Theorem shscomi

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

	Ref	Expression
Hypotheses	shincl.1	⊢ 𝐴 ∈ Sℋ
	shincl.2	⊢ 𝐵 ∈ Sℋ
Assertion	shscomi	⊢ ( 𝐴 +ℋ 𝐵 ) = ( 𝐵 +ℋ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		shincl.1	⊢ 𝐴 ∈ Sℋ
2		shincl.2	⊢ 𝐵 ∈ Sℋ
3		shscom	⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐴 +ℋ 𝐵 ) = ( 𝐵 +ℋ 𝐴 ) )
4	1 2 3	mp2an	⊢ ( 𝐴 +ℋ 𝐵 ) = ( 𝐵 +ℋ 𝐴 )