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	$$⊢H\in {\mathbf{S}}_{\mathscr{H}}\to \left(\left(A\in H\wedge B\in \perp \left(H\right)\right)\to A{\cdot }_{\mathrm{ih}}B=0\right)$$

Proof

Step	Hyp	Ref	Expression
1		shss	$$⊢H\in {\mathbf{S}}_{\mathscr{H}}\to H\subseteq \mathscr{H}$$
2		ocorth	$$⊢H\subseteq \mathscr{H}\to \left(\left(A\in H\wedge B\in \perp \left(H\right)\right)\to A{\cdot }_{\mathrm{ih}}B=0\right)$$
3	1 2	syl	$$⊢H\in {\mathbf{S}}_{\mathscr{H}}\to \left(\left(A\in H\wedge B\in \perp \left(H\right)\right)\to A{\cdot }_{\mathrm{ih}}B=0\right)$$