Step | Hyp | Ref
| Expression |
1 | | tgqioo.1 |
. 2
⊢ 𝑄 = (topGen‘((,) “
(ℚ × ℚ))) |
2 | | iooex 9878 |
. . . 4
⊢ (,)
∈ V |
3 | 2 | imaex 4976 |
. . 3
⊢ ((,)
“ (ℚ × ℚ)) ∈ V |
4 | | imassrn 4974 |
. . 3
⊢ ((,)
“ (ℚ × ℚ)) ⊆ ran (,) |
5 | | ioof 9942 |
. . . . . 6
⊢
(,):(ℝ* × ℝ*)⟶𝒫
ℝ |
6 | | ffn 5357 |
. . . . . 6
⊢
((,):(ℝ* × ℝ*)⟶𝒫
ℝ → (,) Fn (ℝ* ×
ℝ*)) |
7 | 5, 6 | ax-mp 5 |
. . . . 5
⊢ (,) Fn
(ℝ* × ℝ*) |
8 | | simpll 527 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → 𝑥 ∈ ℝ*) |
9 | | elioo1 9882 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → (𝑧 ∈ (𝑥(,)𝑦) ↔ (𝑧 ∈ ℝ* ∧ 𝑥 < 𝑧 ∧ 𝑧 < 𝑦))) |
10 | 9 | biimpa 296 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → (𝑧 ∈ ℝ* ∧ 𝑥 < 𝑧 ∧ 𝑧 < 𝑦)) |
11 | 10 | simp1d 1009 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → 𝑧 ∈ ℝ*) |
12 | 10 | simp2d 1010 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → 𝑥 < 𝑧) |
13 | | qbtwnxr 10228 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ*
∧ 𝑧 ∈
ℝ* ∧ 𝑥
< 𝑧) → ∃𝑢 ∈ ℚ (𝑥 < 𝑢 ∧ 𝑢 < 𝑧)) |
14 | 8, 11, 12, 13 | syl3anc 1238 |
. . . . . . . . 9
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → ∃𝑢 ∈ ℚ (𝑥 < 𝑢 ∧ 𝑢 < 𝑧)) |
15 | | simplr 528 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → 𝑦 ∈ ℝ*) |
16 | 10 | simp3d 1011 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → 𝑧 < 𝑦) |
17 | | qbtwnxr 10228 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ ℝ*
∧ 𝑦 ∈
ℝ* ∧ 𝑧
< 𝑦) → ∃𝑣 ∈ ℚ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦)) |
18 | 11, 15, 16, 17 | syl3anc 1238 |
. . . . . . . . 9
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → ∃𝑣 ∈ ℚ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦)) |
19 | | reeanv 2644 |
. . . . . . . . . 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))) |
20 | | df-ov 5868 |
. . . . . . . . . . . . . 14
⊢ (𝑢(,)𝑣) = ((,)‘〈𝑢, 𝑣〉) |
21 | | opelxpi 4652 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) →
〈𝑢, 𝑣〉 ∈ (ℚ ×
ℚ)) |
22 | 21 | 3ad2ant2 1019 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 〈𝑢, 𝑣〉 ∈ (ℚ ×
ℚ)) |
23 | | ffun 5360 |
. . . . . . . . . . . . . . . . 17
⊢
((,):(ℝ* × ℝ*)⟶𝒫
ℝ → Fun (,)) |
24 | 5, 23 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ Fun
(,) |
25 | | qssre 9603 |
. . . . . . . . . . . . . . . . . . 19
⊢ ℚ
⊆ ℝ |
26 | | ressxr 7975 |
. . . . . . . . . . . . . . . . . . 19
⊢ ℝ
⊆ ℝ* |
27 | 25, 26 | sstri 3162 |
. . . . . . . . . . . . . . . . . 18
⊢ ℚ
⊆ ℝ* |
28 | | xpss12 4727 |
. . . . . . . . . . . . . . . . . 18
⊢ ((ℚ
⊆ ℝ* ∧ ℚ ⊆ ℝ*) →
(ℚ × ℚ) ⊆ (ℝ* ×
ℝ*)) |
29 | 27, 27, 28 | mp2an 426 |
. . . . . . . . . . . . . . . . 17
⊢ (ℚ
× ℚ) ⊆ (ℝ* ×
ℝ*) |
30 | 5 | fdmi 5365 |
. . . . . . . . . . . . . . . . 17
⊢ dom (,) =
(ℝ* × ℝ*) |
31 | 29, 30 | sseqtrri 3188 |
. . . . . . . . . . . . . . . 16
⊢ (ℚ
× ℚ) ⊆ dom (,) |
32 | | funfvima2 5740 |
. . . . . . . . . . . . . . . 16
⊢ ((Fun (,)
∧ (ℚ × ℚ) ⊆ dom (,)) → (〈𝑢, 𝑣〉 ∈ (ℚ × ℚ)
→ ((,)‘〈𝑢,
𝑣〉) ∈ ((,)
“ (ℚ × ℚ)))) |
33 | 24, 31, 32 | mp2an 426 |
. . . . . . . . . . . . . . 15
⊢
(〈𝑢, 𝑣〉 ∈ (ℚ ×
ℚ) → ((,)‘〈𝑢, 𝑣〉) ∈ ((,) “ (ℚ ×
ℚ))) |
34 | 22, 33 | syl 14 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → ((,)‘〈𝑢, 𝑣〉) ∈ ((,) “ (ℚ ×
ℚ))) |
35 | 20, 34 | eqeltrid 2262 |
. . . . . . . . . . . . 13
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → (𝑢(,)𝑣) ∈ ((,) “ (ℚ ×
ℚ))) |
36 | 11 | 3ad2ant1 1018 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑧 ∈ ℝ*) |
37 | | simp3lr 1069 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑢 < 𝑧) |
38 | | simp3rl 1070 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑧 < 𝑣) |
39 | | simp2l 1023 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑢 ∈ ℚ) |
40 | 27, 39 | sselid 3151 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑢 ∈ ℝ*) |
41 | | simp2r 1024 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑣 ∈ ℚ) |
42 | 27, 41 | sselid 3151 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑣 ∈ ℝ*) |
43 | | elioo1 9882 |
. . . . . . . . . . . . . . 15
⊢ ((𝑢 ∈ ℝ*
∧ 𝑣 ∈
ℝ*) → (𝑧 ∈ (𝑢(,)𝑣) ↔ (𝑧 ∈ ℝ* ∧ 𝑢 < 𝑧 ∧ 𝑧 < 𝑣))) |
44 | 40, 42, 43 | syl2anc 411 |
. . . . . . . . . . . . . 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))) |
45 | 36, 37, 38, 44 | mpbir3and 1180 |
. . . . . . . . . . . . 13
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑧 ∈ (𝑢(,)𝑣)) |
46 | 8 | 3ad2ant1 1018 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑥 ∈ ℝ*) |
47 | | simp3ll 1068 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑥 < 𝑢) |
48 | 46, 40, 47 | xrltled 9770 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑥 ≤ 𝑢) |
49 | | iooss1 9887 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ*
∧ 𝑥 ≤ 𝑢) → (𝑢(,)𝑣) ⊆ (𝑥(,)𝑣)) |
50 | 46, 48, 49 | syl2anc 411 |
. . . . . . . . . . . . . 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 | 15 | 3ad2ant1 1018 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑦 ∈ ℝ*) |
52 | | simp3rr 1071 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑣 < 𝑦) |
53 | 42, 51, 52 | xrltled 9770 |
. . . . . . . . . . . . . . 15
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → 𝑣 ≤ 𝑦) |
54 | | iooss2 9888 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ ℝ*
∧ 𝑣 ≤ 𝑦) → (𝑥(,)𝑣) ⊆ (𝑥(,)𝑦)) |
55 | 51, 53, 54 | syl2anc 411 |
. . . . . . . . . . . . . 14
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → (𝑥(,)𝑣) ⊆ (𝑥(,)𝑦)) |
56 | 50, 55 | sstrd 3163 |
. . . . . . . . . . . . 13
⊢ ((((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) ∧ (𝑢 ∈ ℚ ∧ 𝑣 ∈ ℚ) ∧ ((𝑥 < 𝑢 ∧ 𝑢 < 𝑧) ∧ (𝑧 < 𝑣 ∧ 𝑣 < 𝑦))) → (𝑢(,)𝑣) ⊆ (𝑥(,)𝑦)) |
57 | | eleq2 2239 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = (𝑢(,)𝑣) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (𝑢(,)𝑣))) |
58 | | sseq1 3176 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = (𝑢(,)𝑣) → (𝑤 ⊆ (𝑥(,)𝑦) ↔ (𝑢(,)𝑣) ⊆ (𝑥(,)𝑦))) |
59 | 57, 58 | anbi12d 473 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = (𝑢(,)𝑣) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥(,)𝑦)) ↔ (𝑧 ∈ (𝑢(,)𝑣) ∧ (𝑢(,)𝑣) ⊆ (𝑥(,)𝑦)))) |
60 | 59 | rspcev 2839 |
. . . . . . . . . . . . 13
⊢ (((𝑢(,)𝑣) ∈ ((,) “ (ℚ ×
ℚ)) ∧ (𝑧 ∈
(𝑢(,)𝑣) ∧ (𝑢(,)𝑣) ⊆ (𝑥(,)𝑦))) → ∃𝑤 ∈ ((,) “ (ℚ ×
ℚ))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥(,)𝑦))) |
61 | 35, 45, 56, 60 | syl12anc 1236 |
. . . . . . . . . . . 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))) |
62 | 61 | 3exp 1202 |
. . . . . . . . . . 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))))) |
63 | 62 | rexlimdvv 2599 |
. . . . . . . . . 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)))) |
64 | 19, 63 | syl5bir 153 |
. . . . . . . . 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)))) |
65 | 14, 18, 64 | mp2and 433 |
. . . . . . . 8
⊢ (((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) ∧ 𝑧 ∈ (𝑥(,)𝑦)) → ∃𝑤 ∈ ((,) “ (ℚ ×
ℚ))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥(,)𝑦))) |
66 | 65 | ralrimiva 2548 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → ∀𝑧 ∈ (𝑥(,)𝑦)∃𝑤 ∈ ((,) “ (ℚ ×
ℚ))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥(,)𝑦))) |
67 | | qtopbas 13593 |
. . . . . . . 8
⊢ ((,)
“ (ℚ × ℚ)) ∈ TopBases |
68 | | eltg2b 13125 |
. . . . . . . 8
⊢ (((,)
“ (ℚ × ℚ)) ∈ TopBases → ((𝑥(,)𝑦) ∈ (topGen‘((,) “ (ℚ
× ℚ))) ↔ ∀𝑧 ∈ (𝑥(,)𝑦)∃𝑤 ∈ ((,) “ (ℚ ×
ℚ))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥(,)𝑦)))) |
69 | 67, 68 | ax-mp 5 |
. . . . . . 7
⊢ ((𝑥(,)𝑦) ∈ (topGen‘((,) “ (ℚ
× ℚ))) ↔ ∀𝑧 ∈ (𝑥(,)𝑦)∃𝑤 ∈ ((,) “ (ℚ ×
ℚ))(𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥(,)𝑦))) |
70 | 66, 69 | sylibr 134 |
. . . . . 6
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 ∈
ℝ*) → (𝑥(,)𝑦) ∈ (topGen‘((,) “ (ℚ
× ℚ)))) |
71 | 70 | rgen2a 2529 |
. . . . 5
⊢
∀𝑥 ∈
ℝ* ∀𝑦 ∈ ℝ* (𝑥(,)𝑦) ∈ (topGen‘((,) “ (ℚ
× ℚ))) |
72 | | ffnov 5969 |
. . . . 5
⊢
((,):(ℝ* ×
ℝ*)⟶(topGen‘((,) “ (ℚ ×
ℚ))) ↔ ((,) Fn (ℝ* × ℝ*)
∧ ∀𝑥 ∈
ℝ* ∀𝑦 ∈ ℝ* (𝑥(,)𝑦) ∈ (topGen‘((,) “ (ℚ
× ℚ))))) |
73 | 7, 71, 72 | mpbir2an 942 |
. . . 4
⊢
(,):(ℝ* ×
ℝ*)⟶(topGen‘((,) “ (ℚ ×
ℚ))) |
74 | | frn 5366 |
. . . 4
⊢
((,):(ℝ* ×
ℝ*)⟶(topGen‘((,) “ (ℚ ×
ℚ))) → ran (,) ⊆ (topGen‘((,) “ (ℚ ×
ℚ)))) |
75 | 73, 74 | ax-mp 5 |
. . 3
⊢ ran (,)
⊆ (topGen‘((,) “ (ℚ × ℚ))) |
76 | | 2basgeng 13153 |
. . 3
⊢ ((((,)
“ (ℚ × ℚ)) ∈ V ∧ ((,) “ (ℚ ×
ℚ)) ⊆ ran (,) ∧ ran (,) ⊆ (topGen‘((,) “
(ℚ × ℚ)))) → (topGen‘((,) “ (ℚ ×
ℚ))) = (topGen‘ran (,))) |
77 | 3, 4, 75, 76 | mp3an 1337 |
. 2
⊢
(topGen‘((,) “ (ℚ × ℚ))) =
(topGen‘ran (,)) |
78 | 1, 77 | eqtr2i 2197 |
1
⊢
(topGen‘ran (,)) = 𝑄 |