Theorem obsocv

Description: An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015)

	Ref	Expression
Hypotheses	obsocv.z	⊢ 0 = ( 0g ‘ 𝑊 )
	obsocv.o	⊢ ⊥ = ( ocv ‘ 𝑊 )
Assertion	obsocv	⊢ ( 𝐵 ∈ ( OBasis ‘ 𝑊 ) → ( ⊥ ‘ 𝐵 ) = { 0 } )

Proof

Step	Hyp	Ref	Expression
1		obsocv.z	⊢ 0 = ( 0g ‘ 𝑊 )
2		obsocv.o	⊢ ⊥ = ( ocv ‘ 𝑊 )
3		eqid	⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
4		eqid	⊢ ( ·𝑖 ‘ 𝑊 ) = ( ·𝑖 ‘ 𝑊 )
5		eqid	⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
6		eqid	⊢ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) = ( 1r ‘ ( Scalar ‘ 𝑊 ) )
7		eqid	⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) )
8	3 4 5 6 7 2 1	isobs	⊢ ( 𝐵 ∈ ( OBasis ‘ 𝑊 ) ↔ ( 𝑊 ∈ PreHil ∧ 𝐵 ⊆ ( Base ‘ 𝑊 ) ∧ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = if ( 𝑥 = 𝑦 , ( 1r ‘ ( Scalar ‘ 𝑊 ) ) , ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ ( ⊥ ‘ 𝐵 ) = { 0 } ) ) )
9	8	simp3bi	⊢ ( 𝐵 ∈ ( OBasis ‘ 𝑊 ) → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = if ( 𝑥 = 𝑦 , ( 1r ‘ ( Scalar ‘ 𝑊 ) ) , ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ ( ⊥ ‘ 𝐵 ) = { 0 } ) )
10	9	simprd	⊢ ( 𝐵 ∈ ( OBasis ‘ 𝑊 ) → ( ⊥ ‘ 𝐵 ) = { 0 } )