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 ) )