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Metamath Proof Explorer

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 ∈ S_ℋ → ( 𝐴 ∈ ( ⊥ ‘ 𝐻 ) ↔ ( 𝐴 ∈ ℋ ∧ ∀ 𝑥 ∈ 𝐻 ( 𝐴 ·_ih 𝑥 ) = 0 ) ) )

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