Step |
Hyp |
Ref |
Expression |
1 |
|
xrge0iifhmeo.1 |
⊢ 𝐹 = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 = 0 , +∞ , - ( log ‘ 𝑥 ) ) ) |
2 |
|
xrge0iifhmeo.k |
⊢ 𝐽 = ( ( ordTop ‘ ≤ ) ↾t ( 0 [,] +∞ ) ) |
3 |
|
letsr |
⊢ ≤ ∈ TosetRel |
4 |
|
tsrps |
⊢ ( ≤ ∈ TosetRel → ≤ ∈ PosetRel ) |
5 |
3 4
|
ax-mp |
⊢ ≤ ∈ PosetRel |
6 |
5
|
elexi |
⊢ ≤ ∈ V |
7 |
6
|
inex1 |
⊢ ( ≤ ∩ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) ∈ V |
8 |
|
cnvps |
⊢ ( ≤ ∈ PosetRel → ◡ ≤ ∈ PosetRel ) |
9 |
5 8
|
ax-mp |
⊢ ◡ ≤ ∈ PosetRel |
10 |
9
|
elexi |
⊢ ◡ ≤ ∈ V |
11 |
10
|
inex1 |
⊢ ( ◡ ≤ ∩ ( ( 0 [,] +∞ ) × ( 0 [,] +∞ ) ) ) ∈ V |
12 |
1
|
xrge0iifiso |
⊢ 𝐹 Isom < , ◡ < ( ( 0 [,] 1 ) , ( 0 [,] +∞ ) ) |
13 |
|
iccssxr |
⊢ ( 0 [,] 1 ) ⊆ ℝ* |
14 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
15 |
|
gtiso |
⊢ ( ( ( 0 [,] 1 ) ⊆ ℝ* ∧ ( 0 [,] +∞ ) ⊆ ℝ* ) → ( 𝐹 Isom < , ◡ < ( ( 0 [,] 1 ) , ( 0 [,] +∞ ) ) ↔ 𝐹 Isom ≤ , ◡ ≤ ( ( 0 [,] 1 ) , ( 0 [,] +∞ ) ) ) ) |
16 |
13 14 15
|
mp2an |
⊢ ( 𝐹 Isom < , ◡ < ( ( 0 [,] 1 ) , ( 0 [,] +∞ ) ) ↔ 𝐹 Isom ≤ , ◡ ≤ ( ( 0 [,] 1 ) , ( 0 [,] +∞ ) ) ) |
17 |
12 16
|
mpbi |
⊢ 𝐹 Isom ≤ , ◡ ≤ ( ( 0 [,] 1 ) , ( 0 [,] +∞ ) ) |
18 |
|
isores1 |
⊢ ( 𝐹 Isom ≤ , ◡ ≤ ( ( 0 [,] 1 ) , ( 0 [,] +∞ ) ) ↔ 𝐹 Isom ( ≤ ∩ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) , ◡ ≤ ( ( 0 [,] 1 ) , ( 0 [,] +∞ ) ) ) |
19 |
17 18
|
mpbi |
⊢ 𝐹 Isom ( ≤ ∩ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) , ◡ ≤ ( ( 0 [,] 1 ) , ( 0 [,] +∞ ) ) |
20 |
|
isores2 |
⊢ ( 𝐹 Isom ( ≤ ∩ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) , ◡ ≤ ( ( 0 [,] 1 ) , ( 0 [,] +∞ ) ) ↔ 𝐹 Isom ( ≤ ∩ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) , ( ◡ ≤ ∩ ( ( 0 [,] +∞ ) × ( 0 [,] +∞ ) ) ) ( ( 0 [,] 1 ) , ( 0 [,] +∞ ) ) ) |
21 |
19 20
|
mpbi |
⊢ 𝐹 Isom ( ≤ ∩ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) , ( ◡ ≤ ∩ ( ( 0 [,] +∞ ) × ( 0 [,] +∞ ) ) ) ( ( 0 [,] 1 ) , ( 0 [,] +∞ ) ) |
22 |
|
ledm |
⊢ ℝ* = dom ≤ |
23 |
22
|
psssdm |
⊢ ( ( ≤ ∈ PosetRel ∧ ( 0 [,] 1 ) ⊆ ℝ* ) → dom ( ≤ ∩ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) = ( 0 [,] 1 ) ) |
24 |
5 13 23
|
mp2an |
⊢ dom ( ≤ ∩ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) = ( 0 [,] 1 ) |
25 |
24
|
eqcomi |
⊢ ( 0 [,] 1 ) = dom ( ≤ ∩ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) |
26 |
|
lern |
⊢ ℝ* = ran ≤ |
27 |
|
df-rn |
⊢ ran ≤ = dom ◡ ≤ |
28 |
26 27
|
eqtri |
⊢ ℝ* = dom ◡ ≤ |
29 |
28
|
psssdm |
⊢ ( ( ◡ ≤ ∈ PosetRel ∧ ( 0 [,] +∞ ) ⊆ ℝ* ) → dom ( ◡ ≤ ∩ ( ( 0 [,] +∞ ) × ( 0 [,] +∞ ) ) ) = ( 0 [,] +∞ ) ) |
30 |
9 14 29
|
mp2an |
⊢ dom ( ◡ ≤ ∩ ( ( 0 [,] +∞ ) × ( 0 [,] +∞ ) ) ) = ( 0 [,] +∞ ) |
31 |
30
|
eqcomi |
⊢ ( 0 [,] +∞ ) = dom ( ◡ ≤ ∩ ( ( 0 [,] +∞ ) × ( 0 [,] +∞ ) ) ) |
32 |
25 31
|
ordthmeo |
⊢ ( ( ( ≤ ∩ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) ∈ V ∧ ( ◡ ≤ ∩ ( ( 0 [,] +∞ ) × ( 0 [,] +∞ ) ) ) ∈ V ∧ 𝐹 Isom ( ≤ ∩ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) , ( ◡ ≤ ∩ ( ( 0 [,] +∞ ) × ( 0 [,] +∞ ) ) ) ( ( 0 [,] 1 ) , ( 0 [,] +∞ ) ) ) → 𝐹 ∈ ( ( ordTop ‘ ( ≤ ∩ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) ) Homeo ( ordTop ‘ ( ◡ ≤ ∩ ( ( 0 [,] +∞ ) × ( 0 [,] +∞ ) ) ) ) ) ) |
33 |
7 11 21 32
|
mp3an |
⊢ 𝐹 ∈ ( ( ordTop ‘ ( ≤ ∩ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) ) Homeo ( ordTop ‘ ( ◡ ≤ ∩ ( ( 0 [,] +∞ ) × ( 0 [,] +∞ ) ) ) ) ) |
34 |
|
dfii5 |
⊢ II = ( ordTop ‘ ( ≤ ∩ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) ) |
35 |
|
iccss2 |
⊢ ( ( 𝑥 ∈ ( 0 [,] +∞ ) ∧ 𝑦 ∈ ( 0 [,] +∞ ) ) → ( 𝑥 [,] 𝑦 ) ⊆ ( 0 [,] +∞ ) ) |
36 |
14 35
|
cnvordtrestixx |
⊢ ( ( ordTop ‘ ≤ ) ↾t ( 0 [,] +∞ ) ) = ( ordTop ‘ ( ◡ ≤ ∩ ( ( 0 [,] +∞ ) × ( 0 [,] +∞ ) ) ) ) |
37 |
2 36
|
eqtri |
⊢ 𝐽 = ( ordTop ‘ ( ◡ ≤ ∩ ( ( 0 [,] +∞ ) × ( 0 [,] +∞ ) ) ) ) |
38 |
34 37
|
oveq12i |
⊢ ( II Homeo 𝐽 ) = ( ( ordTop ‘ ( ≤ ∩ ( ( 0 [,] 1 ) × ( 0 [,] 1 ) ) ) ) Homeo ( ordTop ‘ ( ◡ ≤ ∩ ( ( 0 [,] +∞ ) × ( 0 [,] +∞ ) ) ) ) ) |
39 |
33 38
|
eleqtrri |
⊢ 𝐹 ∈ ( II Homeo 𝐽 ) |