Step |
Hyp |
Ref |
Expression |
1 |
|
rrnheibor.1 |
⊢ 𝑋 = ( ℝ ↑m 𝐼 ) |
2 |
|
rrnheibor.2 |
⊢ 𝑀 = ( ( ℝn ‘ 𝐼 ) ↾ ( 𝑌 × 𝑌 ) ) |
3 |
|
rrnheibor.3 |
⊢ 𝑇 = ( MetOpen ‘ 𝑀 ) |
4 |
|
rrnheibor.4 |
⊢ 𝑈 = ( MetOpen ‘ ( ℝn ‘ 𝐼 ) ) |
5 |
1
|
rrnmet |
⊢ ( 𝐼 ∈ Fin → ( ℝn ‘ 𝐼 ) ∈ ( Met ‘ 𝑋 ) ) |
6 |
|
metres2 |
⊢ ( ( ( ℝn ‘ 𝐼 ) ∈ ( Met ‘ 𝑋 ) ∧ 𝑌 ⊆ 𝑋 ) → ( ( ℝn ‘ 𝐼 ) ↾ ( 𝑌 × 𝑌 ) ) ∈ ( Met ‘ 𝑌 ) ) |
7 |
2 6
|
eqeltrid |
⊢ ( ( ( ℝn ‘ 𝐼 ) ∈ ( Met ‘ 𝑋 ) ∧ 𝑌 ⊆ 𝑋 ) → 𝑀 ∈ ( Met ‘ 𝑌 ) ) |
8 |
5 7
|
sylan |
⊢ ( ( 𝐼 ∈ Fin ∧ 𝑌 ⊆ 𝑋 ) → 𝑀 ∈ ( Met ‘ 𝑌 ) ) |
9 |
8
|
biantrurd |
⊢ ( ( 𝐼 ∈ Fin ∧ 𝑌 ⊆ 𝑋 ) → ( 𝑇 ∈ Comp ↔ ( 𝑀 ∈ ( Met ‘ 𝑌 ) ∧ 𝑇 ∈ Comp ) ) ) |
10 |
3
|
heibor |
⊢ ( ( 𝑀 ∈ ( Met ‘ 𝑌 ) ∧ 𝑇 ∈ Comp ) ↔ ( 𝑀 ∈ ( CMet ‘ 𝑌 ) ∧ 𝑀 ∈ ( TotBnd ‘ 𝑌 ) ) ) |
11 |
9 10
|
bitrdi |
⊢ ( ( 𝐼 ∈ Fin ∧ 𝑌 ⊆ 𝑋 ) → ( 𝑇 ∈ Comp ↔ ( 𝑀 ∈ ( CMet ‘ 𝑌 ) ∧ 𝑀 ∈ ( TotBnd ‘ 𝑌 ) ) ) ) |
12 |
2
|
eleq1i |
⊢ ( 𝑀 ∈ ( CMet ‘ 𝑌 ) ↔ ( ( ℝn ‘ 𝐼 ) ↾ ( 𝑌 × 𝑌 ) ) ∈ ( CMet ‘ 𝑌 ) ) |
13 |
1
|
rrncms |
⊢ ( 𝐼 ∈ Fin → ( ℝn ‘ 𝐼 ) ∈ ( CMet ‘ 𝑋 ) ) |
14 |
13
|
adantr |
⊢ ( ( 𝐼 ∈ Fin ∧ 𝑌 ⊆ 𝑋 ) → ( ℝn ‘ 𝐼 ) ∈ ( CMet ‘ 𝑋 ) ) |
15 |
4
|
cmetss |
⊢ ( ( ℝn ‘ 𝐼 ) ∈ ( CMet ‘ 𝑋 ) → ( ( ( ℝn ‘ 𝐼 ) ↾ ( 𝑌 × 𝑌 ) ) ∈ ( CMet ‘ 𝑌 ) ↔ 𝑌 ∈ ( Clsd ‘ 𝑈 ) ) ) |
16 |
14 15
|
syl |
⊢ ( ( 𝐼 ∈ Fin ∧ 𝑌 ⊆ 𝑋 ) → ( ( ( ℝn ‘ 𝐼 ) ↾ ( 𝑌 × 𝑌 ) ) ∈ ( CMet ‘ 𝑌 ) ↔ 𝑌 ∈ ( Clsd ‘ 𝑈 ) ) ) |
17 |
12 16
|
syl5bb |
⊢ ( ( 𝐼 ∈ Fin ∧ 𝑌 ⊆ 𝑋 ) → ( 𝑀 ∈ ( CMet ‘ 𝑌 ) ↔ 𝑌 ∈ ( Clsd ‘ 𝑈 ) ) ) |
18 |
1 2
|
rrntotbnd |
⊢ ( 𝐼 ∈ Fin → ( 𝑀 ∈ ( TotBnd ‘ 𝑌 ) ↔ 𝑀 ∈ ( Bnd ‘ 𝑌 ) ) ) |
19 |
18
|
adantr |
⊢ ( ( 𝐼 ∈ Fin ∧ 𝑌 ⊆ 𝑋 ) → ( 𝑀 ∈ ( TotBnd ‘ 𝑌 ) ↔ 𝑀 ∈ ( Bnd ‘ 𝑌 ) ) ) |
20 |
17 19
|
anbi12d |
⊢ ( ( 𝐼 ∈ Fin ∧ 𝑌 ⊆ 𝑋 ) → ( ( 𝑀 ∈ ( CMet ‘ 𝑌 ) ∧ 𝑀 ∈ ( TotBnd ‘ 𝑌 ) ) ↔ ( 𝑌 ∈ ( Clsd ‘ 𝑈 ) ∧ 𝑀 ∈ ( Bnd ‘ 𝑌 ) ) ) ) |
21 |
11 20
|
bitrd |
⊢ ( ( 𝐼 ∈ Fin ∧ 𝑌 ⊆ 𝑋 ) → ( 𝑇 ∈ Comp ↔ ( 𝑌 ∈ ( Clsd ‘ 𝑈 ) ∧ 𝑀 ∈ ( Bnd ‘ 𝑌 ) ) ) ) |