Theorem ocvss

Description: The orthocomplement of a subset is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015)

	Ref	Expression
Hypotheses	ocvss.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	ocvss.o	⊢ ⊥ = ( ocv ‘ 𝑊 )
Assertion	ocvss	⊢ ( ⊥ ‘ 𝑆 ) ⊆ 𝑉

Proof

Step	Hyp	Ref	Expression
1		ocvss.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		ocvss.o	⊢ ⊥ = ( ocv ‘ 𝑊 )
3		eqid	⊢ ( ·𝑖 ‘ 𝑊 ) = ( ·𝑖 ‘ 𝑊 )
4		eqid	⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
5		eqid	⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) )
6	1 3 4 5 2	elocv	⊢ ( 𝑥 ∈ ( ⊥ ‘ 𝑆 ) ↔ ( 𝑆 ⊆ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( ·𝑖 ‘ 𝑊 ) 𝑦 ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ) )
7	6	simp2bi	⊢ ( 𝑥 ∈ ( ⊥ ‘ 𝑆 ) → 𝑥 ∈ 𝑉 )
8	7	ssriv	⊢ ( ⊥ ‘ 𝑆 ) ⊆ 𝑉