Metamath Proof Explorer


Theorem iscnrm3

Description: A completely normal topology is a topology in which two separated sets can be separated by open neighborhoods. (Contributed by Zhi Wang, 5-Sep-2024)

Ref Expression
Assertion iscnrm3 ( 𝐽 ∈ CNrm ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛𝐽𝑚𝐽 ( 𝑠𝑛𝑡𝑚 ∧ ( 𝑛𝑚 ) = ∅ ) ) ) )

Proof

Step Hyp Ref Expression
1 eqid 𝐽 = 𝐽
2 1 iscnrm ( 𝐽 ∈ CNrm ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑧 ∈ 𝒫 𝐽 ( 𝐽t 𝑧 ) ∈ Nrm ) )
3 iscnrm3lem1 ( 𝐽 ∈ Top → ( ∀ 𝑧 ∈ 𝒫 𝐽𝑐 ∈ ( Clsd ‘ ( 𝐽t 𝑧 ) ) ∀ 𝑑 ∈ ( Clsd ‘ ( 𝐽t 𝑧 ) ) ( ( 𝑐𝑑 ) = ∅ → ∃ 𝑙 ∈ ( 𝐽t 𝑧 ) ∃ 𝑘 ∈ ( 𝐽t 𝑧 ) ( 𝑐𝑙𝑑𝑘 ∧ ( 𝑙𝑘 ) = ∅ ) ) ↔ ∀ 𝑧 ∈ 𝒫 𝐽 ( ( 𝐽t 𝑧 ) ∈ Top ∧ ∀ 𝑐 ∈ ( Clsd ‘ ( 𝐽t 𝑧 ) ) ∀ 𝑑 ∈ ( Clsd ‘ ( 𝐽t 𝑧 ) ) ( ( 𝑐𝑑 ) = ∅ → ∃ 𝑙 ∈ ( 𝐽t 𝑧 ) ∃ 𝑘 ∈ ( 𝐽t 𝑧 ) ( 𝑐𝑙𝑑𝑘 ∧ ( 𝑙𝑘 ) = ∅ ) ) ) ) )
4 isnrm3 ( ( 𝐽t 𝑧 ) ∈ Nrm ↔ ( ( 𝐽t 𝑧 ) ∈ Top ∧ ∀ 𝑐 ∈ ( Clsd ‘ ( 𝐽t 𝑧 ) ) ∀ 𝑑 ∈ ( Clsd ‘ ( 𝐽t 𝑧 ) ) ( ( 𝑐𝑑 ) = ∅ → ∃ 𝑙 ∈ ( 𝐽t 𝑧 ) ∃ 𝑘 ∈ ( 𝐽t 𝑧 ) ( 𝑐𝑙𝑑𝑘 ∧ ( 𝑙𝑘 ) = ∅ ) ) ) )
5 4 ralbii ( ∀ 𝑧 ∈ 𝒫 𝐽 ( 𝐽t 𝑧 ) ∈ Nrm ↔ ∀ 𝑧 ∈ 𝒫 𝐽 ( ( 𝐽t 𝑧 ) ∈ Top ∧ ∀ 𝑐 ∈ ( Clsd ‘ ( 𝐽t 𝑧 ) ) ∀ 𝑑 ∈ ( Clsd ‘ ( 𝐽t 𝑧 ) ) ( ( 𝑐𝑑 ) = ∅ → ∃ 𝑙 ∈ ( 𝐽t 𝑧 ) ∃ 𝑘 ∈ ( 𝐽t 𝑧 ) ( 𝑐𝑙𝑑𝑘 ∧ ( 𝑙𝑘 ) = ∅ ) ) ) )
6 3 5 bitr4di ( 𝐽 ∈ Top → ( ∀ 𝑧 ∈ 𝒫 𝐽𝑐 ∈ ( Clsd ‘ ( 𝐽t 𝑧 ) ) ∀ 𝑑 ∈ ( Clsd ‘ ( 𝐽t 𝑧 ) ) ( ( 𝑐𝑑 ) = ∅ → ∃ 𝑙 ∈ ( 𝐽t 𝑧 ) ∃ 𝑘 ∈ ( 𝐽t 𝑧 ) ( 𝑐𝑙𝑑𝑘 ∧ ( 𝑙𝑘 ) = ∅ ) ) ↔ ∀ 𝑧 ∈ 𝒫 𝐽 ( 𝐽t 𝑧 ) ∈ Nrm ) )
7 iscnrm3r ( 𝐽 ∈ Top → ( ∀ 𝑧 ∈ 𝒫 𝐽𝑐 ∈ ( Clsd ‘ ( 𝐽t 𝑧 ) ) ∀ 𝑑 ∈ ( Clsd ‘ ( 𝐽t 𝑧 ) ) ( ( 𝑐𝑑 ) = ∅ → ∃ 𝑙 ∈ ( 𝐽t 𝑧 ) ∃ 𝑘 ∈ ( 𝐽t 𝑧 ) ( 𝑐𝑙𝑑𝑘 ∧ ( 𝑙𝑘 ) = ∅ ) ) → ( ( 𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽 ) → ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛𝐽𝑚𝐽 ( 𝑠𝑛𝑡𝑚 ∧ ( 𝑛𝑚 ) = ∅ ) ) ) ) )
8 iscnrm3l ( 𝐽 ∈ Top → ( ∀ 𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛𝐽𝑚𝐽 ( 𝑠𝑛𝑡𝑚 ∧ ( 𝑛𝑚 ) = ∅ ) ) → ( ( 𝑧 ∈ 𝒫 𝐽𝑐 ∈ ( Clsd ‘ ( 𝐽t 𝑧 ) ) ∧ 𝑑 ∈ ( Clsd ‘ ( 𝐽t 𝑧 ) ) ) → ( ( 𝑐𝑑 ) = ∅ → ∃ 𝑙 ∈ ( 𝐽t 𝑧 ) ∃ 𝑘 ∈ ( 𝐽t 𝑧 ) ( 𝑐𝑙𝑑𝑘 ∧ ( 𝑙𝑘 ) = ∅ ) ) ) ) )
9 7 8 iscnrm3lem2 ( 𝐽 ∈ Top → ( ∀ 𝑧 ∈ 𝒫 𝐽𝑐 ∈ ( Clsd ‘ ( 𝐽t 𝑧 ) ) ∀ 𝑑 ∈ ( Clsd ‘ ( 𝐽t 𝑧 ) ) ( ( 𝑐𝑑 ) = ∅ → ∃ 𝑙 ∈ ( 𝐽t 𝑧 ) ∃ 𝑘 ∈ ( 𝐽t 𝑧 ) ( 𝑐𝑙𝑑𝑘 ∧ ( 𝑙𝑘 ) = ∅ ) ) ↔ ∀ 𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛𝐽𝑚𝐽 ( 𝑠𝑛𝑡𝑚 ∧ ( 𝑛𝑚 ) = ∅ ) ) ) )
10 6 9 bitr3d ( 𝐽 ∈ Top → ( ∀ 𝑧 ∈ 𝒫 𝐽 ( 𝐽t 𝑧 ) ∈ Nrm ↔ ∀ 𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛𝐽𝑚𝐽 ( 𝑠𝑛𝑡𝑚 ∧ ( 𝑛𝑚 ) = ∅ ) ) ) )
11 10 pm5.32i ( ( 𝐽 ∈ Top ∧ ∀ 𝑧 ∈ 𝒫 𝐽 ( 𝐽t 𝑧 ) ∈ Nrm ) ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛𝐽𝑚𝐽 ( 𝑠𝑛𝑡𝑚 ∧ ( 𝑛𝑚 ) = ∅ ) ) ) )
12 2 11 bitri ( 𝐽 ∈ CNrm ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑠 ∈ 𝒫 𝐽𝑡 ∈ 𝒫 𝐽 ( ( ( 𝑠 ∩ ( ( cls ‘ 𝐽 ) ‘ 𝑡 ) ) = ∅ ∧ ( ( ( cls ‘ 𝐽 ) ‘ 𝑠 ) ∩ 𝑡 ) = ∅ ) → ∃ 𝑛𝐽𝑚𝐽 ( 𝑠𝑛𝑡𝑚 ∧ ( 𝑛𝑚 ) = ∅ ) ) ) )