Step |
Hyp |
Ref |
Expression |
1 |
|
rrexthaus.1 |
⊢ 𝐾 = ( TopOpen ‘ 𝑅 ) |
2 |
|
rrextnrg |
⊢ ( 𝑅 ∈ ℝExt → 𝑅 ∈ NrmRing ) |
3 |
|
nrgngp |
⊢ ( 𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp ) |
4 |
|
ngpxms |
⊢ ( 𝑅 ∈ NrmGrp → 𝑅 ∈ ∞MetSp ) |
5 |
2 3 4
|
3syl |
⊢ ( 𝑅 ∈ ℝExt → 𝑅 ∈ ∞MetSp ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
7 |
|
eqid |
⊢ ( ( dist ‘ 𝑅 ) ↾ ( ( Base ‘ 𝑅 ) × ( Base ‘ 𝑅 ) ) ) = ( ( dist ‘ 𝑅 ) ↾ ( ( Base ‘ 𝑅 ) × ( Base ‘ 𝑅 ) ) ) |
8 |
1 6 7
|
xmstopn |
⊢ ( 𝑅 ∈ ∞MetSp → 𝐾 = ( MetOpen ‘ ( ( dist ‘ 𝑅 ) ↾ ( ( Base ‘ 𝑅 ) × ( Base ‘ 𝑅 ) ) ) ) ) |
9 |
5 8
|
syl |
⊢ ( 𝑅 ∈ ℝExt → 𝐾 = ( MetOpen ‘ ( ( dist ‘ 𝑅 ) ↾ ( ( Base ‘ 𝑅 ) × ( Base ‘ 𝑅 ) ) ) ) ) |
10 |
6 7
|
xmsxmet |
⊢ ( 𝑅 ∈ ∞MetSp → ( ( dist ‘ 𝑅 ) ↾ ( ( Base ‘ 𝑅 ) × ( Base ‘ 𝑅 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑅 ) ) ) |
11 |
|
eqid |
⊢ ( MetOpen ‘ ( ( dist ‘ 𝑅 ) ↾ ( ( Base ‘ 𝑅 ) × ( Base ‘ 𝑅 ) ) ) ) = ( MetOpen ‘ ( ( dist ‘ 𝑅 ) ↾ ( ( Base ‘ 𝑅 ) × ( Base ‘ 𝑅 ) ) ) ) |
12 |
11
|
methaus |
⊢ ( ( ( dist ‘ 𝑅 ) ↾ ( ( Base ‘ 𝑅 ) × ( Base ‘ 𝑅 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑅 ) ) → ( MetOpen ‘ ( ( dist ‘ 𝑅 ) ↾ ( ( Base ‘ 𝑅 ) × ( Base ‘ 𝑅 ) ) ) ) ∈ Haus ) |
13 |
5 10 12
|
3syl |
⊢ ( 𝑅 ∈ ℝExt → ( MetOpen ‘ ( ( dist ‘ 𝑅 ) ↾ ( ( Base ‘ 𝑅 ) × ( Base ‘ 𝑅 ) ) ) ) ∈ Haus ) |
14 |
9 13
|
eqeltrd |
⊢ ( 𝑅 ∈ ℝExt → 𝐾 ∈ Haus ) |