Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
2 |
1
|
iscnrm |
⊢ ( 𝐽 ∈ CNrm ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑧 ∈ 𝒫 ∪ 𝐽 ( 𝐽 ↾t 𝑧 ) ∈ Nrm ) ) |
3 |
|
iscnrm3lem1 |
⊢ ( 𝐽 ∈ Top → ( ∀ 𝑧 ∈ 𝒫 ∪ 𝐽 ∀ 𝑐 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑧 ) ) ∀ 𝑑 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑧 ) ) ( ( 𝑐 ∩ 𝑑 ) = ∅ → ∃ 𝑙 ∈ ( 𝐽 ↾t 𝑧 ) ∃ 𝑘 ∈ ( 𝐽 ↾t 𝑧 ) ( 𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ ( 𝑙 ∩ 𝑘 ) = ∅ ) ) ↔ ∀ 𝑧 ∈ 𝒫 ∪ 𝐽 ( ( 𝐽 ↾t 𝑧 ) ∈ Top ∧ ∀ 𝑐 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑧 ) ) ∀ 𝑑 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑧 ) ) ( ( 𝑐 ∩ 𝑑 ) = ∅ → ∃ 𝑙 ∈ ( 𝐽 ↾t 𝑧 ) ∃ 𝑘 ∈ ( 𝐽 ↾t 𝑧 ) ( 𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ ( 𝑙 ∩ 𝑘 ) = ∅ ) ) ) ) ) |
4 |
|
isnrm3 |
⊢ ( ( 𝐽 ↾t 𝑧 ) ∈ Nrm ↔ ( ( 𝐽 ↾t 𝑧 ) ∈ Top ∧ ∀ 𝑐 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑧 ) ) ∀ 𝑑 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑧 ) ) ( ( 𝑐 ∩ 𝑑 ) = ∅ → ∃ 𝑙 ∈ ( 𝐽 ↾t 𝑧 ) ∃ 𝑘 ∈ ( 𝐽 ↾t 𝑧 ) ( 𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ ( 𝑙 ∩ 𝑘 ) = ∅ ) ) ) ) |
5 |
4
|
ralbii |
⊢ ( ∀ 𝑧 ∈ 𝒫 ∪ 𝐽 ( 𝐽 ↾t 𝑧 ) ∈ Nrm ↔ ∀ 𝑧 ∈ 𝒫 ∪ 𝐽 ( ( 𝐽 ↾t 𝑧 ) ∈ Top ∧ ∀ 𝑐 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑧 ) ) ∀ 𝑑 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑧 ) ) ( ( 𝑐 ∩ 𝑑 ) = ∅ → ∃ 𝑙 ∈ ( 𝐽 ↾t 𝑧 ) ∃ 𝑘 ∈ ( 𝐽 ↾t 𝑧 ) ( 𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ ( 𝑙 ∩ 𝑘 ) = ∅ ) ) ) ) |
6 |
3 5
|
bitr4di |
⊢ ( 𝐽 ∈ Top → ( ∀ 𝑧 ∈ 𝒫 ∪ 𝐽 ∀ 𝑐 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑧 ) ) ∀ 𝑑 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑧 ) ) ( ( 𝑐 ∩ 𝑑 ) = ∅ → ∃ 𝑙 ∈ ( 𝐽 ↾t 𝑧 ) ∃ 𝑘 ∈ ( 𝐽 ↾t 𝑧 ) ( 𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ ( 𝑙 ∩ 𝑘 ) = ∅ ) ) ↔ ∀ 𝑧 ∈ 𝒫 ∪ 𝐽 ( 𝐽 ↾t 𝑧 ) ∈ Nrm ) ) |
7 |
|
iscnrm3r |
⊢ ( 𝐽 ∈ Top → ( ∀ 𝑧 ∈ 𝒫 ∪ 𝐽 ∀ 𝑐 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑧 ) ) ∀ 𝑑 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑧 ) ) ( ( 𝑐 ∩ 𝑑 ) = ∅ → ∃ 𝑙 ∈ ( 𝐽 ↾t 𝑧 ) ∃ 𝑘 ∈ ( 𝐽 ↾t 𝑧 ) ( 𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ ( 𝑙 ∩ 𝑘 ) = ∅ ) ) → ( ( 𝑠 ∈ 𝒫 ∪ 𝐽 ∧ 𝑡 ∈ 𝒫 ∪ 𝐽 ) → ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) ) ) |
8 |
|
iscnrm3l |
⊢ ( 𝐽 ∈ Top → ( ∀ 𝑠 ∈ 𝒫 ∪ 𝐽 ∀ 𝑡 ∈ 𝒫 ∪ 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) → ( ( 𝑧 ∈ 𝒫 ∪ 𝐽 ∧ 𝑐 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑧 ) ) ∧ 𝑑 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑧 ) ) ) → ( ( 𝑐 ∩ 𝑑 ) = ∅ → ∃ 𝑙 ∈ ( 𝐽 ↾t 𝑧 ) ∃ 𝑘 ∈ ( 𝐽 ↾t 𝑧 ) ( 𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ ( 𝑙 ∩ 𝑘 ) = ∅ ) ) ) ) ) |
9 |
7 8
|
iscnrm3lem2 |
⊢ ( 𝐽 ∈ Top → ( ∀ 𝑧 ∈ 𝒫 ∪ 𝐽 ∀ 𝑐 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑧 ) ) ∀ 𝑑 ∈ ( Clsd ‘ ( 𝐽 ↾t 𝑧 ) ) ( ( 𝑐 ∩ 𝑑 ) = ∅ → ∃ 𝑙 ∈ ( 𝐽 ↾t 𝑧 ) ∃ 𝑘 ∈ ( 𝐽 ↾t 𝑧 ) ( 𝑐 ⊆ 𝑙 ∧ 𝑑 ⊆ 𝑘 ∧ ( 𝑙 ∩ 𝑘 ) = ∅ ) ) ↔ ∀ 𝑠 ∈ 𝒫 ∪ 𝐽 ∀ 𝑡 ∈ 𝒫 ∪ 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) ) |
10 |
6 9
|
bitr3d |
⊢ ( 𝐽 ∈ Top → ( ∀ 𝑧 ∈ 𝒫 ∪ 𝐽 ( 𝐽 ↾t 𝑧 ) ∈ Nrm ↔ ∀ 𝑠 ∈ 𝒫 ∪ 𝐽 ∀ 𝑡 ∈ 𝒫 ∪ 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) ) |
11 |
10
|
pm5.32i |
⊢ ( ( 𝐽 ∈ Top ∧ ∀ 𝑧 ∈ 𝒫 ∪ 𝐽 ( 𝐽 ↾t 𝑧 ) ∈ Nrm ) ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑠 ∈ 𝒫 ∪ 𝐽 ∀ 𝑡 ∈ 𝒫 ∪ 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) ) |
12 |
2 11
|
bitri |
⊢ ( 𝐽 ∈ CNrm ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑠 ∈ 𝒫 ∪ 𝐽 ∀ 𝑡 ∈ 𝒫 ∪ 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑠 ⊆ 𝑛 ∧ 𝑡 ⊆ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) ) |