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Metamath Proof Explorer

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 ) )