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Axiom ax-hvass 27987
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 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Detailed syntax breakdown of Axiom ax-hvass
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 chil 27904 . . . 4 class
31, 2wcel 2030 . . 3 wff 𝐴 ∈ ℋ
4 cB . . . 4 class 𝐵
54, 2wcel 2030 . . 3 wff 𝐵 ∈ ℋ
6 cC . . . 4 class 𝐶
76, 2wcel 2030 . . 3 wff 𝐶 ∈ ℋ
83, 5, 7w3a 1054 . 2 wff (𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)
9 cva 27905 . . . . 5 class +
101, 4, 9co 6690 . . . 4 class (𝐴 + 𝐵)
1110, 6, 9co 6690 . . 3 class ((𝐴 + 𝐵) + 𝐶)
124, 6, 9co 6690 . . . 4 class (𝐵 + 𝐶)
131, 12, 9co 6690 . . 3 class (𝐴 + (𝐵 + 𝐶))
1411, 13wceq 1523 . 2 wff ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
158, 14wi 4 1 wff ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Colors of variables: wff setvar class
This axiom is referenced by:  hvadd32  28019  hvadd12  28020  hvadd4  28021  hvpncan  28024  hvaddsubass  28026  hvassi  28038  hilablo  28145  spanunsni  28566  hoaddassi  28763
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