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Metamath Proof Explorer

Theorem sh0

Description: The zero vector belongs to any subspace of a Hilbert space. (Contributed by NM, 11-Oct-1999) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

Ref Expression
Assertion sh0 ⊢ H ∈ S ℋ → 0 ℎ ∈ H

Proof

Step Hyp Ref Expression
1 issh ⊢ H ∈ S ℋ ↔ H ⊆ ℋ ∧ 0 ℎ ∈ H ∧ + ℎ H × H ⊆ H ∧ ⋅ ℎ ℂ × H ⊆ H
2 1 simplbi ⊢ H ∈ S ℋ → H ⊆ ℋ ∧ 0 ℎ ∈ H
3 2 simprd ⊢ H ∈ S ℋ → 0 ℎ ∈ H