Axiom ax-hvmul0 11350

Description: Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 11365 and hvsubval 11356).

Assertion

Ref	Expression
ax-hvmul0	|- (A e. ~H -> (0 .h A) = 0h)

Detailed syntax breakdown of Axiom ax-hvmul0

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2		chil 11258	. . 3 class ~H
3	1, 2	wcel 1588	. 2 wff A e. ~H
4		cc0 6752	. . . 4 class 0
5		csm 11260	. . . 4 class .h
6	4, 1, 5	co 4981	. . 3 class (0 .h A)
7		c0v 11261	. . 3 class 0h
8	6, 7	wceq 1586	. 2 wff (0 .h A) = 0h
9	3, 8	wi 3	1 wff (A e. ~H -> (0 .h A) = 0h)

This axiom is referenced by:  hvmul0 11363  hvmul0or 11364  hvsubid 11365  hi01 11432  h1de2ctlem 11945  spansneleq 11960  h1datomi 11971

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