Theorem shocorth

Description: Members of a subspace and its complement are orthogonal. (Contributed by NM, 10-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	shocorth	⊢ ( 𝐻 ∈ Sℋ → ( ( 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ ( ⊥ ‘ 𝐻 ) ) → ( 𝐴 ·ih 𝐵 ) = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		shss	⊢ ( 𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ )
2		ocorth	⊢ ( 𝐻 ⊆ ℋ → ( ( 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ ( ⊥ ‘ 𝐻 ) ) → ( 𝐴 ·ih 𝐵 ) = 0 ) )
3	1 2	syl	⊢ ( 𝐻 ∈ Sℋ → ( ( 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ ( ⊥ ‘ 𝐻 ) ) → ( 𝐴 ·ih 𝐵 ) = 0 ) )