Axiom ax-hvcom 31029

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

Assertion

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

Detailed syntax breakdown of Axiom ax-hvcom

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		chba 30947	. . . 4 class ℋ
3	1, 2	wcel 2105	. . 3 wff 𝐴 ∈ ℋ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2105	. . 3 wff 𝐵 ∈ ℋ
6	3, 5	wa 395	. 2 wff (𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ)
7		cva 30948	. . . 4 class +ℎ
8	1, 4, 7	co 7430	. . 3 class (𝐴 +ℎ 𝐵)
9	4, 1, 7	co 7430	. . 3 class (𝐵 +ℎ 𝐴)
10	8, 9	wceq 1536	. 2 wff (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴)
11	6, 10	wi 4	1 wff ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴))

This axiom is referenced by:  hvcomi  31047  hvaddlid  31051  hvadd32  31062  hvadd12  31063  hvpncan2  31068  hvsub32  31073  hvaddcan2  31099  hilablo  31188  hhssabloi  31290  shscom  31347  pjhtheu2  31444  pjpjpre  31447  pjpo  31456  spanunsni  31607  chscllem4  31668  hoaddcomi  31800  pjimai  32204  superpos  32382  sumdmdii  32443  cdj3lem3  32466  cdj3lem3b  32468