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Metamath Proof Explorer

Theorem hvcomi

Description: Commutation of vector addition. (Contributed by NM, 3-Sep-1999) (New usage is discouraged.)

Ref Expression
Hypotheses hvaddcl.1 ⊢ 𝐴 ∈ ℋ
hvaddcl.2 ⊢ 𝐵 ∈ ℋ
Assertion hvcomi ( 𝐴 +ℎ 𝐵 ) = ( 𝐵 +ℎ 𝐴 )

Proof

Step Hyp Ref Expression
1 hvaddcl.1 ⊢ 𝐴 ∈ ℋ
2 hvaddcl.2 ⊢ 𝐵 ∈ ℋ
3 ax-hvcom ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 +ℎ 𝐵 ) = ( 𝐵 +ℎ 𝐴 ) )
4 1 2 3 mp2an ⊢ ( 𝐴 +ℎ 𝐵 ) = ( 𝐵 +ℎ 𝐴 )