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Metamath Proof Explorer

Axiom ax-hvcom

Description: Vector addition is commutative. (Contributed by NM, 3-Sep-1999) (New usage is discouraged.)

Ref Expression
Assertion ax-hvcom ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 +ℎ 𝐵 ) = ( 𝐵 +ℎ 𝐴 ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA ⊢ 𝐴
1 chba ⊢ ℋ
2 0 1 wcel ⊢ 𝐴 ∈ ℋ
3 cB ⊢ 𝐵
4 3 1 wcel ⊢ 𝐵 ∈ ℋ
5 2 4 wa ⊢ ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ )
6 cva ⊢ +ℎ
7 0 3 6 co ⊢ ( 𝐴 +ℎ 𝐵 )
8 3 0 6 co ⊢ ( 𝐵 +ℎ 𝐴 )
9 7 8 wceq ⊢ ( 𝐴 +ℎ 𝐵 ) = ( 𝐵 +ℎ 𝐴 )
10 5 9 wi ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 +ℎ 𝐵 ) = ( 𝐵 +ℎ 𝐴 ) )