3anc 1318 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (0[,]1) ≈ ((𝑥
− (𝑦 / 2))[,](𝑥 + (𝑦 / 2)))) |
28 | | domentr 7901 |
. . . . . . . . . . . . 13
⊢
((𝒫 ℕ ≼ (0[,]1) ∧ (0[,]1) ≈ ((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2)))) → 𝒫 ℕ ≼
((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2)))) |
29 | 13, 27, 28 | sylancr 694 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ 𝒫 ℕ ≼ ((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2)))) |
30 | | ovex 6577 |
. . . . . . . . . . . . 13
⊢ ((𝑥 − 𝑦)(,)(𝑥 + 𝑦)) ∈ V |
31 | | rpre 11715 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ ℝ+
→ 𝑦 ∈
ℝ) |
32 | | resubcl 10224 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 − 𝑦) ∈ ℝ) |
33 | 31, 32 | sylan2 490 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 − 𝑦) ∈
ℝ) |
34 | 33 | rexrd 9968 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 − 𝑦) ∈
ℝ*) |
35 | | readdcl 9898 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ) |
36 | 31, 35 | sylan2 490 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 + 𝑦) ∈
ℝ) |
37 | 36 | rexrd 9968 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 + 𝑦) ∈
ℝ*) |
38 | 20 | recnd 9947 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ 𝑥 ∈
ℂ) |
39 | 15 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑦 / 2) ∈
ℝ) |
40 | 39 | recnd 9947 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑦 / 2) ∈
ℂ) |
41 | 38, 40, 40 | subsub4d 10302 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ ((𝑥 − (𝑦 / 2)) − (𝑦 / 2)) = (𝑥 − ((𝑦 / 2) + (𝑦 / 2)))) |
42 | 31 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ 𝑦 ∈
ℝ) |
43 | 42 | recnd 9947 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ 𝑦 ∈
ℂ) |
44 | 43 | 2halvesd 11155 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ ((𝑦 / 2) + (𝑦 / 2)) = 𝑦) |
45 | 44 | oveq2d 6565 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 − ((𝑦 / 2) + (𝑦 / 2))) = (𝑥 − 𝑦)) |
46 | 41, 45 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ ((𝑥 − (𝑦 / 2)) − (𝑦 / 2)) = (𝑥 − 𝑦)) |
47 | 14 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑦 / 2) ∈
ℝ+) |
48 | 17, 47 | ltsubrpd 11780 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ ((𝑥 − (𝑦 / 2)) − (𝑦 / 2)) < (𝑥 − (𝑦 / 2))) |
49 | 46, 48 | eqbrtrrd 4607 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 − 𝑦) < (𝑥 − (𝑦 / 2))) |
50 | | ltaddrp 11743 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 + (𝑦 / 2)) ∈ ℝ ∧ (𝑦 / 2) ∈
ℝ+) → (𝑥 + (𝑦 / 2)) < ((𝑥 + (𝑦 / 2)) + (𝑦 / 2))) |
51 | 19, 47, 50 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 + (𝑦 / 2)) < ((𝑥 + (𝑦 / 2)) + (𝑦 / 2))) |
52 | 38, 40, 40 | addassd 9941 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ ((𝑥 + (𝑦 / 2)) + (𝑦 / 2)) = (𝑥 + ((𝑦 / 2) + (𝑦 / 2)))) |
53 | 44 | oveq2d 6565 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 + ((𝑦 / 2) + (𝑦 / 2))) = (𝑥 + 𝑦)) |
54 | 52, 53 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ ((𝑥 + (𝑦 / 2)) + (𝑦 / 2)) = (𝑥 + 𝑦)) |
55 | 51, 54 | breqtrd 4609 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 + (𝑦 / 2)) < (𝑥 + 𝑦)) |
56 | | iccssioo 12113 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 − 𝑦) ∈ ℝ* ∧ (𝑥 + 𝑦) ∈ ℝ*) ∧ ((𝑥 − 𝑦) < (𝑥 − (𝑦 / 2)) ∧ (𝑥 + (𝑦 / 2)) < (𝑥 + 𝑦))) → ((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2))) ⊆ ((𝑥 − 𝑦)(,)(𝑥 + 𝑦))) |
57 | 34, 37, 49, 55, 56 | syl22anc 1319 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ ((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2))) ⊆ ((𝑥 − 𝑦)(,)(𝑥 + 𝑦))) |
58 | | ssdomg 7887 |
. . . . . . . . . . . . 13
⊢ (((𝑥 − 𝑦)(,)(𝑥 + 𝑦)) ∈ V → (((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2))) ⊆ ((𝑥 − 𝑦)(,)(𝑥 + 𝑦)) → ((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2))) ≼ ((𝑥 − 𝑦)(,)(𝑥 + 𝑦)))) |
59 | 30, 57, 58 | mpsyl 66 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ ((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2))) ≼ ((𝑥 − 𝑦)(,)(𝑥 + 𝑦))) |
60 | | domtr 7895 |
. . . . . . . . . . . 12
⊢
((𝒫 ℕ ≼ ((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2))) ∧ ((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2))) ≼ ((𝑥 − 𝑦)(,)(𝑥 + 𝑦))) → 𝒫 ℕ ≼ ((𝑥 − 𝑦)(,)(𝑥 + 𝑦))) |
61 | 29, 59, 60 | syl2anc 691 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ 𝒫 ℕ ≼ ((𝑥 − 𝑦)(,)(𝑥 + 𝑦))) |
62 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢ ((abs
∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − )
↾ (ℝ × ℝ)) |
63 | 62 | bl2ioo 22403 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥(ball‘((abs ∘
− ) ↾ (ℝ × ℝ)))𝑦) = ((𝑥 − 𝑦)(,)(𝑥 + 𝑦))) |
64 | 31, 63 | sylan2 490 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥(ball‘((abs
∘ − ) ↾ (ℝ × ℝ)))𝑦) = ((𝑥 − 𝑦)(,)(𝑥 + 𝑦))) |
65 | 61, 64 | breqtrrd 4611 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ 𝒫 ℕ ≼ (𝑥(ball‘((abs ∘ − ) ↾
(ℝ × ℝ)))𝑦)) |
66 | 12, 65 | sylan 487 |
. . . . . . . . 9
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) → 𝒫
ℕ ≼ (𝑥(ball‘((abs ∘ − ) ↾
(ℝ × ℝ)))𝑦)) |
67 | 66 | adantr 480 |
. . . . . . . 8
⊢ ((((𝐴 ∈ (topGen‘ran (,))
∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) ∧ (𝑥(ball‘((abs ∘
− ) ↾ (ℝ × ℝ)))𝑦) ⊆ 𝐴) → 𝒫 ℕ ≼ (𝑥(ball‘((abs ∘
− ) ↾ (ℝ × ℝ)))𝑦)) |
68 | | simplll 794 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ (topGen‘ran (,))
∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) ∧ (𝑥(ball‘((abs ∘
− ) ↾ (ℝ × ℝ)))𝑦) ⊆ 𝐴) → 𝐴 ∈ (topGen‘ran
(,))) |
69 | | simpr 476 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ (topGen‘ran (,))
∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) ∧ (𝑥(ball‘((abs ∘
− ) ↾ (ℝ × ℝ)))𝑦) ⊆ 𝐴) → (𝑥(ball‘((abs ∘ − ) ↾
(ℝ × ℝ)))𝑦) ⊆ 𝐴) |
70 | | ssdomg 7887 |
. . . . . . . . 9
⊢ (𝐴 ∈ (topGen‘ran (,))
→ ((𝑥(ball‘((abs
∘ − ) ↾ (ℝ × ℝ)))𝑦) ⊆ 𝐴 → (𝑥(ball‘((abs ∘ − ) ↾
(ℝ × ℝ)))𝑦) ≼ 𝐴)) |
71 | 68, 69, 70 | sylc 63 |
. . . . . . . 8
⊢ ((((𝐴 ∈ (topGen‘ran (,))
∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) ∧ (𝑥(ball‘((abs ∘
− ) ↾ (ℝ × ℝ)))𝑦) ⊆ 𝐴) → (𝑥(ball‘((abs ∘ − ) ↾
(ℝ × ℝ)))𝑦) ≼ 𝐴) |
72 | | domtr 7895 |
. . . . . . . 8
⊢
((𝒫 ℕ ≼ (𝑥(ball‘((abs ∘ − ) ↾
(ℝ × ℝ)))𝑦) ∧ (𝑥(ball‘((abs ∘ − ) ↾
(ℝ × ℝ)))𝑦) ≼ 𝐴) → 𝒫 ℕ ≼ 𝐴) |
73 | 67, 71, 72 | syl2anc 691 |
. . . . . . 7
⊢ ((((𝐴 ∈ (topGen‘ran (,))
∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) ∧ (𝑥(ball‘((abs ∘
− ) ↾ (ℝ × ℝ)))𝑦) ⊆ 𝐴) → 𝒫 ℕ ≼ 𝐴) |
74 | | eqid 2610 |
. . . . . . . . . 10
⊢
(MetOpen‘((abs ∘ − ) ↾ (ℝ ×
ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ ×
ℝ))) |
75 | 62, 74 | tgioo 22407 |
. . . . . . . . 9
⊢
(topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾
(ℝ × ℝ))) |
76 | 75 | eleq2i 2680 |
. . . . . . . 8
⊢ (𝐴 ∈ (topGen‘ran (,))
↔ 𝐴 ∈
(MetOpen‘((abs ∘ − ) ↾ (ℝ ×
ℝ)))) |
77 | 62 | rexmet 22402 |
. . . . . . . . 9
⊢ ((abs
∘ − ) ↾ (ℝ × ℝ)) ∈
(∞Met‘ℝ) |
78 | 74 | mopni2 22108 |
. . . . . . . . 9
⊢ ((((abs
∘ − ) ↾ (ℝ × ℝ)) ∈
(∞Met‘ℝ) ∧ 𝐴 ∈ (MetOpen‘((abs ∘ −
) ↾ (ℝ × ℝ))) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ+ (𝑥(ball‘((abs ∘
− ) ↾ (ℝ × ℝ)))𝑦) ⊆ 𝐴) |
79 | 77, 78 | mp3an1 1403 |
. . . . . . . 8
⊢ ((𝐴 ∈ (MetOpen‘((abs
∘ − ) ↾ (ℝ × ℝ))) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ+ (𝑥(ball‘((abs ∘
− ) ↾ (ℝ × ℝ)))𝑦) ⊆ 𝐴) |
80 | 76, 79 | sylanb 488 |
. . . . . . 7
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ+
(𝑥(ball‘((abs ∘
− ) ↾ (ℝ × ℝ)))𝑦) ⊆ 𝐴) |
81 | 73, 80 | r19.29a 3060 |
. . . . . 6
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝑥 ∈ 𝐴) → 𝒫 ℕ
≼ 𝐴) |
82 | 81 | ex 449 |
. . . . 5
⊢ (𝐴 ∈ (topGen‘ran (,))
→ (𝑥 ∈ 𝐴 → 𝒫 ℕ
≼ 𝐴)) |
83 | 82 | exlimdv 1848 |
. . . 4
⊢ (𝐴 ∈ (topGen‘ran (,))
→ (∃𝑥 𝑥 ∈ 𝐴 → 𝒫 ℕ ≼ 𝐴)) |
84 | 11, 83 | syl5bi 231 |
. . 3
⊢ (𝐴 ∈ (topGen‘ran (,))
→ (𝐴 ≠ ∅
→ 𝒫 ℕ ≼ 𝐴)) |
85 | 84 | imp 444 |
. 2
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅)
→ 𝒫 ℕ ≼ 𝐴) |
86 | | sbth 7965 |
. 2
⊢ ((𝐴 ≼ 𝒫 ℕ ∧
𝒫 ℕ ≼ 𝐴) → 𝐴 ≈ 𝒫 ℕ) |
87 | 10, 85, 86 | syl2anc 691 |
1
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅)
→ 𝐴 ≈ 𝒫
ℕ) |