Step | Hyp | Ref
| Expression |
1 | | tgqioo.1 |
. 2
⊢ 𝑄 = (topGen‘((,) “
(ℚ × ℚ))) |
2 | | imassrn 5477 |
. . 3
⊢ ((,)
“ (ℚ × ℚ)) ⊆ ran (,) |
3 | | ioof 12271 |
. . . . . 6
⊢
(,):(ℝ* × ℝ*)⟶𝒫
ℝ |
4 | | ffn 6045 |
. . . . . 6
⊢
((,):(ℝ* × ℝ*)⟶𝒫
ℝ → (,) Fn (ℝ* ×
ℝ*)) |
5 | 3, 4 | ax-mp 5 |
. . . . 5
⊢ (,) Fn
(ℝ* × ℝ*) |
6 | | simpll 790 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → 𝑥 ∈ ℝ*) |
7 | | elioo1 12215 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → (𝑧 ∈ (𝑥(,)𝑦) ↔ (𝑧 ∈ ℝ* ∧ 𝑥 < 𝑧 ∧ 𝑧 < 𝑦))) |
8 | 7 | biimpa 501 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → (𝑧 ∈ ℝ* ∧ 𝑥 < 𝑧 ∧ 𝑧 < 𝑦)) |
9 | 8 | simp1d 1073 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → 𝑧 ∈ ℝ*) |
10 | 8 | simp2d 1074 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → 𝑥 < 𝑧) |
11 | | qbtwnxr 12031 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ*
∧ 𝑧 ∈
ℝ* ∧ 𝑥
< 𝑧) → ∃𝑢 ∈ ℚ (𝑥 < 𝑢 ∧ 𝑢 < 𝑧)) |
12 | 6, 9, 10, 11 | syl3anc 1326 |
. . . . . . . . 9
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → ∃𝑢 ∈ ℚ (𝑥 < 𝑢 ∧ 𝑢 < 𝑧)) |
13 | | simplr 792 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → 𝑦 ∈ ℝ*) |
14 | 8 | simp3d 1075 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → 𝑧 < 𝑦) |
15 | | qbtwnxr 12031 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ ℝ*
∧ 𝑦 ∈
ℝ* ∧ 𝑧
< 𝑦) → ∃𝑣 ∈ ℚ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦)) |
16 | 9, 13, 14, 15 | syl3anc 1326 |
. . . . . . . . 9
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → ∃𝑣 ∈ ℚ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦)) |
17 | | reeanv 3107 |
. . . . . . . . . 10
⊢
(∃𝑢 ∈
ℚ ∃𝑣 ∈
ℚ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦)) ↔ (∃𝑢 ∈ ℚ (𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ ∃𝑣 ∈ ℚ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) |
18 | | df-ov 6653 |
. . . . . . . . . . . . . 14
⊢ (𝑢(,)𝑣) = ((,)‘〈𝑢, 𝑣〉) |
19 | | opelxpi 5148 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) →
〈𝑢, 𝑣〉 ∈ (ℚ ×
ℚ)) |
20 | 19 | 3ad2ant2 1083 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 〈𝑢, 𝑣〉 ∈ (ℚ ×
ℚ)) |
21 | | ffun 6048 |
. . . . . . . . . . . . . . . . 17
⊢
((,):(ℝ* × ℝ*)⟶𝒫
ℝ → Fun (,)) |
22 | 3, 21 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ Fun
(,) |
23 | | qssre 11798 |
. . . . . . . . . . . . . . . . . . 19
⊢ ℚ
⊆ ℝ |
24 | | ressxr 10083 |
. . . . . . . . . . . . . . . . . . 19
⊢ ℝ
⊆ ℝ* |
25 | 23, 24 | sstri 3612 |
. . . . . . . . . . . . . . . . . 18
⊢ ℚ
⊆ ℝ* |
26 | | xpss12 5225 |
. . . . . . . . . . . . . . . . . 18
⊢ ((ℚ
⊆ ℝ* ∧ ℚ ⊆ ℝ*) →
(ℚ × ℚ) ⊆ (ℝ* ×
ℝ*)) |
27 | 25, 25, 26 | mp2an 708 |
. . . . . . . . . . . . . . . . 17
⊢ (ℚ
× ℚ) ⊆ (ℝ* ×
ℝ*) |
28 | 3 | fdmi 6052 |
. . . . . . . . . . . . . . . . 17
⊢ dom (,) =
(ℝ* × ℝ*) |
29 | 27, 28 | sseqtr4i 3638 |
. . . . . . . . . . . . . . . 16
⊢ (ℚ
× ℚ) ⊆ dom (,) |
30 | | funfvima2 6493 |
. . . . . . . . . . . . . . . 16
⊢ ((Fun (,)
∧ (ℚ × ℚ) ⊆ dom (,)) → (〈𝑢, 𝑣〉 ∈ (ℚ × ℚ)
→ ((,)‘〈𝑢,
𝑣〉) ∈ ((,)
“ (ℚ × ℚ)))) |
31 | 22, 29, 30 | mp2an 708 |
. . . . . . . . . . . . . . 15
⊢
(〈𝑢, 𝑣〉 ∈ (ℚ ×
ℚ) → ((,)‘〈𝑢, 𝑣〉) ∈ ((,) “ (ℚ ×
ℚ))) |
32 | 20, 31 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → ((,)‘〈𝑢, 𝑣〉) ∈ ((,) “ (ℚ ×
ℚ))) |
33 | 18, 32 | syl5eqel 2705 |
. . . . . . . . . . . . 13
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → (𝑢(,)𝑣) ∈ ((,) “ (ℚ ×
ℚ))) |
34 | 9 | 3ad2ant1 1082 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑧 ∈ ℝ*) |
35 | | simp3lr 1133 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑢 < 𝑧) |
36 | | simp3rl 1134 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑧 < 𝑣) |
37 | | simp2l 1087 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑢 ∈ ℚ) |
38 | 25, 37 | sseldi 3601 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑢 ∈ ℝ*) |
39 | | simp2r 1088 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑣 ∈ ℚ) |
40 | 25, 39 | sseldi 3601 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑣 ∈ ℝ*) |
41 | | elioo1 12215 |
. . . . . . . . . . . . . . 15
⊢ ((𝑢 ∈ ℝ*
∧ 𝑣 ∈
ℝ*) → (𝑧 ∈ (𝑢(,)𝑣) ↔ (𝑧 ∈ ℝ* ∧ 𝑢 < 𝑧 ∧ 𝑧 < 𝑣))) |
42 | 38, 40, 41 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → (𝑧 ∈ (𝑢(,)𝑣) ↔ (𝑧 ∈ ℝ* ∧ 𝑢 < 𝑧 ∧ 𝑧 < 𝑣))) |
43 | 34, 35, 36, 42 | mpbir3and 1245 |
. . . . . . . . . . . . 13
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑧 ∈ (𝑢(,)𝑣)) |
44 | 6 | 3ad2ant1 1082 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑥 ∈ ℝ*) |
45 | | simp3ll 1132 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑥 < 𝑢) |
46 | | xrltle 11982 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℝ*
∧ 𝑢 ∈
ℝ*) → (𝑥 < 𝑢 → 𝑥 ≤ 𝑢)) |
47 | 44, 38, 46 | syl2anc 693 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → (𝑥 < 𝑢 → 𝑥 ≤ 𝑢)) |
48 | 45, 47 | mpd 15 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑥 ≤ 𝑢) |
49 | | iooss1 12210 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ*
∧ 𝑥 ≤ 𝑢) → (𝑢(,)𝑣) ⊆ (𝑥(,)𝑣)) |
50 | 44, 48, 49 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → (𝑢(,)𝑣) ⊆ (𝑥(,)𝑣)) |
51 | 13 | 3ad2ant1 1082 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑦 ∈ ℝ*) |
52 | | simp3rr 1135 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑣 < 𝑦) |
53 | | xrltle 11982 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑣 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → (𝑣 < 𝑦 → 𝑣 ≤ 𝑦)) |
54 | 40, 51, 53 | syl2anc 693 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → (𝑣 < 𝑦 → 𝑣 ≤ 𝑦)) |
55 | 52, 54 | mpd 15 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑣 ≤ 𝑦) |
56 | | iooss2 12211 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ ℝ*
∧ 𝑣 ≤ 𝑦) → (𝑥(,)𝑣) ⊆ (𝑥(,)𝑦)) |
57 | 51, 55, 56 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → (𝑥(,)𝑣) ⊆ (𝑥(,)𝑦)) |
58 | 50, 57 | sstrd 3613 |
. . . . . . . . . . . . 13
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → (𝑢(,)𝑣) ⊆ (𝑥(,)𝑦)) |
59 | | eleq2 2690 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = (𝑢(,)𝑣) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝑢(,)𝑣))) |
60 | | sseq1 3626 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = (𝑢(,)𝑣) → (𝑤 ⊆ (𝑥(,)𝑦) ↔ (𝑢(,)𝑣) ⊆ (𝑥(,)𝑦))) |
61 | 59, 60 | anbi12d 747 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = (𝑢(,)𝑣) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥(,)𝑦)) ↔ (𝑧 ∈ (𝑢(,)𝑣) ∧ (𝑢(,)𝑣) ⊆ (𝑥(,)𝑦)))) |
62 | 61 | rspcev 3309 |
. . . . . . . . . . . . 13
⊢ (((𝑢(,)𝑣) ∈ ((,) “ (ℚ ×
ℚ)) ∧ (𝑧 ∈
(𝑢(,)𝑣) ∧ (𝑢(,)𝑣) ⊆ (𝑥(,)𝑦))) → ∃𝑤 ∈ ((,) “ (ℚ ×
ℚ))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥(,)𝑦))) |
63 | 33, 43, 58, 62 | syl12anc 1324 |
. . . . . . . . . . . 12
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → ∃𝑤 ∈ ((,) “ (ℚ ×
ℚ))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥(,)𝑦))) |
64 | 63 | 3exp 1264 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → ((𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) → (((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦)) → ∃𝑤 ∈ ((,) “ (ℚ ×
ℚ))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥(,)𝑦))))) |
65 | 64 | rexlimdvv 3037 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → (∃𝑢 ∈ ℚ ∃𝑣 ∈ ℚ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦)) → ∃𝑤 ∈ ((,) “ (ℚ ×
ℚ))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥(,)𝑦)))) |
66 | 17, 65 | syl5bir 233 |
. . . . . . . . 9
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → ((∃𝑢 ∈ ℚ (𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ ∃𝑣 ∈ ℚ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦)) → ∃𝑤 ∈ ((,) “ (ℚ ×
ℚ))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥(,)𝑦)))) |
67 | 12, 16, 66 | mp2and 715 |
. . . . . . . 8
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → ∃𝑤 ∈ ((,) “ (ℚ ×
ℚ))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥(,)𝑦))) |
68 | 67 | ralrimiva 2966 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → ∀𝑧 ∈ (𝑥(,)𝑦)∃𝑤 ∈ ((,) “ (ℚ ×
ℚ))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥(,)𝑦))) |
69 | | qtopbas 22563 |
. . . . . . . 8
⊢ ((,)
“ (ℚ × ℚ)) ∈ TopBases |
70 | | eltg2b 20763 |
. . . . . . . 8
⊢ (((,)
“ (ℚ × ℚ)) ∈ TopBases → ((𝑥(,)𝑦) ∈ (topGen‘((,) “ (ℚ
× ℚ))) ↔ ∀𝑧 ∈ (𝑥(,)𝑦)∃𝑤 ∈ ((,) “ (ℚ ×
ℚ))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥(,)𝑦)))) |
71 | 69, 70 | ax-mp 5 |
. . . . . . 7
⊢ ((𝑥(,)𝑦) ∈ (topGen‘((,) “ (ℚ
× ℚ))) ↔ ∀𝑧 ∈ (𝑥(,)𝑦)∃𝑤 ∈ ((,) “ (ℚ ×
ℚ))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥(,)𝑦))) |
72 | 68, 71 | sylibr 224 |
. . . . . 6
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → (𝑥(,)𝑦) ∈ (topGen‘((,) “ (ℚ
× ℚ)))) |
73 | 72 | rgen2a 2977 |
. . . . 5
⊢
∀𝑥 ∈
ℝ* ∀𝑦 ∈ ℝ* (𝑥(,)𝑦) ∈ (topGen‘((,) “ (ℚ
× ℚ))) |
74 | | ffnov 6764 |
. . . . 5
⊢
((,):(ℝ* ×
ℝ*)⟶(topGen‘((,) “ (ℚ ×
ℚ))) ↔ ((,) Fn (ℝ* × ℝ*)
∧ ∀𝑥 ∈
ℝ* ∀𝑦 ∈ ℝ* (𝑥(,)𝑦) ∈ (topGen‘((,) “ (ℚ
× ℚ))))) |
75 | 5, 73, 74 | mpbir2an 955 |
. . . 4
⊢
(,):(ℝ* ×
ℝ*)⟶(topGen‘((,) “ (ℚ ×
ℚ))) |
76 | | frn 6053 |
. . . 4
⊢
((,):(ℝ* ×
ℝ*)⟶(topGen‘((,) “ (ℚ ×
ℚ))) → ran (,) ⊆ (topGen‘((,) “ (ℚ ×
ℚ)))) |
77 | 75, 76 | ax-mp 5 |
. . 3
⊢ ran (,)
⊆ (topGen‘((,) “ (ℚ × ℚ))) |
78 | | 2basgen 20794 |
. . 3
⊢ ((((,)
“ (ℚ × ℚ)) ⊆ ran (,) ∧ ran (,) ⊆
(topGen‘((,) “ (ℚ × ℚ)))) →
(topGen‘((,) “ (ℚ × ℚ))) = (topGen‘ran
(,))) |
79 | 2, 77, 78 | mp2an 708 |
. 2
⊢
(topGen‘((,) “ (ℚ × ℚ))) =
(topGen‘ran (,)) |
80 | 1, 79 | eqtr2i 2645 |
1
⊢
(topGen‘ran (,)) = 𝑄 |