| Step | Hyp | Ref
| Expression |
| 1 | | reex 11246 |
. . . 4
⊢ ℝ
∈ V |
| 2 | | elssuni 4937 |
. . . . 5
⊢ (𝐴 ∈ (topGen‘ran (,))
→ 𝐴 ⊆ ∪ (topGen‘ran (,))) |
| 3 | | uniretop 24783 |
. . . . 5
⊢ ℝ =
∪ (topGen‘ran (,)) |
| 4 | 2, 3 | sseqtrrdi 4025 |
. . . 4
⊢ (𝐴 ∈ (topGen‘ran (,))
→ 𝐴 ⊆
ℝ) |
| 5 | | ssdomg 9040 |
. . . 4
⊢ (ℝ
∈ V → (𝐴 ⊆
ℝ → 𝐴 ≼
ℝ)) |
| 6 | 1, 4, 5 | mpsyl 68 |
. . 3
⊢ (𝐴 ∈ (topGen‘ran (,))
→ 𝐴 ≼
ℝ) |
| 7 | | rpnnen 16263 |
. . 3
⊢ ℝ
≈ 𝒫 ℕ |
| 8 | | domentr 9053 |
. . 3
⊢ ((𝐴 ≼ ℝ ∧ ℝ
≈ 𝒫 ℕ) → 𝐴 ≼ 𝒫 ℕ) |
| 9 | 6, 7, 8 | sylancl 586 |
. 2
⊢ (𝐴 ∈ (topGen‘ran (,))
→ 𝐴 ≼ 𝒫
ℕ) |
| 10 | | n0 4353 |
. . . 4
⊢ (𝐴 ≠ ∅ ↔
∃𝑥 𝑥 ∈ 𝐴) |
| 11 | 4 | sselda 3983 |
. . . . . . . . 9
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ) |
| 12 | | rpnnen2 16262 |
. . . . . . . . . . . 12
⊢ 𝒫
ℕ ≼ (0[,]1) |
| 13 | | rphalfcl 13062 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ ℝ+
→ (𝑦 / 2) ∈
ℝ+) |
| 14 | 13 | rpred 13077 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ ℝ+
→ (𝑦 / 2) ∈
ℝ) |
| 15 | | resubcl 11573 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ (𝑦 / 2) ∈ ℝ) →
(𝑥 − (𝑦 / 2)) ∈
ℝ) |
| 16 | 14, 15 | sylan2 593 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 − (𝑦 / 2)) ∈
ℝ) |
| 17 | | readdcl 11238 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ (𝑦 / 2) ∈ ℝ) →
(𝑥 + (𝑦 / 2)) ∈ ℝ) |
| 18 | 14, 17 | sylan2 593 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 + (𝑦 / 2)) ∈
ℝ) |
| 19 | | simpl 482 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ 𝑥 ∈
ℝ) |
| 20 | | ltsubrp 13071 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ (𝑦 / 2) ∈
ℝ+) → (𝑥 − (𝑦 / 2)) < 𝑥) |
| 21 | 13, 20 | sylan2 593 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 − (𝑦 / 2)) < 𝑥) |
| 22 | | ltaddrp 13072 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ (𝑦 / 2) ∈
ℝ+) → 𝑥 < (𝑥 + (𝑦 / 2))) |
| 23 | 13, 22 | sylan2 593 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ 𝑥 < (𝑥 + (𝑦 / 2))) |
| 24 | 16, 19, 18, 21, 23 | lttrd 11422 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 − (𝑦 / 2)) < (𝑥 + (𝑦 / 2))) |
| 25 | | iccen 13537 |
. . . . . . . . . . . . 13
⊢ (((𝑥 − (𝑦 / 2)) ∈ ℝ ∧ (𝑥 + (𝑦 / 2)) ∈ ℝ ∧ (𝑥 − (𝑦 / 2)) < (𝑥 + (𝑦 / 2))) → (0[,]1) ≈ ((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2)))) |
| 26 | 16, 18, 24, 25 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (0[,]1) ≈ ((𝑥
− (𝑦 / 2))[,](𝑥 + (𝑦 / 2)))) |
| 27 | | domentr 9053 |
. . . . . . . . . . . 12
⊢
((𝒫 ℕ ≼ (0[,]1) ∧ (0[,]1) ≈ ((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2)))) → 𝒫 ℕ ≼
((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2)))) |
| 28 | 12, 26, 27 | sylancr 587 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ 𝒫 ℕ ≼ ((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2)))) |
| 29 | | ovex 7464 |
. . . . . . . . . . . 12
⊢ ((𝑥 − 𝑦)(,)(𝑥 + 𝑦)) ∈ V |
| 30 | | rpre 13043 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ ℝ+
→ 𝑦 ∈
ℝ) |
| 31 | | resubcl 11573 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 − 𝑦) ∈ ℝ) |
| 32 | 30, 31 | sylan2 593 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 − 𝑦) ∈
ℝ) |
| 33 | 32 | rexrd 11311 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 − 𝑦) ∈
ℝ*) |
| 34 | | readdcl 11238 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ) |
| 35 | 30, 34 | sylan2 593 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 + 𝑦) ∈
ℝ) |
| 36 | 35 | rexrd 11311 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 + 𝑦) ∈
ℝ*) |
| 37 | 19 | recnd 11289 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ 𝑥 ∈
ℂ) |
| 38 | 14 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑦 / 2) ∈
ℝ) |
| 39 | 38 | recnd 11289 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑦 / 2) ∈
ℂ) |
| 40 | 37, 39, 39 | subsub4d 11651 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ ((𝑥 − (𝑦 / 2)) − (𝑦 / 2)) = (𝑥 − ((𝑦 / 2) + (𝑦 / 2)))) |
| 41 | 30 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ 𝑦 ∈
ℝ) |
| 42 | 41 | recnd 11289 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ 𝑦 ∈
ℂ) |
| 43 | 42 | 2halvesd 12512 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ ((𝑦 / 2) + (𝑦 / 2)) = 𝑦) |
| 44 | 43 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 − ((𝑦 / 2) + (𝑦 / 2))) = (𝑥 − 𝑦)) |
| 45 | 40, 44 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ ((𝑥 − (𝑦 / 2)) − (𝑦 / 2)) = (𝑥 − 𝑦)) |
| 46 | 13 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑦 / 2) ∈
ℝ+) |
| 47 | 16, 46 | ltsubrpd 13109 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ ((𝑥 − (𝑦 / 2)) − (𝑦 / 2)) < (𝑥 − (𝑦 / 2))) |
| 48 | 45, 47 | eqbrtrrd 5167 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 − 𝑦) < (𝑥 − (𝑦 / 2))) |
| 49 | 18, 46 | ltaddrpd 13110 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 + (𝑦 / 2)) < ((𝑥 + (𝑦 / 2)) + (𝑦 / 2))) |
| 50 | 37, 39, 39 | addassd 11283 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ ((𝑥 + (𝑦 / 2)) + (𝑦 / 2)) = (𝑥 + ((𝑦 / 2) + (𝑦 / 2)))) |
| 51 | 43 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 + ((𝑦 / 2) + (𝑦 / 2))) = (𝑥 + 𝑦)) |
| 52 | 50, 51 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ ((𝑥 + (𝑦 / 2)) + (𝑦 / 2)) = (𝑥 + 𝑦)) |
| 53 | 49, 52 | breqtrd 5169 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥 + (𝑦 / 2)) < (𝑥 + 𝑦)) |
| 54 | | iccssioo 13456 |
. . . . . . . . . . . . 13
⊢ ((((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))) |
| 55 | 33, 36, 48, 53, 54 | syl22anc 839 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ ((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2))) ⊆ ((𝑥 − 𝑦)(,)(𝑥 + 𝑦))) |
| 56 | | ssdomg 9040 |
. . . . . . . . . . . 12
⊢ (((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)))) |
| 57 | 29, 55, 56 | mpsyl 68 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ ((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2))) ≼ ((𝑥 − 𝑦)(,)(𝑥 + 𝑦))) |
| 58 | | domtr 9047 |
. . . . . . . . . . 11
⊢
((𝒫 ℕ ≼ ((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2))) ∧ ((𝑥 − (𝑦 / 2))[,](𝑥 + (𝑦 / 2))) ≼ ((𝑥 − 𝑦)(,)(𝑥 + 𝑦))) → 𝒫 ℕ ≼ ((𝑥 − 𝑦)(,)(𝑥 + 𝑦))) |
| 59 | 28, 57, 58 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ 𝒫 ℕ ≼ ((𝑥 − 𝑦)(,)(𝑥 + 𝑦))) |
| 60 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ ((abs
∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − )
↾ (ℝ × ℝ)) |
| 61 | 60 | bl2ioo 24813 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥(ball‘((abs ∘
− ) ↾ (ℝ × ℝ)))𝑦) = ((𝑥 − 𝑦)(,)(𝑥 + 𝑦))) |
| 62 | 30, 61 | sylan2 593 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ (𝑥(ball‘((abs
∘ − ) ↾ (ℝ × ℝ)))𝑦) = ((𝑥 − 𝑦)(,)(𝑥 + 𝑦))) |
| 63 | 59, 62 | breqtrrd 5171 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ+)
→ 𝒫 ℕ ≼ (𝑥(ball‘((abs ∘ − ) ↾
(ℝ × ℝ)))𝑦)) |
| 64 | 11, 63 | sylan 580 |
. . . . . . . 8
⊢ (((𝐴 ∈ (topGen‘ran (,))
∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) → 𝒫
ℕ ≼ (𝑥(ball‘((abs ∘ − ) ↾
(ℝ × ℝ)))𝑦)) |
| 65 | | simplll 775 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ (topGen‘ran (,))
∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) ∧ (𝑥(ball‘((abs ∘
− ) ↾ (ℝ × ℝ)))𝑦) ⊆ 𝐴) → 𝐴 ∈ (topGen‘ran
(,))) |
| 66 | | simpr 484 |
. . . . . . . . 9
⊢ ((((𝐴 ∈ (topGen‘ran (,))
∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) ∧ (𝑥(ball‘((abs ∘
− ) ↾ (ℝ × ℝ)))𝑦) ⊆ 𝐴) → (𝑥(ball‘((abs ∘ − ) ↾
(ℝ × ℝ)))𝑦) ⊆ 𝐴) |
| 67 | | ssdomg 9040 |
. . . . . . . . 9
⊢ (𝐴 ∈ (topGen‘ran (,))
→ ((𝑥(ball‘((abs
∘ − ) ↾ (ℝ × ℝ)))𝑦) ⊆ 𝐴 → (𝑥(ball‘((abs ∘ − ) ↾
(ℝ × ℝ)))𝑦) ≼ 𝐴)) |
| 68 | 65, 66, 67 | sylc 65 |
. . . . . . . 8
⊢ ((((𝐴 ∈ (topGen‘ran (,))
∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) ∧ (𝑥(ball‘((abs ∘
− ) ↾ (ℝ × ℝ)))𝑦) ⊆ 𝐴) → (𝑥(ball‘((abs ∘ − ) ↾
(ℝ × ℝ)))𝑦) ≼ 𝐴) |
| 69 | | domtr 9047 |
. . . . . . . 8
⊢
((𝒫 ℕ ≼ (𝑥(ball‘((abs ∘ − ) ↾
(ℝ × ℝ)))𝑦) ∧ (𝑥(ball‘((abs ∘ − ) ↾
(ℝ × ℝ)))𝑦) ≼ 𝐴) → 𝒫 ℕ ≼ 𝐴) |
| 70 | 64, 68, 69 | syl2an2r 685 |
. . . . . . 7
⊢ ((((𝐴 ∈ (topGen‘ran (,))
∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ+) ∧ (𝑥(ball‘((abs ∘
− ) ↾ (ℝ × ℝ)))𝑦) ⊆ 𝐴) → 𝒫 ℕ ≼ 𝐴) |
| 71 | | eqid 2737 |
. . . . . . . . . 10
⊢
(MetOpen‘((abs ∘ − ) ↾ (ℝ ×
ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ ×
ℝ))) |
| 72 | 60, 71 | tgioo 24817 |
. . . . . . . . 9
⊢
(topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾
(ℝ × ℝ))) |
| 73 | 72 | eleq2i 2833 |
. . . . . . . 8
⊢ (𝐴 ∈ (topGen‘ran (,))
↔ 𝐴 ∈
(MetOpen‘((abs ∘ − ) ↾ (ℝ ×
ℝ)))) |
| 74 | 60 | rexmet 24812 |
. . . . . . . . 9
⊢ ((abs
∘ − ) ↾ (ℝ × ℝ)) ∈
(∞Met‘ℝ) |
| 75 | 71 | mopni2 24506 |
. . . . . . . . 9
⊢ ((((abs
∘ − ) ↾ (ℝ × ℝ)) ∈
(∞Met‘ℝ) ∧ 𝐴 ∈ (MetOpen‘((abs ∘ −
) ↾ (ℝ × ℝ))) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ+ (𝑥(ball‘((abs ∘
− ) ↾ (ℝ × ℝ)))𝑦) ⊆ 𝐴) |
| 76 | 74, 75 | mp3an1 1450 |
. . . . . . . 8
⊢ ((𝐴 ∈ (MetOpen‘((abs
∘ − ) ↾ (ℝ × ℝ))) ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ+ (𝑥(ball‘((abs ∘
− ) ↾ (ℝ × ℝ)))𝑦) ⊆ 𝐴) |
| 77 | 73, 76 | sylanb 581 |
. . . . . . 7
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ+
(𝑥(ball‘((abs ∘
− ) ↾ (ℝ × ℝ)))𝑦) ⊆ 𝐴) |
| 78 | 70, 77 | r19.29a 3162 |
. . . . . 6
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝑥 ∈ 𝐴) → 𝒫 ℕ
≼ 𝐴) |
| 79 | 78 | ex 412 |
. . . . 5
⊢ (𝐴 ∈ (topGen‘ran (,))
→ (𝑥 ∈ 𝐴 → 𝒫 ℕ
≼ 𝐴)) |
| 80 | 79 | exlimdv 1933 |
. . . 4
⊢ (𝐴 ∈ (topGen‘ran (,))
→ (∃𝑥 𝑥 ∈ 𝐴 → 𝒫 ℕ ≼ 𝐴)) |
| 81 | 10, 80 | biimtrid 242 |
. . 3
⊢ (𝐴 ∈ (topGen‘ran (,))
→ (𝐴 ≠ ∅
→ 𝒫 ℕ ≼ 𝐴)) |
| 82 | 81 | imp 406 |
. 2
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅)
→ 𝒫 ℕ ≼ 𝐴) |
| 83 | | sbth 9133 |
. 2
⊢ ((𝐴 ≼ 𝒫 ℕ ∧
𝒫 ℕ ≼ 𝐴) → 𝐴 ≈ 𝒫 ℕ) |
| 84 | 9, 82, 83 | syl2an2r 685 |
1
⊢ ((𝐴 ∈ (topGen‘ran (,))
∧ 𝐴 ≠ ∅)
→ 𝐴 ≈ 𝒫
ℕ) |