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Metamath Proof Explorer

Theorem cfilresi

Description: A Cauchy filter on a metric subspace extends to a Cauchy filter in the larger space. (Contributed by Mario Carneiro, 15-Oct-2015)

Ref Expression
Assertion cfilresi ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → ( 𝑋 filGen 𝐹 ) ∈ ( CauFil ‘ 𝐷 ) )

Proof

Step Hyp Ref Expression
1 xmetres ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ∈ ( ∞Met ‘ ( 𝑋 ∩ 𝑌 ) ) )
2 iscfil2 ⊢ ( ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ∈ ( ∞Met ‘ ( 𝑋 ∩ 𝑌 ) ) → ( 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ↔ ( 𝐹 ∈ ( Fil ‘ ( 𝑋 ∩ 𝑌 ) ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) 𝑣 ) < 𝑥 ) ) )
3 2 simplbda ⊢ ( ( ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ∈ ( ∞Met ‘ ( 𝑋 ∩ 𝑌 ) ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) 𝑣 ) < 𝑥 )
4 1 3 sylan ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) 𝑣 ) < 𝑥 )
5 cfilfil ⊢ ( ( ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ∈ ( ∞Met ‘ ( 𝑋 ∩ 𝑌 ) ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → 𝐹 ∈ ( Fil ‘ ( 𝑋 ∩ 𝑌 ) ) )
6 1 5 sylan ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → 𝐹 ∈ ( Fil ‘ ( 𝑋 ∩ 𝑌 ) ) )
7 filelss ⊢ ( ( 𝐹 ∈ ( Fil ‘ ( 𝑋 ∩ 𝑌 ) ) ∧ 𝑦 ∈ 𝐹 ) → 𝑦 ⊆ ( 𝑋 ∩ 𝑌 ) )
8 6 7 sylan ⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) ∧ 𝑦 ∈ 𝐹 ) → 𝑦 ⊆ ( 𝑋 ∩ 𝑌 ) )
9 inss2 ⊢ ( 𝑋 ∩ 𝑌 ) ⊆ 𝑌
10 8 9 sstrdi ⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) ∧ 𝑦 ∈ 𝐹 ) → 𝑦 ⊆ 𝑌 )
11 10 sselda ⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) ∧ 𝑦 ∈ 𝐹 ) ∧ 𝑢 ∈ 𝑦 ) → 𝑢 ∈ 𝑌 )
12 10 sselda ⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) ∧ 𝑦 ∈ 𝐹 ) ∧ 𝑣 ∈ 𝑦 ) → 𝑣 ∈ 𝑌 )
13 11 12 anim12dan ⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) ∧ 𝑦 ∈ 𝐹 ) ∧ ( 𝑢 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦 ) ) → ( 𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌 ) )
14 ovres ⊢ ( ( 𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌 ) → ( 𝑢 ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) 𝑣 ) = ( 𝑢 𝐷 𝑣 ) )
15 13 14 syl ⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) ∧ 𝑦 ∈ 𝐹 ) ∧ ( 𝑢 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦 ) ) → ( 𝑢 ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) 𝑣 ) = ( 𝑢 𝐷 𝑣 ) )
16 15 breq1d ⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) ∧ 𝑦 ∈ 𝐹 ) ∧ ( 𝑢 ∈ 𝑦 ∧ 𝑣 ∈ 𝑦 ) ) → ( ( 𝑢 ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) 𝑣 ) < 𝑥 ↔ ( 𝑢 𝐷 𝑣 ) < 𝑥 ) )
17 16 2ralbidva ⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) ∧ 𝑦 ∈ 𝐹 ) → ( ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) 𝑣 ) < 𝑥 ↔ ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 𝐷 𝑣 ) < 𝑥 ) )
18 17 rexbidva ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → ( ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) 𝑣 ) < 𝑥 ↔ ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 𝐷 𝑣 ) < 𝑥 ) )
19 18 ralbidv ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) 𝑣 ) < 𝑥 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 𝐷 𝑣 ) < 𝑥 ) )
20 4 19 mpbid ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 𝐷 𝑣 ) < 𝑥 )
21 filfbas ⊢ ( 𝐹 ∈ ( Fil ‘ ( 𝑋 ∩ 𝑌 ) ) → 𝐹 ∈ ( fBas ‘ ( 𝑋 ∩ 𝑌 ) ) )
22 6 21 syl ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → 𝐹 ∈ ( fBas ‘ ( 𝑋 ∩ 𝑌 ) ) )
23 filsspw ⊢ ( 𝐹 ∈ ( Fil ‘ ( 𝑋 ∩ 𝑌 ) ) → 𝐹 ⊆ 𝒫 ( 𝑋 ∩ 𝑌 ) )
24 6 23 syl ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → 𝐹 ⊆ 𝒫 ( 𝑋 ∩ 𝑌 ) )
25 inss1 ⊢ ( 𝑋 ∩ 𝑌 ) ⊆ 𝑋
26 25 sspwi ⊢ 𝒫 ( 𝑋 ∩ 𝑌 ) ⊆ 𝒫 𝑋
27 24 26 sstrdi ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → 𝐹 ⊆ 𝒫 𝑋 )
28 elfvdm ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝑋 ∈ dom ∞Met )
29 28 adantr ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → 𝑋 ∈ dom ∞Met )
30 fbasweak ⊢ ( ( 𝐹 ∈ ( fBas ‘ ( 𝑋 ∩ 𝑌 ) ) ∧ 𝐹 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ dom ∞Met ) → 𝐹 ∈ ( fBas ‘ 𝑋 ) )
31 22 27 29 30 syl3anc ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → 𝐹 ∈ ( fBas ‘ 𝑋 ) )
32 fgcfil ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( fBas ‘ 𝑋 ) ) → ( ( 𝑋 filGen 𝐹 ) ∈ ( CauFil ‘ 𝐷 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 𝐷 𝑣 ) < 𝑥 ) )
33 31 32 syldan ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → ( ( 𝑋 filGen 𝐹 ) ∈ ( CauFil ‘ 𝐷 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ∀ 𝑢 ∈ 𝑦 ∀ 𝑣 ∈ 𝑦 ( 𝑢 𝐷 𝑣 ) < 𝑥 ) )
34 20 33 mpbird ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ ( 𝐷 ↾ ( 𝑌 × 𝑌 ) ) ) ) → ( 𝑋 filGen 𝐹 ) ∈ ( CauFil ‘ 𝐷 ) )