Step |
Hyp |
Ref |
Expression |
1 |
|
uniretop |
⊢ ℝ = ∪ ( topGen ‘ ran (,) ) |
2 |
|
rehaus |
⊢ ( topGen ‘ ran (,) ) ∈ Haus |
3 |
2
|
a1i |
⊢ ( ⊤ → ( topGen ‘ ran (,) ) ∈ Haus ) |
4 |
|
rerrext |
⊢ ℝfld ∈ ℝExt |
5 |
|
eqid |
⊢ ( topGen ‘ ran (,) ) = ( topGen ‘ ran (,) ) |
6 |
|
retopn |
⊢ ( topGen ‘ ran (,) ) = ( TopOpen ‘ ℝfld ) |
7 |
5 6
|
rrhcne |
⊢ ( ℝfld ∈ ℝExt → ( ℝHom ‘ ℝfld ) ∈ ( ( topGen ‘ ran (,) ) Cn ( topGen ‘ ran (,) ) ) ) |
8 |
4 7
|
mp1i |
⊢ ( ⊤ → ( ℝHom ‘ ℝfld ) ∈ ( ( topGen ‘ ran (,) ) Cn ( topGen ‘ ran (,) ) ) ) |
9 |
|
retop |
⊢ ( topGen ‘ ran (,) ) ∈ Top |
10 |
1
|
toptopon |
⊢ ( ( topGen ‘ ran (,) ) ∈ Top ↔ ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ) |
11 |
9 10
|
mpbi |
⊢ ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) |
12 |
|
idcn |
⊢ ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) → ( I ↾ ℝ ) ∈ ( ( topGen ‘ ran (,) ) Cn ( topGen ‘ ran (,) ) ) ) |
13 |
11 12
|
ax-mp |
⊢ ( I ↾ ℝ ) ∈ ( ( topGen ‘ ran (,) ) Cn ( topGen ‘ ran (,) ) ) |
14 |
13
|
a1i |
⊢ ( ⊤ → ( I ↾ ℝ ) ∈ ( ( topGen ‘ ran (,) ) Cn ( topGen ‘ ran (,) ) ) ) |
15 |
9
|
a1i |
⊢ ( ⊤ → ( topGen ‘ ran (,) ) ∈ Top ) |
16 |
|
f1oi |
⊢ ( I ↾ ℚ ) : ℚ –1-1-onto→ ℚ |
17 |
|
f1of |
⊢ ( ( I ↾ ℚ ) : ℚ –1-1-onto→ ℚ → ( I ↾ ℚ ) : ℚ ⟶ ℚ ) |
18 |
16 17
|
ax-mp |
⊢ ( I ↾ ℚ ) : ℚ ⟶ ℚ |
19 |
|
qssre |
⊢ ℚ ⊆ ℝ |
20 |
|
fss |
⊢ ( ( ( I ↾ ℚ ) : ℚ ⟶ ℚ ∧ ℚ ⊆ ℝ ) → ( I ↾ ℚ ) : ℚ ⟶ ℝ ) |
21 |
18 19 20
|
mp2an |
⊢ ( I ↾ ℚ ) : ℚ ⟶ ℝ |
22 |
21
|
a1i |
⊢ ( ⊤ → ( I ↾ ℚ ) : ℚ ⟶ ℝ ) |
23 |
19
|
a1i |
⊢ ( ⊤ → ℚ ⊆ ℝ ) |
24 |
|
qdensere |
⊢ ( ( cls ‘ ( topGen ‘ ran (,) ) ) ‘ ℚ ) = ℝ |
25 |
24
|
a1i |
⊢ ( ⊤ → ( ( cls ‘ ( topGen ‘ ran (,) ) ) ‘ ℚ ) = ℝ ) |
26 |
9
|
a1i |
⊢ ( ( ( 𝑥 ∈ ℝ ∧ 𝑎 ∈ ( topGen ‘ ran (,) ) ) ∧ 𝑥 ∈ 𝑎 ) → ( topGen ‘ ran (,) ) ∈ Top ) |
27 |
|
simplr |
⊢ ( ( ( 𝑥 ∈ ℝ ∧ 𝑎 ∈ ( topGen ‘ ran (,) ) ) ∧ 𝑥 ∈ 𝑎 ) → 𝑎 ∈ ( topGen ‘ ran (,) ) ) |
28 |
|
simpr |
⊢ ( ( ( 𝑥 ∈ ℝ ∧ 𝑎 ∈ ( topGen ‘ ran (,) ) ) ∧ 𝑥 ∈ 𝑎 ) → 𝑥 ∈ 𝑎 ) |
29 |
|
opnneip |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ Top ∧ 𝑎 ∈ ( topGen ‘ ran (,) ) ∧ 𝑥 ∈ 𝑎 ) → 𝑎 ∈ ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ) |
30 |
26 27 28 29
|
syl3anc |
⊢ ( ( ( 𝑥 ∈ ℝ ∧ 𝑎 ∈ ( topGen ‘ ran (,) ) ) ∧ 𝑥 ∈ 𝑎 ) → 𝑎 ∈ ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ) |
31 |
|
fvex |
⊢ ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ∈ V |
32 |
|
qex |
⊢ ℚ ∈ V |
33 |
|
elrestr |
⊢ ( ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ∈ V ∧ ℚ ∈ V ∧ 𝑎 ∈ ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ) → ( 𝑎 ∩ ℚ ) ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ) |
34 |
31 32 33
|
mp3an12 |
⊢ ( 𝑎 ∈ ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) → ( 𝑎 ∩ ℚ ) ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ) |
35 |
30 34
|
syl |
⊢ ( ( ( 𝑥 ∈ ℝ ∧ 𝑎 ∈ ( topGen ‘ ran (,) ) ) ∧ 𝑥 ∈ 𝑎 ) → ( 𝑎 ∩ ℚ ) ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ) |
36 |
|
inss2 |
⊢ ( 𝑎 ∩ ℚ ) ⊆ ℚ |
37 |
|
resiima |
⊢ ( ( 𝑎 ∩ ℚ ) ⊆ ℚ → ( ( I ↾ ℚ ) “ ( 𝑎 ∩ ℚ ) ) = ( 𝑎 ∩ ℚ ) ) |
38 |
36 37
|
ax-mp |
⊢ ( ( I ↾ ℚ ) “ ( 𝑎 ∩ ℚ ) ) = ( 𝑎 ∩ ℚ ) |
39 |
|
inss1 |
⊢ ( 𝑎 ∩ ℚ ) ⊆ 𝑎 |
40 |
38 39
|
eqsstri |
⊢ ( ( I ↾ ℚ ) “ ( 𝑎 ∩ ℚ ) ) ⊆ 𝑎 |
41 |
40
|
a1i |
⊢ ( ( ( 𝑥 ∈ ℝ ∧ 𝑎 ∈ ( topGen ‘ ran (,) ) ) ∧ 𝑥 ∈ 𝑎 ) → ( ( I ↾ ℚ ) “ ( 𝑎 ∩ ℚ ) ) ⊆ 𝑎 ) |
42 |
|
imaeq2 |
⊢ ( 𝑏 = ( 𝑎 ∩ ℚ ) → ( ( I ↾ ℚ ) “ 𝑏 ) = ( ( I ↾ ℚ ) “ ( 𝑎 ∩ ℚ ) ) ) |
43 |
42
|
sseq1d |
⊢ ( 𝑏 = ( 𝑎 ∩ ℚ ) → ( ( ( I ↾ ℚ ) “ 𝑏 ) ⊆ 𝑎 ↔ ( ( I ↾ ℚ ) “ ( 𝑎 ∩ ℚ ) ) ⊆ 𝑎 ) ) |
44 |
43
|
rspcev |
⊢ ( ( ( 𝑎 ∩ ℚ ) ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ∧ ( ( I ↾ ℚ ) “ ( 𝑎 ∩ ℚ ) ) ⊆ 𝑎 ) → ∃ 𝑏 ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ( ( I ↾ ℚ ) “ 𝑏 ) ⊆ 𝑎 ) |
45 |
35 41 44
|
syl2anc |
⊢ ( ( ( 𝑥 ∈ ℝ ∧ 𝑎 ∈ ( topGen ‘ ran (,) ) ) ∧ 𝑥 ∈ 𝑎 ) → ∃ 𝑏 ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ( ( I ↾ ℚ ) “ 𝑏 ) ⊆ 𝑎 ) |
46 |
45
|
ex |
⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑎 ∈ ( topGen ‘ ran (,) ) ) → ( 𝑥 ∈ 𝑎 → ∃ 𝑏 ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ( ( I ↾ ℚ ) “ 𝑏 ) ⊆ 𝑎 ) ) |
47 |
46
|
ralrimiva |
⊢ ( 𝑥 ∈ ℝ → ∀ 𝑎 ∈ ( topGen ‘ ran (,) ) ( 𝑥 ∈ 𝑎 → ∃ 𝑏 ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ( ( I ↾ ℚ ) “ 𝑏 ) ⊆ 𝑎 ) ) |
48 |
47
|
ancli |
⊢ ( 𝑥 ∈ ℝ → ( 𝑥 ∈ ℝ ∧ ∀ 𝑎 ∈ ( topGen ‘ ran (,) ) ( 𝑥 ∈ 𝑎 → ∃ 𝑏 ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ( ( I ↾ ℚ ) “ 𝑏 ) ⊆ 𝑎 ) ) ) |
49 |
24
|
eleq2i |
⊢ ( 𝑥 ∈ ( ( cls ‘ ( topGen ‘ ran (,) ) ) ‘ ℚ ) ↔ 𝑥 ∈ ℝ ) |
50 |
49
|
biimpri |
⊢ ( 𝑥 ∈ ℝ → 𝑥 ∈ ( ( cls ‘ ( topGen ‘ ran (,) ) ) ‘ ℚ ) ) |
51 |
|
trnei |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ℚ ⊆ ℝ ∧ 𝑥 ∈ ℝ ) → ( 𝑥 ∈ ( ( cls ‘ ( topGen ‘ ran (,) ) ) ‘ ℚ ) ↔ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ∈ ( Fil ‘ ℚ ) ) ) |
52 |
11 19 51
|
mp3an12 |
⊢ ( 𝑥 ∈ ℝ → ( 𝑥 ∈ ( ( cls ‘ ( topGen ‘ ran (,) ) ) ‘ ℚ ) ↔ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ∈ ( Fil ‘ ℚ ) ) ) |
53 |
50 52
|
mpbid |
⊢ ( 𝑥 ∈ ℝ → ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ∈ ( Fil ‘ ℚ ) ) |
54 |
|
isflf |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ∈ ( Fil ‘ ℚ ) ∧ ( I ↾ ℚ ) : ℚ ⟶ ℝ ) → ( 𝑥 ∈ ( ( ( topGen ‘ ran (,) ) fLimf ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ) ‘ ( I ↾ ℚ ) ) ↔ ( 𝑥 ∈ ℝ ∧ ∀ 𝑎 ∈ ( topGen ‘ ran (,) ) ( 𝑥 ∈ 𝑎 → ∃ 𝑏 ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ( ( I ↾ ℚ ) “ 𝑏 ) ⊆ 𝑎 ) ) ) ) |
55 |
11 21 54
|
mp3an13 |
⊢ ( ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ∈ ( Fil ‘ ℚ ) → ( 𝑥 ∈ ( ( ( topGen ‘ ran (,) ) fLimf ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ) ‘ ( I ↾ ℚ ) ) ↔ ( 𝑥 ∈ ℝ ∧ ∀ 𝑎 ∈ ( topGen ‘ ran (,) ) ( 𝑥 ∈ 𝑎 → ∃ 𝑏 ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ( ( I ↾ ℚ ) “ 𝑏 ) ⊆ 𝑎 ) ) ) ) |
56 |
53 55
|
syl |
⊢ ( 𝑥 ∈ ℝ → ( 𝑥 ∈ ( ( ( topGen ‘ ran (,) ) fLimf ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ) ‘ ( I ↾ ℚ ) ) ↔ ( 𝑥 ∈ ℝ ∧ ∀ 𝑎 ∈ ( topGen ‘ ran (,) ) ( 𝑥 ∈ 𝑎 → ∃ 𝑏 ∈ ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ( ( I ↾ ℚ ) “ 𝑏 ) ⊆ 𝑎 ) ) ) ) |
57 |
48 56
|
mpbird |
⊢ ( 𝑥 ∈ ℝ → 𝑥 ∈ ( ( ( topGen ‘ ran (,) ) fLimf ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ) ‘ ( I ↾ ℚ ) ) ) |
58 |
57
|
ne0d |
⊢ ( 𝑥 ∈ ℝ → ( ( ( topGen ‘ ran (,) ) fLimf ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ) ‘ ( I ↾ ℚ ) ) ≠ ∅ ) |
59 |
58
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ ) → ( ( ( topGen ‘ ran (,) ) fLimf ( ( ( nei ‘ ( topGen ‘ ran (,) ) ) ‘ { 𝑥 } ) ↾t ℚ ) ) ‘ ( I ↾ ℚ ) ) ≠ ∅ ) |
60 |
|
recusp |
⊢ ℝfld ∈ CUnifSp |
61 |
|
cuspusp |
⊢ ( ℝfld ∈ CUnifSp → ℝfld ∈ UnifSp ) |
62 |
60 61
|
ax-mp |
⊢ ℝfld ∈ UnifSp |
63 |
6
|
uspreg |
⊢ ( ( ℝfld ∈ UnifSp ∧ ( topGen ‘ ran (,) ) ∈ Haus ) → ( topGen ‘ ran (,) ) ∈ Reg ) |
64 |
62 2 63
|
mp2an |
⊢ ( topGen ‘ ran (,) ) ∈ Reg |
65 |
64
|
a1i |
⊢ ( ⊤ → ( topGen ‘ ran (,) ) ∈ Reg ) |
66 |
|
resabs1 |
⊢ ( ℚ ⊆ ℝ → ( ( I ↾ ℝ ) ↾ ℚ ) = ( I ↾ ℚ ) ) |
67 |
19 66
|
ax-mp |
⊢ ( ( I ↾ ℝ ) ↾ ℚ ) = ( I ↾ ℚ ) |
68 |
1
|
cnrest |
⊢ ( ( ( I ↾ ℝ ) ∈ ( ( topGen ‘ ran (,) ) Cn ( topGen ‘ ran (,) ) ) ∧ ℚ ⊆ ℝ ) → ( ( I ↾ ℝ ) ↾ ℚ ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ℚ ) Cn ( topGen ‘ ran (,) ) ) ) |
69 |
13 19 68
|
mp2an |
⊢ ( ( I ↾ ℝ ) ↾ ℚ ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ℚ ) Cn ( topGen ‘ ran (,) ) ) |
70 |
67 69
|
eqeltrri |
⊢ ( I ↾ ℚ ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ℚ ) Cn ( topGen ‘ ran (,) ) ) |
71 |
70
|
a1i |
⊢ ( ⊤ → ( I ↾ ℚ ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ℚ ) Cn ( topGen ‘ ran (,) ) ) ) |
72 |
1 1 15 3 22 23 25 59 65 71
|
cnextfres1 |
⊢ ( ⊤ → ( ( ( ( topGen ‘ ran (,) ) CnExt ( topGen ‘ ran (,) ) ) ‘ ( I ↾ ℚ ) ) ↾ ℚ ) = ( I ↾ ℚ ) ) |
73 |
72
|
mptru |
⊢ ( ( ( ( topGen ‘ ran (,) ) CnExt ( topGen ‘ ran (,) ) ) ‘ ( I ↾ ℚ ) ) ↾ ℚ ) = ( I ↾ ℚ ) |
74 |
|
recms |
⊢ ℝfld ∈ CMetSp |
75 |
74
|
elexi |
⊢ ℝfld ∈ V |
76 |
5 6
|
rrhval |
⊢ ( ℝfld ∈ V → ( ℝHom ‘ ℝfld ) = ( ( ( topGen ‘ ran (,) ) CnExt ( topGen ‘ ran (,) ) ) ‘ ( ℚHom ‘ ℝfld ) ) ) |
77 |
75 76
|
ax-mp |
⊢ ( ℝHom ‘ ℝfld ) = ( ( ( topGen ‘ ran (,) ) CnExt ( topGen ‘ ran (,) ) ) ‘ ( ℚHom ‘ ℝfld ) ) |
78 |
|
qqhre |
⊢ ( ℚHom ‘ ℝfld ) = ( I ↾ ℚ ) |
79 |
78
|
fveq2i |
⊢ ( ( ( topGen ‘ ran (,) ) CnExt ( topGen ‘ ran (,) ) ) ‘ ( ℚHom ‘ ℝfld ) ) = ( ( ( topGen ‘ ran (,) ) CnExt ( topGen ‘ ran (,) ) ) ‘ ( I ↾ ℚ ) ) |
80 |
77 79
|
eqtri |
⊢ ( ℝHom ‘ ℝfld ) = ( ( ( topGen ‘ ran (,) ) CnExt ( topGen ‘ ran (,) ) ) ‘ ( I ↾ ℚ ) ) |
81 |
80
|
reseq1i |
⊢ ( ( ℝHom ‘ ℝfld ) ↾ ℚ ) = ( ( ( ( topGen ‘ ran (,) ) CnExt ( topGen ‘ ran (,) ) ) ‘ ( I ↾ ℚ ) ) ↾ ℚ ) |
82 |
73 81 67
|
3eqtr4i |
⊢ ( ( ℝHom ‘ ℝfld ) ↾ ℚ ) = ( ( I ↾ ℝ ) ↾ ℚ ) |
83 |
82
|
a1i |
⊢ ( ⊤ → ( ( ℝHom ‘ ℝfld ) ↾ ℚ ) = ( ( I ↾ ℝ ) ↾ ℚ ) ) |
84 |
1 3 8 14 83 23 25
|
hauseqcn |
⊢ ( ⊤ → ( ℝHom ‘ ℝfld ) = ( I ↾ ℝ ) ) |
85 |
84
|
mptru |
⊢ ( ℝHom ‘ ℝfld ) = ( I ↾ ℝ ) |