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Metamath Proof Explorer

Theorem shscomi

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

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

Proof

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