Axiom ax-hvass 31425

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

Assertion

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

Detailed syntax breakdown of Axiom ax-hvass

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		chba 31342	. . . 4 class ℋ
3	1, 2	wcel 2146	. . 3 wff 𝐴 ∈ ℋ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2146	. . 3 wff 𝐵 ∈ ℋ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2146	. . 3 wff 𝐶 ∈ ℋ
8	3, 5, 7	w3a 1103	. 2 wff (𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)
9		cva 31343	. . . . 5 class +ℎ
10	1, 4, 9	co 7419	. . . 4 class (𝐴 +ℎ 𝐵)
11	10, 6, 9	co 7419	. . 3 class ((𝐴 +ℎ 𝐵) +ℎ 𝐶)
12	4, 6, 9	co 7419	. . . 4 class (𝐵 +ℎ 𝐶)
13	1, 12, 9	co 7419	. . 3 class (𝐴 +ℎ (𝐵 +ℎ 𝐶))
14	11, 13	wceq 1570	. 2 wff ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶))
15	8, 14	wi 4	1 wff ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶)))

This axiom is used by:  hvadd32  31457  hvadd12  31458  hvadd4  31459  hvpncan  31462  hvaddsubass  31464  hvassi  31476  hilablo  31583  spanunsni  32002  hoaddassi  32199