Theorem shocel

Description: Membership in orthogonal complement of H subspace. (Contributed by NM, 9-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	shocel	⊢ ( 𝐻 ∈ Sℋ → ( 𝐴 ∈ ( ⊥ ‘ 𝐻 ) ↔ ( 𝐴 ∈ ℋ ∧ ∀ 𝑥 ∈ 𝐻 ( 𝐴 ·ih 𝑥 ) = 0 ) ) )

Proof

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