Step |
Hyp |
Ref |
Expression |
1 |
|
icopnfhmeo.f |
⊢ 𝐹 = ( 𝑥 ∈ ( 0 [,) 1 ) ↦ ( 𝑥 / ( 1 − 𝑥 ) ) ) |
2 |
|
icopnfhmeo.j |
⊢ 𝐽 = ( TopOpen ‘ ℂfld ) |
3 |
1
|
icopnfcnv |
⊢ ( 𝐹 : ( 0 [,) 1 ) –1-1-onto→ ( 0 [,) +∞ ) ∧ ◡ 𝐹 = ( 𝑦 ∈ ( 0 [,) +∞ ) ↦ ( 𝑦 / ( 1 + 𝑦 ) ) ) ) |
4 |
3
|
simpli |
⊢ 𝐹 : ( 0 [,) 1 ) –1-1-onto→ ( 0 [,) +∞ ) |
5 |
|
0re |
⊢ 0 ∈ ℝ |
6 |
|
1xr |
⊢ 1 ∈ ℝ* |
7 |
|
elico2 |
⊢ ( ( 0 ∈ ℝ ∧ 1 ∈ ℝ* ) → ( 𝑥 ∈ ( 0 [,) 1 ) ↔ ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 < 1 ) ) ) |
8 |
5 6 7
|
mp2an |
⊢ ( 𝑥 ∈ ( 0 [,) 1 ) ↔ ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ∧ 𝑥 < 1 ) ) |
9 |
8
|
simp1bi |
⊢ ( 𝑥 ∈ ( 0 [,) 1 ) → 𝑥 ∈ ℝ ) |
10 |
9
|
ssriv |
⊢ ( 0 [,) 1 ) ⊆ ℝ |
11 |
10
|
sseli |
⊢ ( 𝑧 ∈ ( 0 [,) 1 ) → 𝑧 ∈ ℝ ) |
12 |
11
|
adantr |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → 𝑧 ∈ ℝ ) |
13 |
|
elico2 |
⊢ ( ( 0 ∈ ℝ ∧ 1 ∈ ℝ* ) → ( 𝑤 ∈ ( 0 [,) 1 ) ↔ ( 𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 < 1 ) ) ) |
14 |
5 6 13
|
mp2an |
⊢ ( 𝑤 ∈ ( 0 [,) 1 ) ↔ ( 𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 < 1 ) ) |
15 |
14
|
simp3bi |
⊢ ( 𝑤 ∈ ( 0 [,) 1 ) → 𝑤 < 1 ) |
16 |
10
|
sseli |
⊢ ( 𝑤 ∈ ( 0 [,) 1 ) → 𝑤 ∈ ℝ ) |
17 |
|
1re |
⊢ 1 ∈ ℝ |
18 |
|
difrp |
⊢ ( ( 𝑤 ∈ ℝ ∧ 1 ∈ ℝ ) → ( 𝑤 < 1 ↔ ( 1 − 𝑤 ) ∈ ℝ+ ) ) |
19 |
16 17 18
|
sylancl |
⊢ ( 𝑤 ∈ ( 0 [,) 1 ) → ( 𝑤 < 1 ↔ ( 1 − 𝑤 ) ∈ ℝ+ ) ) |
20 |
15 19
|
mpbid |
⊢ ( 𝑤 ∈ ( 0 [,) 1 ) → ( 1 − 𝑤 ) ∈ ℝ+ ) |
21 |
20
|
rpregt0d |
⊢ ( 𝑤 ∈ ( 0 [,) 1 ) → ( ( 1 − 𝑤 ) ∈ ℝ ∧ 0 < ( 1 − 𝑤 ) ) ) |
22 |
21
|
adantl |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( ( 1 − 𝑤 ) ∈ ℝ ∧ 0 < ( 1 − 𝑤 ) ) ) |
23 |
16
|
adantl |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → 𝑤 ∈ ℝ ) |
24 |
|
elico2 |
⊢ ( ( 0 ∈ ℝ ∧ 1 ∈ ℝ* ) → ( 𝑧 ∈ ( 0 [,) 1 ) ↔ ( 𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 < 1 ) ) ) |
25 |
5 6 24
|
mp2an |
⊢ ( 𝑧 ∈ ( 0 [,) 1 ) ↔ ( 𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 < 1 ) ) |
26 |
25
|
simp3bi |
⊢ ( 𝑧 ∈ ( 0 [,) 1 ) → 𝑧 < 1 ) |
27 |
|
difrp |
⊢ ( ( 𝑧 ∈ ℝ ∧ 1 ∈ ℝ ) → ( 𝑧 < 1 ↔ ( 1 − 𝑧 ) ∈ ℝ+ ) ) |
28 |
11 17 27
|
sylancl |
⊢ ( 𝑧 ∈ ( 0 [,) 1 ) → ( 𝑧 < 1 ↔ ( 1 − 𝑧 ) ∈ ℝ+ ) ) |
29 |
26 28
|
mpbid |
⊢ ( 𝑧 ∈ ( 0 [,) 1 ) → ( 1 − 𝑧 ) ∈ ℝ+ ) |
30 |
29
|
adantr |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( 1 − 𝑧 ) ∈ ℝ+ ) |
31 |
30
|
rpregt0d |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( ( 1 − 𝑧 ) ∈ ℝ ∧ 0 < ( 1 − 𝑧 ) ) ) |
32 |
|
lt2mul2div |
⊢ ( ( ( 𝑧 ∈ ℝ ∧ ( ( 1 − 𝑤 ) ∈ ℝ ∧ 0 < ( 1 − 𝑤 ) ) ) ∧ ( 𝑤 ∈ ℝ ∧ ( ( 1 − 𝑧 ) ∈ ℝ ∧ 0 < ( 1 − 𝑧 ) ) ) ) → ( ( 𝑧 · ( 1 − 𝑤 ) ) < ( 𝑤 · ( 1 − 𝑧 ) ) ↔ ( 𝑧 / ( 1 − 𝑧 ) ) < ( 𝑤 / ( 1 − 𝑤 ) ) ) ) |
33 |
12 22 23 31 32
|
syl22anc |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( ( 𝑧 · ( 1 − 𝑤 ) ) < ( 𝑤 · ( 1 − 𝑧 ) ) ↔ ( 𝑧 / ( 1 − 𝑧 ) ) < ( 𝑤 / ( 1 − 𝑤 ) ) ) ) |
34 |
12 23
|
remulcld |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( 𝑧 · 𝑤 ) ∈ ℝ ) |
35 |
12 23 34
|
ltsub1d |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( 𝑧 < 𝑤 ↔ ( 𝑧 − ( 𝑧 · 𝑤 ) ) < ( 𝑤 − ( 𝑧 · 𝑤 ) ) ) ) |
36 |
12
|
recnd |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → 𝑧 ∈ ℂ ) |
37 |
|
1cnd |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → 1 ∈ ℂ ) |
38 |
23
|
recnd |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → 𝑤 ∈ ℂ ) |
39 |
36 37 38
|
subdid |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( 𝑧 · ( 1 − 𝑤 ) ) = ( ( 𝑧 · 1 ) − ( 𝑧 · 𝑤 ) ) ) |
40 |
36
|
mulid1d |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( 𝑧 · 1 ) = 𝑧 ) |
41 |
40
|
oveq1d |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( ( 𝑧 · 1 ) − ( 𝑧 · 𝑤 ) ) = ( 𝑧 − ( 𝑧 · 𝑤 ) ) ) |
42 |
39 41
|
eqtrd |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( 𝑧 · ( 1 − 𝑤 ) ) = ( 𝑧 − ( 𝑧 · 𝑤 ) ) ) |
43 |
38 37 36
|
subdid |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( 𝑤 · ( 1 − 𝑧 ) ) = ( ( 𝑤 · 1 ) − ( 𝑤 · 𝑧 ) ) ) |
44 |
38
|
mulid1d |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( 𝑤 · 1 ) = 𝑤 ) |
45 |
38 36
|
mulcomd |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( 𝑤 · 𝑧 ) = ( 𝑧 · 𝑤 ) ) |
46 |
44 45
|
oveq12d |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( ( 𝑤 · 1 ) − ( 𝑤 · 𝑧 ) ) = ( 𝑤 − ( 𝑧 · 𝑤 ) ) ) |
47 |
43 46
|
eqtrd |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( 𝑤 · ( 1 − 𝑧 ) ) = ( 𝑤 − ( 𝑧 · 𝑤 ) ) ) |
48 |
42 47
|
breq12d |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( ( 𝑧 · ( 1 − 𝑤 ) ) < ( 𝑤 · ( 1 − 𝑧 ) ) ↔ ( 𝑧 − ( 𝑧 · 𝑤 ) ) < ( 𝑤 − ( 𝑧 · 𝑤 ) ) ) ) |
49 |
35 48
|
bitr4d |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( 𝑧 < 𝑤 ↔ ( 𝑧 · ( 1 − 𝑤 ) ) < ( 𝑤 · ( 1 − 𝑧 ) ) ) ) |
50 |
|
id |
⊢ ( 𝑥 = 𝑧 → 𝑥 = 𝑧 ) |
51 |
|
oveq2 |
⊢ ( 𝑥 = 𝑧 → ( 1 − 𝑥 ) = ( 1 − 𝑧 ) ) |
52 |
50 51
|
oveq12d |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 / ( 1 − 𝑥 ) ) = ( 𝑧 / ( 1 − 𝑧 ) ) ) |
53 |
|
ovex |
⊢ ( 𝑧 / ( 1 − 𝑧 ) ) ∈ V |
54 |
52 1 53
|
fvmpt |
⊢ ( 𝑧 ∈ ( 0 [,) 1 ) → ( 𝐹 ‘ 𝑧 ) = ( 𝑧 / ( 1 − 𝑧 ) ) ) |
55 |
|
id |
⊢ ( 𝑥 = 𝑤 → 𝑥 = 𝑤 ) |
56 |
|
oveq2 |
⊢ ( 𝑥 = 𝑤 → ( 1 − 𝑥 ) = ( 1 − 𝑤 ) ) |
57 |
55 56
|
oveq12d |
⊢ ( 𝑥 = 𝑤 → ( 𝑥 / ( 1 − 𝑥 ) ) = ( 𝑤 / ( 1 − 𝑤 ) ) ) |
58 |
|
ovex |
⊢ ( 𝑤 / ( 1 − 𝑤 ) ) ∈ V |
59 |
57 1 58
|
fvmpt |
⊢ ( 𝑤 ∈ ( 0 [,) 1 ) → ( 𝐹 ‘ 𝑤 ) = ( 𝑤 / ( 1 − 𝑤 ) ) ) |
60 |
54 59
|
breqan12d |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( ( 𝐹 ‘ 𝑧 ) < ( 𝐹 ‘ 𝑤 ) ↔ ( 𝑧 / ( 1 − 𝑧 ) ) < ( 𝑤 / ( 1 − 𝑤 ) ) ) ) |
61 |
33 49 60
|
3bitr4d |
⊢ ( ( 𝑧 ∈ ( 0 [,) 1 ) ∧ 𝑤 ∈ ( 0 [,) 1 ) ) → ( 𝑧 < 𝑤 ↔ ( 𝐹 ‘ 𝑧 ) < ( 𝐹 ‘ 𝑤 ) ) ) |
62 |
61
|
rgen2 |
⊢ ∀ 𝑧 ∈ ( 0 [,) 1 ) ∀ 𝑤 ∈ ( 0 [,) 1 ) ( 𝑧 < 𝑤 ↔ ( 𝐹 ‘ 𝑧 ) < ( 𝐹 ‘ 𝑤 ) ) |
63 |
|
df-isom |
⊢ ( 𝐹 Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +∞ ) ) ↔ ( 𝐹 : ( 0 [,) 1 ) –1-1-onto→ ( 0 [,) +∞ ) ∧ ∀ 𝑧 ∈ ( 0 [,) 1 ) ∀ 𝑤 ∈ ( 0 [,) 1 ) ( 𝑧 < 𝑤 ↔ ( 𝐹 ‘ 𝑧 ) < ( 𝐹 ‘ 𝑤 ) ) ) ) |
64 |
4 62 63
|
mpbir2an |
⊢ 𝐹 Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +∞ ) ) |
65 |
|
letsr |
⊢ ≤ ∈ TosetRel |
66 |
65
|
elexi |
⊢ ≤ ∈ V |
67 |
66
|
inex1 |
⊢ ( ≤ ∩ ( ( 0 [,) 1 ) × ( 0 [,) 1 ) ) ) ∈ V |
68 |
66
|
inex1 |
⊢ ( ≤ ∩ ( ( 0 [,) +∞ ) × ( 0 [,) +∞ ) ) ) ∈ V |
69 |
|
icossxr |
⊢ ( 0 [,) 1 ) ⊆ ℝ* |
70 |
|
icossxr |
⊢ ( 0 [,) +∞ ) ⊆ ℝ* |
71 |
|
leiso |
⊢ ( ( ( 0 [,) 1 ) ⊆ ℝ* ∧ ( 0 [,) +∞ ) ⊆ ℝ* ) → ( 𝐹 Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +∞ ) ) ↔ 𝐹 Isom ≤ , ≤ ( ( 0 [,) 1 ) , ( 0 [,) +∞ ) ) ) ) |
72 |
69 70 71
|
mp2an |
⊢ ( 𝐹 Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +∞ ) ) ↔ 𝐹 Isom ≤ , ≤ ( ( 0 [,) 1 ) , ( 0 [,) +∞ ) ) ) |
73 |
64 72
|
mpbi |
⊢ 𝐹 Isom ≤ , ≤ ( ( 0 [,) 1 ) , ( 0 [,) +∞ ) ) |
74 |
|
isores1 |
⊢ ( 𝐹 Isom ≤ , ≤ ( ( 0 [,) 1 ) , ( 0 [,) +∞ ) ) ↔ 𝐹 Isom ( ≤ ∩ ( ( 0 [,) 1 ) × ( 0 [,) 1 ) ) ) , ≤ ( ( 0 [,) 1 ) , ( 0 [,) +∞ ) ) ) |
75 |
73 74
|
mpbi |
⊢ 𝐹 Isom ( ≤ ∩ ( ( 0 [,) 1 ) × ( 0 [,) 1 ) ) ) , ≤ ( ( 0 [,) 1 ) , ( 0 [,) +∞ ) ) |
76 |
|
isores2 |
⊢ ( 𝐹 Isom ( ≤ ∩ ( ( 0 [,) 1 ) × ( 0 [,) 1 ) ) ) , ≤ ( ( 0 [,) 1 ) , ( 0 [,) +∞ ) ) ↔ 𝐹 Isom ( ≤ ∩ ( ( 0 [,) 1 ) × ( 0 [,) 1 ) ) ) , ( ≤ ∩ ( ( 0 [,) +∞ ) × ( 0 [,) +∞ ) ) ) ( ( 0 [,) 1 ) , ( 0 [,) +∞ ) ) ) |
77 |
75 76
|
mpbi |
⊢ 𝐹 Isom ( ≤ ∩ ( ( 0 [,) 1 ) × ( 0 [,) 1 ) ) ) , ( ≤ ∩ ( ( 0 [,) +∞ ) × ( 0 [,) +∞ ) ) ) ( ( 0 [,) 1 ) , ( 0 [,) +∞ ) ) |
78 |
|
tsrps |
⊢ ( ≤ ∈ TosetRel → ≤ ∈ PosetRel ) |
79 |
65 78
|
ax-mp |
⊢ ≤ ∈ PosetRel |
80 |
|
ledm |
⊢ ℝ* = dom ≤ |
81 |
80
|
psssdm |
⊢ ( ( ≤ ∈ PosetRel ∧ ( 0 [,) 1 ) ⊆ ℝ* ) → dom ( ≤ ∩ ( ( 0 [,) 1 ) × ( 0 [,) 1 ) ) ) = ( 0 [,) 1 ) ) |
82 |
79 69 81
|
mp2an |
⊢ dom ( ≤ ∩ ( ( 0 [,) 1 ) × ( 0 [,) 1 ) ) ) = ( 0 [,) 1 ) |
83 |
82
|
eqcomi |
⊢ ( 0 [,) 1 ) = dom ( ≤ ∩ ( ( 0 [,) 1 ) × ( 0 [,) 1 ) ) ) |
84 |
80
|
psssdm |
⊢ ( ( ≤ ∈ PosetRel ∧ ( 0 [,) +∞ ) ⊆ ℝ* ) → dom ( ≤ ∩ ( ( 0 [,) +∞ ) × ( 0 [,) +∞ ) ) ) = ( 0 [,) +∞ ) ) |
85 |
79 70 84
|
mp2an |
⊢ dom ( ≤ ∩ ( ( 0 [,) +∞ ) × ( 0 [,) +∞ ) ) ) = ( 0 [,) +∞ ) |
86 |
85
|
eqcomi |
⊢ ( 0 [,) +∞ ) = dom ( ≤ ∩ ( ( 0 [,) +∞ ) × ( 0 [,) +∞ ) ) ) |
87 |
83 86
|
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 [,) +∞ ) ) ) ) ) ) |
88 |
67 68 77 87
|
mp3an |
⊢ 𝐹 ∈ ( ( ordTop ‘ ( ≤ ∩ ( ( 0 [,) 1 ) × ( 0 [,) 1 ) ) ) ) Homeo ( ordTop ‘ ( ≤ ∩ ( ( 0 [,) +∞ ) × ( 0 [,) +∞ ) ) ) ) ) |
89 |
|
eqid |
⊢ ( ordTop ‘ ≤ ) = ( ordTop ‘ ≤ ) |
90 |
2 89
|
xrrest2 |
⊢ ( ( 0 [,) 1 ) ⊆ ℝ → ( 𝐽 ↾t ( 0 [,) 1 ) ) = ( ( ordTop ‘ ≤ ) ↾t ( 0 [,) 1 ) ) ) |
91 |
10 90
|
ax-mp |
⊢ ( 𝐽 ↾t ( 0 [,) 1 ) ) = ( ( ordTop ‘ ≤ ) ↾t ( 0 [,) 1 ) ) |
92 |
|
iccssico2 |
⊢ ( ( 𝑥 ∈ ( 0 [,) 1 ) ∧ 𝑦 ∈ ( 0 [,) 1 ) ) → ( 𝑥 [,] 𝑦 ) ⊆ ( 0 [,) 1 ) ) |
93 |
69 92
|
ordtrestixx |
⊢ ( ( ordTop ‘ ≤ ) ↾t ( 0 [,) 1 ) ) = ( ordTop ‘ ( ≤ ∩ ( ( 0 [,) 1 ) × ( 0 [,) 1 ) ) ) ) |
94 |
91 93
|
eqtri |
⊢ ( 𝐽 ↾t ( 0 [,) 1 ) ) = ( ordTop ‘ ( ≤ ∩ ( ( 0 [,) 1 ) × ( 0 [,) 1 ) ) ) ) |
95 |
|
rge0ssre |
⊢ ( 0 [,) +∞ ) ⊆ ℝ |
96 |
2 89
|
xrrest2 |
⊢ ( ( 0 [,) +∞ ) ⊆ ℝ → ( 𝐽 ↾t ( 0 [,) +∞ ) ) = ( ( ordTop ‘ ≤ ) ↾t ( 0 [,) +∞ ) ) ) |
97 |
95 96
|
ax-mp |
⊢ ( 𝐽 ↾t ( 0 [,) +∞ ) ) = ( ( ordTop ‘ ≤ ) ↾t ( 0 [,) +∞ ) ) |
98 |
|
iccssico2 |
⊢ ( ( 𝑥 ∈ ( 0 [,) +∞ ) ∧ 𝑦 ∈ ( 0 [,) +∞ ) ) → ( 𝑥 [,] 𝑦 ) ⊆ ( 0 [,) +∞ ) ) |
99 |
70 98
|
ordtrestixx |
⊢ ( ( ordTop ‘ ≤ ) ↾t ( 0 [,) +∞ ) ) = ( ordTop ‘ ( ≤ ∩ ( ( 0 [,) +∞ ) × ( 0 [,) +∞ ) ) ) ) |
100 |
97 99
|
eqtri |
⊢ ( 𝐽 ↾t ( 0 [,) +∞ ) ) = ( ordTop ‘ ( ≤ ∩ ( ( 0 [,) +∞ ) × ( 0 [,) +∞ ) ) ) ) |
101 |
94 100
|
oveq12i |
⊢ ( ( 𝐽 ↾t ( 0 [,) 1 ) ) Homeo ( 𝐽 ↾t ( 0 [,) +∞ ) ) ) = ( ( ordTop ‘ ( ≤ ∩ ( ( 0 [,) 1 ) × ( 0 [,) 1 ) ) ) ) Homeo ( ordTop ‘ ( ≤ ∩ ( ( 0 [,) +∞ ) × ( 0 [,) +∞ ) ) ) ) ) |
102 |
88 101
|
eleqtrri |
⊢ 𝐹 ∈ ( ( 𝐽 ↾t ( 0 [,) 1 ) ) Homeo ( 𝐽 ↾t ( 0 [,) +∞ ) ) ) |
103 |
64 102
|
pm3.2i |
⊢ ( 𝐹 Isom < , < ( ( 0 [,) 1 ) , ( 0 [,) +∞ ) ) ∧ 𝐹 ∈ ( ( 𝐽 ↾t ( 0 [,) 1 ) ) Homeo ( 𝐽 ↾t ( 0 [,) +∞ ) ) ) ) |