Step |
Hyp |
Ref |
Expression |
1 |
|
simprl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) → 𝑥 ∈ ( topGen ‘ ran (,) ) ) |
2 |
|
inss1 |
⊢ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ 𝑥 |
3 |
|
simprr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) → 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) |
4 |
2 3
|
sselid |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) → 𝑦 ∈ 𝑥 ) |
5 |
|
tg2 |
⊢ ( ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ 𝑥 ) → ∃ 𝑧 ∈ ran (,) ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) ) |
6 |
1 4 5
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) → ∃ 𝑧 ∈ ran (,) ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) ) |
7 |
|
ioof |
⊢ (,) : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ |
8 |
|
ffn |
⊢ ( (,) : ( ℝ* × ℝ* ) ⟶ 𝒫 ℝ → (,) Fn ( ℝ* × ℝ* ) ) |
9 |
|
ovelrn |
⊢ ( (,) Fn ( ℝ* × ℝ* ) → ( 𝑧 ∈ ran (,) ↔ ∃ 𝑎 ∈ ℝ* ∃ 𝑏 ∈ ℝ* 𝑧 = ( 𝑎 (,) 𝑏 ) ) ) |
10 |
7 8 9
|
mp2b |
⊢ ( 𝑧 ∈ ran (,) ↔ ∃ 𝑎 ∈ ℝ* ∃ 𝑏 ∈ ℝ* 𝑧 = ( 𝑎 (,) 𝑏 ) ) |
11 |
|
inss1 |
⊢ ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ 𝑧 |
12 |
|
simprrr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) ∧ ( 𝑧 = ( 𝑎 (,) 𝑏 ) ∧ ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) ) ) → 𝑧 ⊆ 𝑥 ) |
13 |
11 12
|
sstrid |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) ∧ ( 𝑧 = ( 𝑎 (,) 𝑏 ) ∧ ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) ) ) → ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ 𝑥 ) |
14 |
|
simprrl |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) ∧ ( 𝑧 = ( 𝑎 (,) 𝑏 ) ∧ ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) ) ) → 𝑦 ∈ 𝑧 ) |
15 |
|
simprl |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) ∧ ( 𝑧 = ( 𝑎 (,) 𝑏 ) ∧ ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) ) ) → 𝑧 = ( 𝑎 (,) 𝑏 ) ) |
16 |
15
|
ineq1d |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) ∧ ( 𝑧 = ( 𝑎 (,) 𝑏 ) ∧ ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) ) ) → ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ) |
17 |
16
|
oveq2d |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) ∧ ( 𝑧 = ( 𝑎 (,) 𝑏 ) ∧ ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) ) ) → ( ( topGen ‘ ran (,) ) ↾t ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ) = ( ( topGen ‘ ran (,) ) ↾t ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ) ) |
18 |
|
ioosconn |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( 𝑎 (,) 𝑏 ) ) ∈ SConn |
19 |
|
ioossre |
⊢ ( 𝑎 (,) 𝑏 ) ⊆ ℝ |
20 |
|
eqid |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( 𝑎 (,) 𝑏 ) ) = ( ( topGen ‘ ran (,) ) ↾t ( 𝑎 (,) 𝑏 ) ) |
21 |
20
|
resconn |
⊢ ( ( 𝑎 (,) 𝑏 ) ⊆ ℝ → ( ( ( topGen ‘ ran (,) ) ↾t ( 𝑎 (,) 𝑏 ) ) ∈ SConn ↔ ( ( topGen ‘ ran (,) ) ↾t ( 𝑎 (,) 𝑏 ) ) ∈ Conn ) ) |
22 |
|
reconn |
⊢ ( ( 𝑎 (,) 𝑏 ) ⊆ ℝ → ( ( ( topGen ‘ ran (,) ) ↾t ( 𝑎 (,) 𝑏 ) ) ∈ Conn ↔ ∀ 𝑢 ∈ ( 𝑎 (,) 𝑏 ) ∀ 𝑣 ∈ ( 𝑎 (,) 𝑏 ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) ) ) |
23 |
21 22
|
bitrd |
⊢ ( ( 𝑎 (,) 𝑏 ) ⊆ ℝ → ( ( ( topGen ‘ ran (,) ) ↾t ( 𝑎 (,) 𝑏 ) ) ∈ SConn ↔ ∀ 𝑢 ∈ ( 𝑎 (,) 𝑏 ) ∀ 𝑣 ∈ ( 𝑎 (,) 𝑏 ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) ) ) |
24 |
19 23
|
ax-mp |
⊢ ( ( ( topGen ‘ ran (,) ) ↾t ( 𝑎 (,) 𝑏 ) ) ∈ SConn ↔ ∀ 𝑢 ∈ ( 𝑎 (,) 𝑏 ) ∀ 𝑣 ∈ ( 𝑎 (,) 𝑏 ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) ) |
25 |
18 24
|
mpbi |
⊢ ∀ 𝑢 ∈ ( 𝑎 (,) 𝑏 ) ∀ 𝑣 ∈ ( 𝑎 (,) 𝑏 ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) |
26 |
|
inss1 |
⊢ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ ( 𝑎 (,) 𝑏 ) |
27 |
|
ssralv |
⊢ ( ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ ( 𝑎 (,) 𝑏 ) → ( ∀ 𝑣 ∈ ( 𝑎 (,) 𝑏 ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) → ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) ) ) |
28 |
27
|
ralimdv |
⊢ ( ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ ( 𝑎 (,) 𝑏 ) → ( ∀ 𝑢 ∈ ( 𝑎 (,) 𝑏 ) ∀ 𝑣 ∈ ( 𝑎 (,) 𝑏 ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) → ∀ 𝑢 ∈ ( 𝑎 (,) 𝑏 ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) ) ) |
29 |
|
ssralv |
⊢ ( ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ ( 𝑎 (,) 𝑏 ) → ( ∀ 𝑢 ∈ ( 𝑎 (,) 𝑏 ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) → ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) ) ) |
30 |
28 29
|
syld |
⊢ ( ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ ( 𝑎 (,) 𝑏 ) → ( ∀ 𝑢 ∈ ( 𝑎 (,) 𝑏 ) ∀ 𝑣 ∈ ( 𝑎 (,) 𝑏 ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) → ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) ) ) |
31 |
26 30
|
ax-mp |
⊢ ( ∀ 𝑢 ∈ ( 𝑎 (,) 𝑏 ) ∀ 𝑣 ∈ ( 𝑎 (,) 𝑏 ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) → ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) ) |
32 |
25 31
|
mp1i |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) ∧ ( 𝑧 = ( 𝑎 (,) 𝑏 ) ∧ ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) ) ) → ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) ) |
33 |
|
inss2 |
⊢ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ ( 𝐴 [,] 𝐵 ) |
34 |
|
iccconn |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( topGen ‘ ran (,) ) ↾t ( 𝐴 [,] 𝐵 ) ) ∈ Conn ) |
35 |
|
iccssre |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
36 |
|
reconn |
⊢ ( ( 𝐴 [,] 𝐵 ) ⊆ ℝ → ( ( ( topGen ‘ ran (,) ) ↾t ( 𝐴 [,] 𝐵 ) ) ∈ Conn ↔ ∀ 𝑢 ∈ ( 𝐴 [,] 𝐵 ) ∀ 𝑣 ∈ ( 𝐴 [,] 𝐵 ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝐴 [,] 𝐵 ) ) ) |
37 |
35 36
|
syl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( ( topGen ‘ ran (,) ) ↾t ( 𝐴 [,] 𝐵 ) ) ∈ Conn ↔ ∀ 𝑢 ∈ ( 𝐴 [,] 𝐵 ) ∀ 𝑣 ∈ ( 𝐴 [,] 𝐵 ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝐴 [,] 𝐵 ) ) ) |
38 |
34 37
|
mpbid |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ∀ 𝑢 ∈ ( 𝐴 [,] 𝐵 ) ∀ 𝑣 ∈ ( 𝐴 [,] 𝐵 ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝐴 [,] 𝐵 ) ) |
39 |
38
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) ∧ ( 𝑧 = ( 𝑎 (,) 𝑏 ) ∧ ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) ) ) → ∀ 𝑢 ∈ ( 𝐴 [,] 𝐵 ) ∀ 𝑣 ∈ ( 𝐴 [,] 𝐵 ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝐴 [,] 𝐵 ) ) |
40 |
|
ssralv |
⊢ ( ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ ( 𝐴 [,] 𝐵 ) → ( ∀ 𝑣 ∈ ( 𝐴 [,] 𝐵 ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝐴 [,] 𝐵 ) → ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝐴 [,] 𝐵 ) ) ) |
41 |
40
|
ralimdv |
⊢ ( ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ ( 𝐴 [,] 𝐵 ) → ( ∀ 𝑢 ∈ ( 𝐴 [,] 𝐵 ) ∀ 𝑣 ∈ ( 𝐴 [,] 𝐵 ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝐴 [,] 𝐵 ) → ∀ 𝑢 ∈ ( 𝐴 [,] 𝐵 ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝐴 [,] 𝐵 ) ) ) |
42 |
|
ssralv |
⊢ ( ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ ( 𝐴 [,] 𝐵 ) → ( ∀ 𝑢 ∈ ( 𝐴 [,] 𝐵 ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝐴 [,] 𝐵 ) → ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝐴 [,] 𝐵 ) ) ) |
43 |
41 42
|
syld |
⊢ ( ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ ( 𝐴 [,] 𝐵 ) → ( ∀ 𝑢 ∈ ( 𝐴 [,] 𝐵 ) ∀ 𝑣 ∈ ( 𝐴 [,] 𝐵 ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝐴 [,] 𝐵 ) → ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝐴 [,] 𝐵 ) ) ) |
44 |
33 39 43
|
mpsyl |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) ∧ ( 𝑧 = ( 𝑎 (,) 𝑏 ) ∧ ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) ) ) → ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝐴 [,] 𝐵 ) ) |
45 |
|
ssin |
⊢ ( ( ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) ∧ ( 𝑢 [,] 𝑣 ) ⊆ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝑢 [,] 𝑣 ) ⊆ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ) |
46 |
45
|
2ralbii |
⊢ ( ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) ∧ ( 𝑢 [,] 𝑣 ) ⊆ ( 𝐴 [,] 𝐵 ) ) ↔ ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ) |
47 |
|
r19.26-2 |
⊢ ( ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) ∧ ( 𝑢 [,] 𝑣 ) ⊆ ( 𝐴 [,] 𝐵 ) ) ↔ ( ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) ∧ ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝐴 [,] 𝐵 ) ) ) |
48 |
46 47
|
bitr3i |
⊢ ( ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ↔ ( ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝑎 (,) 𝑏 ) ∧ ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( 𝐴 [,] 𝐵 ) ) ) |
49 |
32 44 48
|
sylanbrc |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) ∧ ( 𝑧 = ( 𝑎 (,) 𝑏 ) ∧ ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) ) ) → ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ) |
50 |
26 19
|
sstri |
⊢ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ ℝ |
51 |
|
eqid |
⊢ ( ( topGen ‘ ran (,) ) ↾t ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ) = ( ( topGen ‘ ran (,) ) ↾t ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ) |
52 |
51
|
resconn |
⊢ ( ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ ℝ → ( ( ( topGen ‘ ran (,) ) ↾t ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ↔ ( ( topGen ‘ ran (,) ) ↾t ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ Conn ) ) |
53 |
|
reconn |
⊢ ( ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ ℝ → ( ( ( topGen ‘ ran (,) ) ↾t ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ Conn ↔ ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ) ) |
54 |
52 53
|
bitrd |
⊢ ( ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ ℝ → ( ( ( topGen ‘ ran (,) ) ↾t ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ↔ ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ) ) |
55 |
50 54
|
ax-mp |
⊢ ( ( ( topGen ‘ ran (,) ) ↾t ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ↔ ∀ 𝑢 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ∀ 𝑣 ∈ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ( 𝑢 [,] 𝑣 ) ⊆ ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ) |
56 |
49 55
|
sylibr |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) ∧ ( 𝑧 = ( 𝑎 (,) 𝑏 ) ∧ ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) ) ) → ( ( topGen ‘ ran (,) ) ↾t ( ( 𝑎 (,) 𝑏 ) ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ) |
57 |
17 56
|
eqeltrd |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) ∧ ( 𝑧 = ( 𝑎 (,) 𝑏 ) ∧ ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) ) ) → ( ( topGen ‘ ran (,) ) ↾t ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ) |
58 |
13 14 57
|
3jca |
⊢ ( ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) ∧ ( 𝑧 = ( 𝑎 (,) 𝑏 ) ∧ ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) ) ) → ( ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑧 ∧ ( ( topGen ‘ ran (,) ) ↾t ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ) ) |
59 |
58
|
exp32 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) → ( 𝑧 = ( 𝑎 (,) 𝑏 ) → ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) → ( ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑧 ∧ ( ( topGen ‘ ran (,) ) ↾t ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ) ) ) ) |
60 |
59
|
rexlimdvw |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) → ( ∃ 𝑏 ∈ ℝ* 𝑧 = ( 𝑎 (,) 𝑏 ) → ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) → ( ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑧 ∧ ( ( topGen ‘ ran (,) ) ↾t ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ) ) ) ) |
61 |
60
|
rexlimdvw |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) → ( ∃ 𝑎 ∈ ℝ* ∃ 𝑏 ∈ ℝ* 𝑧 = ( 𝑎 (,) 𝑏 ) → ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) → ( ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑧 ∧ ( ( topGen ‘ ran (,) ) ↾t ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ) ) ) ) |
62 |
10 61
|
syl5bi |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) → ( 𝑧 ∈ ran (,) → ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) → ( ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑧 ∧ ( ( topGen ‘ ran (,) ) ↾t ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ) ) ) ) |
63 |
62
|
reximdvai |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) → ( ∃ 𝑧 ∈ ran (,) ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) → ∃ 𝑧 ∈ ran (,) ( ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑧 ∧ ( ( topGen ‘ ran (,) ) ↾t ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ) ) ) |
64 |
|
retopbas |
⊢ ran (,) ∈ TopBases |
65 |
|
bastg |
⊢ ( ran (,) ∈ TopBases → ran (,) ⊆ ( topGen ‘ ran (,) ) ) |
66 |
|
ssrexv |
⊢ ( ran (,) ⊆ ( topGen ‘ ran (,) ) → ( ∃ 𝑧 ∈ ran (,) ( ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑧 ∧ ( ( topGen ‘ ran (,) ) ↾t ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ) → ∃ 𝑧 ∈ ( topGen ‘ ran (,) ) ( ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑧 ∧ ( ( topGen ‘ ran (,) ) ↾t ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ) ) ) |
67 |
64 65 66
|
mp2b |
⊢ ( ∃ 𝑧 ∈ ran (,) ( ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑧 ∧ ( ( topGen ‘ ran (,) ) ↾t ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ) → ∃ 𝑧 ∈ ( topGen ‘ ran (,) ) ( ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑧 ∧ ( ( topGen ‘ ran (,) ) ↾t ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ) ) |
68 |
63 67
|
syl6 |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) → ( ∃ 𝑧 ∈ ran (,) ( 𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥 ) → ∃ 𝑧 ∈ ( topGen ‘ ran (,) ) ( ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑧 ∧ ( ( topGen ‘ ran (,) ) ↾t ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ) ) ) |
69 |
6 68
|
mpd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) ∧ ( 𝑥 ∈ ( topGen ‘ ran (,) ) ∧ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ) ) → ∃ 𝑧 ∈ ( topGen ‘ ran (,) ) ( ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑧 ∧ ( ( topGen ‘ ran (,) ) ↾t ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ) ) |
70 |
69
|
ralrimivva |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ∀ 𝑥 ∈ ( topGen ‘ ran (,) ) ∀ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ∃ 𝑧 ∈ ( topGen ‘ ran (,) ) ( ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑧 ∧ ( ( topGen ‘ ran (,) ) ↾t ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ) ) |
71 |
|
retop |
⊢ ( topGen ‘ ran (,) ) ∈ Top |
72 |
|
ovex |
⊢ ( 𝐴 [,] 𝐵 ) ∈ V |
73 |
|
subislly |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ Top ∧ ( 𝐴 [,] 𝐵 ) ∈ V ) → ( ( ( topGen ‘ ran (,) ) ↾t ( 𝐴 [,] 𝐵 ) ) ∈ Locally SConn ↔ ∀ 𝑥 ∈ ( topGen ‘ ran (,) ) ∀ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ∃ 𝑧 ∈ ( topGen ‘ ran (,) ) ( ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑧 ∧ ( ( topGen ‘ ran (,) ) ↾t ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ) ) ) |
74 |
71 72 73
|
mp2an |
⊢ ( ( ( topGen ‘ ran (,) ) ↾t ( 𝐴 [,] 𝐵 ) ) ∈ Locally SConn ↔ ∀ 𝑥 ∈ ( topGen ‘ ran (,) ) ∀ 𝑦 ∈ ( 𝑥 ∩ ( 𝐴 [,] 𝐵 ) ) ∃ 𝑧 ∈ ( topGen ‘ ran (,) ) ( ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑧 ∧ ( ( topGen ‘ ran (,) ) ↾t ( 𝑧 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∈ SConn ) ) |
75 |
70 74
|
sylibr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( topGen ‘ ran (,) ) ↾t ( 𝐴 [,] 𝐵 ) ) ∈ Locally SConn ) |