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Metamath Proof Explorer

Theorem oc0

Description: The zero vector belongs to an orthogonal complement of a Hilbert subspace. (Contributed by NM, 11-Oct-1999) (New usage is discouraged.)

Ref Expression
Assertion oc0 ⊢ H ∈ S ℋ → 0 ℎ ∈ ⊥ ⁡ H

Proof

Step Hyp Ref Expression
1 shocsh ⊢ H ∈ S ℋ → ⊥ ⁡ H ∈ S ℋ
2 sh0 ⊢ ⊥ ⁡ H ∈ S ℋ → 0 ℎ ∈ ⊥ ⁡ H
3 1 2 syl ⊢ H ∈ S ℋ → 0 ℎ ∈ ⊥ ⁡ H