Step |
Hyp |
Ref |
Expression |
1 |
|
tmsbas.k |
⊢ 𝐾 = ( toMetSp ‘ 𝐷 ) |
2 |
1
|
tmsds |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐷 = ( dist ‘ 𝐾 ) ) |
3 |
1
|
tmsbas |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝑋 = ( Base ‘ 𝐾 ) ) |
4 |
3
|
fveq2d |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( ∞Met ‘ 𝑋 ) = ( ∞Met ‘ ( Base ‘ 𝐾 ) ) ) |
5 |
2 4
|
eleq12d |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ↔ ( dist ‘ 𝐾 ) ∈ ( ∞Met ‘ ( Base ‘ 𝐾 ) ) ) ) |
6 |
5
|
ibi |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( dist ‘ 𝐾 ) ∈ ( ∞Met ‘ ( Base ‘ 𝐾 ) ) ) |
7 |
|
ssid |
⊢ ( Base ‘ 𝐾 ) ⊆ ( Base ‘ 𝐾 ) |
8 |
|
xmetres2 |
⊢ ( ( ( dist ‘ 𝐾 ) ∈ ( ∞Met ‘ ( Base ‘ 𝐾 ) ) ∧ ( Base ‘ 𝐾 ) ⊆ ( Base ‘ 𝐾 ) ) → ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝐾 ) ) ) |
9 |
6 7 8
|
sylancl |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝐾 ) ) ) |
10 |
|
xmetf |
⊢ ( ( dist ‘ 𝐾 ) ∈ ( ∞Met ‘ ( Base ‘ 𝐾 ) ) → ( dist ‘ 𝐾 ) : ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ⟶ ℝ* ) |
11 |
|
ffn |
⊢ ( ( dist ‘ 𝐾 ) : ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ⟶ ℝ* → ( dist ‘ 𝐾 ) Fn ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) |
12 |
|
fnresdm |
⊢ ( ( dist ‘ 𝐾 ) Fn ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) → ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) = ( dist ‘ 𝐾 ) ) |
13 |
6 10 11 12
|
4syl |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) = ( dist ‘ 𝐾 ) ) |
14 |
13 2
|
eqtr4d |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) = 𝐷 ) |
15 |
14
|
fveq2d |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( MetOpen ‘ ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ) = ( MetOpen ‘ 𝐷 ) ) |
16 |
|
eqid |
⊢ ( MetOpen ‘ 𝐷 ) = ( MetOpen ‘ 𝐷 ) |
17 |
1 16
|
tmstopn |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( MetOpen ‘ 𝐷 ) = ( TopOpen ‘ 𝐾 ) ) |
18 |
15 17
|
eqtr2d |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( TopOpen ‘ 𝐾 ) = ( MetOpen ‘ ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ) ) |
19 |
|
eqid |
⊢ ( TopOpen ‘ 𝐾 ) = ( TopOpen ‘ 𝐾 ) |
20 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
21 |
|
eqid |
⊢ ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) = ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) |
22 |
19 20 21
|
isxms2 |
⊢ ( 𝐾 ∈ ∞MetSp ↔ ( ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝐾 ) ) ∧ ( TopOpen ‘ 𝐾 ) = ( MetOpen ‘ ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ) ) ) |
23 |
9 18 22
|
sylanbrc |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐾 ∈ ∞MetSp ) |