Theorem css1

Description: The whole space is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015)

	Ref	Expression
Hypotheses	css1.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	css1.c	⊢ 𝐶 = ( ClSubSp ‘ 𝑊 )
Assertion	css1	⊢ ( 𝑊 ∈ PreHil → 𝑉 ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		css1.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		css1.c	⊢ 𝐶 = ( ClSubSp ‘ 𝑊 )
3		eqid	⊢ ( ocv ‘ 𝑊 ) = ( ocv ‘ 𝑊 )
4	1 3	ocv0	⊢ ( ( ocv ‘ 𝑊 ) ‘ ∅ ) = 𝑉
5		0ss	⊢ ∅ ⊆ 𝑉
6	1 2 3	ocvcss	⊢ ( ( 𝑊 ∈ PreHil ∧ ∅ ⊆ 𝑉 ) → ( ( ocv ‘ 𝑊 ) ‘ ∅ ) ∈ 𝐶 )
7	5 6	mpan2	⊢ ( 𝑊 ∈ PreHil → ( ( ocv ‘ 𝑊 ) ‘ ∅ ) ∈ 𝐶 )
8	4 7	eqeltrrid	⊢ ( 𝑊 ∈ PreHil → 𝑉 ∈ 𝐶 )