Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ( topGen ‘ ran (,) ) = ( topGen ‘ ran (,) ) |
2 |
|
eqid |
⊢ ( TopOpen ‘ 𝑅 ) = ( TopOpen ‘ 𝑅 ) |
3 |
1 2
|
rrhval |
⊢ ( 𝑅 ∈ ℝExt → ( ℝHom ‘ 𝑅 ) = ( ( ( topGen ‘ ran (,) ) CnExt ( TopOpen ‘ 𝑅 ) ) ‘ ( ℚHom ‘ 𝑅 ) ) ) |
4 |
3
|
fveq1d |
⊢ ( 𝑅 ∈ ℝExt → ( ( ℝHom ‘ 𝑅 ) ‘ 𝑄 ) = ( ( ( ( topGen ‘ ran (,) ) CnExt ( TopOpen ‘ 𝑅 ) ) ‘ ( ℚHom ‘ 𝑅 ) ) ‘ 𝑄 ) ) |
5 |
4
|
adantr |
⊢ ( ( 𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ ) → ( ( ℝHom ‘ 𝑅 ) ‘ 𝑄 ) = ( ( ( ( topGen ‘ ran (,) ) CnExt ( TopOpen ‘ 𝑅 ) ) ‘ ( ℚHom ‘ 𝑅 ) ) ‘ 𝑄 ) ) |
6 |
|
uniretop |
⊢ ℝ = ∪ ( topGen ‘ ran (,) ) |
7 |
|
eqid |
⊢ ∪ ( TopOpen ‘ 𝑅 ) = ∪ ( TopOpen ‘ 𝑅 ) |
8 |
|
retop |
⊢ ( topGen ‘ ran (,) ) ∈ Top |
9 |
8
|
a1i |
⊢ ( ( 𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ ) → ( topGen ‘ ran (,) ) ∈ Top ) |
10 |
2
|
rrexthaus |
⊢ ( 𝑅 ∈ ℝExt → ( TopOpen ‘ 𝑅 ) ∈ Haus ) |
11 |
10
|
adantr |
⊢ ( ( 𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ ) → ( TopOpen ‘ 𝑅 ) ∈ Haus ) |
12 |
|
qssre |
⊢ ℚ ⊆ ℝ |
13 |
12
|
a1i |
⊢ ( ( 𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ ) → ℚ ⊆ ℝ ) |
14 |
|
rrextnrg |
⊢ ( 𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing ) |
15 |
|
rrextdrg |
⊢ ( 𝑅 ∈ ℝExt → 𝑅 ∈ DivRing ) |
16 |
14 15
|
elind |
⊢ ( 𝑅 ∈ ℝExt → 𝑅 ∈ ( NrmRing ∩ DivRing ) ) |
17 |
|
eqid |
⊢ ( ℤMod ‘ 𝑅 ) = ( ℤMod ‘ 𝑅 ) |
18 |
17
|
rrextnlm |
⊢ ( 𝑅 ∈ ℝExt → ( ℤMod ‘ 𝑅 ) ∈ NrmMod ) |
19 |
|
rrextchr |
⊢ ( 𝑅 ∈ ℝExt → ( chr ‘ 𝑅 ) = 0 ) |
20 |
|
eqid |
⊢ ( ℂfld ↾s ℚ ) = ( ℂfld ↾s ℚ ) |
21 |
|
qqtopn |
⊢ ( ( TopOpen ‘ ℝfld ) ↾t ℚ ) = ( TopOpen ‘ ( ℂfld ↾s ℚ ) ) |
22 |
20 21 17 2
|
qqhcn |
⊢ ( ( 𝑅 ∈ ( NrmRing ∩ DivRing ) ∧ ( ℤMod ‘ 𝑅 ) ∈ NrmMod ∧ ( chr ‘ 𝑅 ) = 0 ) → ( ℚHom ‘ 𝑅 ) ∈ ( ( ( TopOpen ‘ ℝfld ) ↾t ℚ ) Cn ( TopOpen ‘ 𝑅 ) ) ) |
23 |
16 18 19 22
|
syl3anc |
⊢ ( 𝑅 ∈ ℝExt → ( ℚHom ‘ 𝑅 ) ∈ ( ( ( TopOpen ‘ ℝfld ) ↾t ℚ ) Cn ( TopOpen ‘ 𝑅 ) ) ) |
24 |
|
retopn |
⊢ ( topGen ‘ ran (,) ) = ( TopOpen ‘ ℝfld ) |
25 |
24
|
eqcomi |
⊢ ( TopOpen ‘ ℝfld ) = ( topGen ‘ ran (,) ) |
26 |
25
|
oveq1i |
⊢ ( ( TopOpen ‘ ℝfld ) ↾t ℚ ) = ( ( topGen ‘ ran (,) ) ↾t ℚ ) |
27 |
26
|
oveq1i |
⊢ ( ( ( TopOpen ‘ ℝfld ) ↾t ℚ ) Cn ( TopOpen ‘ 𝑅 ) ) = ( ( ( topGen ‘ ran (,) ) ↾t ℚ ) Cn ( TopOpen ‘ 𝑅 ) ) |
28 |
23 27
|
eleqtrdi |
⊢ ( 𝑅 ∈ ℝExt → ( ℚHom ‘ 𝑅 ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ℚ ) Cn ( TopOpen ‘ 𝑅 ) ) ) |
29 |
28
|
adantr |
⊢ ( ( 𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ ) → ( ℚHom ‘ 𝑅 ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ℚ ) Cn ( TopOpen ‘ 𝑅 ) ) ) |
30 |
|
simpr |
⊢ ( ( 𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ ) → 𝑄 ∈ ℚ ) |
31 |
6 7 9 11 13 29 30
|
cnextfres |
⊢ ( ( 𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ ) → ( ( ( ( topGen ‘ ran (,) ) CnExt ( TopOpen ‘ 𝑅 ) ) ‘ ( ℚHom ‘ 𝑅 ) ) ‘ 𝑄 ) = ( ( ℚHom ‘ 𝑅 ) ‘ 𝑄 ) ) |
32 |
5 31
|
eqtrd |
⊢ ( ( 𝑅 ∈ ℝExt ∧ 𝑄 ∈ ℚ ) → ( ( ℝHom ‘ 𝑅 ) ‘ 𝑄 ) = ( ( ℚHom ‘ 𝑅 ) ‘ 𝑄 ) ) |