Axiom ax-hvass 8793

Description: Vector addition is associative.

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		chil 8727	. . . 4 class ℋ
3	1, 2	wcel 955	. . 3 wff A ∈ ℋ
4		cB	. . . 4 class B
5	4, 2	wcel 955	. . 3 wff B ∈ ℋ
6		cC	. . . 4 class C
7	6, 2	wcel 955	. . 3 wff C ∈ ℋ
8	3, 5, 7	w3a 773	. 2 wff (A ∈ ℋ ⋀ B ∈ ℋ ⋀ C ∈ ℋ )
9		cva 8728	. . . . 5 class +h
10	1, 4, 9	co 3948	. . . 4 class (A +h B)
11	10, 6, 9	co 3948	. . 3 class ((A +h B) +h C)
12	4, 6, 9	co 3948	. . . 4 class (B +h C)
13	1, 12, 9	co 3948	. . 3 class (A +h (B +h C))
14	11, 13	wceq 953	. 2 wff ((A +h B) +h C) = (A +h (B +h C))
15	8, 14	wi 3	1 wff ((A ∈ ℋ ⋀ B ∈ ℋ ⋀ C ∈ ℋ ) → ((A +h B) +h C) = (A +h (B +h C)))

This axiom is referenced by:  hvadd23t 8824  hvadd12t 8825  hvadd4t 8826  hvpncant 8829  hvaddsubasst 8831  hvass 8841  hilabl 8948  spanunsn 9419  hoaddass 9619

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