Axiom ax-hvmul0 31159

Description: Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 31175 and hvsubval 31165). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)

Assertion

Ref	Expression
ax-hvmul0	⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ)

Detailed syntax breakdown of Axiom ax-hvmul0

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		chba 31068	. . 3 class ℋ
3	1, 2	wcel 2141	. 2 wff 𝐴 ∈ ℋ
4		cc0 11070	. . . 4 class 0
5		csm 31070	. . . 4 class ·ℎ
6	4, 1, 5	co 7392	. . 3 class (0 ·ℎ 𝐴)
7		c0v 31073	. . 3 class 0ℎ
8	6, 7	wceq 1559	. 2 wff (0 ·ℎ 𝐴) = 0ℎ
9	3, 8	wi 4	1 wff (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ)

This axiom is referenced by:  hvmul0  31173  hvmul0or  31174  hvsubid  31175  hi01  31245  h1de2ctlem  31704  spansneleq  31719  h1datomi  31730