Theorem shscomi

Description: Commutative law for subspace sum. (Contributed by NM, 17-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	shincl.1	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
	shincl.2	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
Assertion	shscomi	$$⊢A{+}_{\mathscr{H}}B=B{+}_{\mathscr{H}}A$$

Proof

Step	Hyp	Ref	Expression
1		shincl.1	$$⊢A\in {\mathbf{S}}_{\mathscr{H}}$$
2		shincl.2	$$⊢B\in {\mathbf{S}}_{\mathscr{H}}$$
3		shscom	$$⊢\left(A\in {\mathbf{S}}_{\mathscr{H}}\wedge B\in {\mathbf{S}}_{\mathscr{H}}\right)\to A{+}_{\mathscr{H}}B=B{+}_{\mathscr{H}}A$$
4	1 2 3	mp2an	$$⊢A{+}_{\mathscr{H}}B=B{+}_{\mathscr{H}}A$$