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Metamath Proof Explorer

Theorem cfilss

Description: A filter finer than a Cauchy filter is Cauchy. (Contributed by Mario Carneiro, 13-Oct-2015)

Ref Expression
Assertion cfilss ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) ∧ ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ) → 𝐺 ∈ ( CauFil ‘ 𝐷 ) )

Proof

Step Hyp Ref Expression
1 simprl ⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) ∧ ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ) → 𝐺 ∈ ( Fil ‘ 𝑋 ) )
2 simprr ⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) ∧ ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ) → 𝐹 ⊆ 𝐺 )
3 iscfil ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐹 ∈ ( CauFil ‘ 𝐷 ) ↔ ( 𝐹 ∈ ( Fil ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) )
4 3 simplbda ⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) )
5 4 adantr ⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) ∧ ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) )
6 ssrexv ⊢ ( 𝐹 ⊆ 𝐺 → ( ∃ 𝑦 ∈ 𝐹 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) → ∃ 𝑦 ∈ 𝐺 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) )
7 6 ralimdv ⊢ ( 𝐹 ⊆ 𝐺 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐹 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐺 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) )
8 2 5 7 sylc ⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) ∧ ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐺 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) )
9 iscfil ⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐺 ∈ ( CauFil ‘ 𝐷 ) ↔ ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐺 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) )
10 9 ad2antrr ⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) ∧ ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ) → ( 𝐺 ∈ ( CauFil ‘ 𝐷 ) ↔ ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐺 ( 𝐷 “ ( 𝑦 × 𝑦 ) ) ⊆ ( 0 [,) 𝑥 ) ) ) )
11 1 8 10 mpbir2and ⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( CauFil ‘ 𝐷 ) ) ∧ ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 ⊆ 𝐺 ) ) → 𝐺 ∈ ( CauFil ‘ 𝐷 ) )