Theorem ocvi

Description: Property of a member of the orthocomplement of a subset. (Contributed by Mario Carneiro, 13-Oct-2015)

	Ref	Expression
Hypotheses	ocvfval.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	ocvfval.i	⊢ , = ( ·𝑖 ‘ 𝑊 )
	ocvfval.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
	ocvfval.z	⊢ 0 = ( 0g ‘ 𝐹 )
	ocvfval.o	⊢ ⊥ = ( ocv ‘ 𝑊 )
Assertion	ocvi	⊢ ( ( 𝐴 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 , 𝐵 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		ocvfval.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		ocvfval.i	⊢ , = ( ·𝑖 ‘ 𝑊 )
3		ocvfval.f	⊢ 𝐹 = ( Scalar ‘ 𝑊 )
4		ocvfval.z	⊢ 0 = ( 0g ‘ 𝐹 )
5		ocvfval.o	⊢ ⊥ = ( ocv ‘ 𝑊 )
6	1 2 3 4 5	elocv	⊢ ( 𝐴 ∈ ( ⊥ ‘ 𝑆 ) ↔ ( 𝑆 ⊆ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝑆 ( 𝐴 , 𝑥 ) = 0 ) )
7	6	simp3bi	⊢ ( 𝐴 ∈ ( ⊥ ‘ 𝑆 ) → ∀ 𝑥 ∈ 𝑆 ( 𝐴 , 𝑥 ) = 0 )
8		oveq2	⊢ ( 𝑥 = 𝐵 → ( 𝐴 , 𝑥 ) = ( 𝐴 , 𝐵 ) )
9	8	eqeq1d	⊢ ( 𝑥 = 𝐵 → ( ( 𝐴 , 𝑥 ) = 0 ↔ ( 𝐴 , 𝐵 ) = 0 ) )
10	9	rspccva	⊢ ( ( ∀ 𝑥 ∈ 𝑆 ( 𝐴 , 𝑥 ) = 0 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 , 𝐵 ) = 0 )
11	7 10	sylan	⊢ ( ( 𝐴 ∈ ( ⊥ ‘ 𝑆 ) ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 , 𝐵 ) = 0 )