Theorem obs2ocv

Description: The double orthocomplement (closure) of an orthonormal basis is the whole space. (Contributed by Mario Carneiro, 23-Oct-2015)

	Ref	Expression
Hypotheses	obs2ocv.o	⊢ ⊥ = ( ocv ‘ 𝑊 )
	obs2ocv.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
Assertion	obs2ocv	⊢ ( 𝐵 ∈ ( OBasis ‘ 𝑊 ) → ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) = 𝑉 )

Proof

Step	Hyp	Ref	Expression
1		obs2ocv.o	⊢ ⊥ = ( ocv ‘ 𝑊 )
2		obs2ocv.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
3		eqid	⊢ ( 0g ‘ 𝑊 ) = ( 0g ‘ 𝑊 )
4	3 1	obsocv	⊢ ( 𝐵 ∈ ( OBasis ‘ 𝑊 ) → ( ⊥ ‘ 𝐵 ) = { ( 0g ‘ 𝑊 ) } )
5	4	fveq2d	⊢ ( 𝐵 ∈ ( OBasis ‘ 𝑊 ) → ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) = ( ⊥ ‘ { ( 0g ‘ 𝑊 ) } ) )
6		obsrcl	⊢ ( 𝐵 ∈ ( OBasis ‘ 𝑊 ) → 𝑊 ∈ PreHil )
7	2 1 3	ocvz	⊢ ( 𝑊 ∈ PreHil → ( ⊥ ‘ { ( 0g ‘ 𝑊 ) } ) = 𝑉 )
8	6 7	syl	⊢ ( 𝐵 ∈ ( OBasis ‘ 𝑊 ) → ( ⊥ ‘ { ( 0g ‘ 𝑊 ) } ) = 𝑉 )
9	5 8	eqtrd	⊢ ( 𝐵 ∈ ( OBasis ‘ 𝑊 ) → ( ⊥ ‘ ( ⊥ ‘ 𝐵 ) ) = 𝑉 )