Axiom ax-hvass

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

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

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		cC	⊢ 𝐶
6	5 1	wcel	⊢ 𝐶 ∈ ℋ
7	2 4 6	w3a	⊢ ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ )
8		cva	⊢ +ℎ
9	0 3 8	co	⊢ ( 𝐴 +ℎ 𝐵 )
10	9 5 8	co	⊢ ( ( 𝐴 +ℎ 𝐵 ) +ℎ 𝐶 )
11	3 5 8	co	⊢ ( 𝐵 +ℎ 𝐶 )
12	0 11 8	co	⊢ ( 𝐴 +ℎ ( 𝐵 +ℎ 𝐶 ) )
13	10 12	wceq	⊢ ( ( 𝐴 +ℎ 𝐵 ) +ℎ 𝐶 ) = ( 𝐴 +ℎ ( 𝐵 +ℎ 𝐶 ) )
14	7 13	wi	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) +ℎ 𝐶 ) = ( 𝐴 +ℎ ( 𝐵 +ℎ 𝐶 ) ) )