Axiom ax-hvmul0 10512

Description: Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 10527 and hvsubval 10518).

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 10420	. . 3 class ~H
3	1, 2	wcel 1300	. 2 wff A e. ~H
4		cc0 6386	. . . 4 class 0
5		csm 10422	. . . 4 class .h
6	4, 1, 5	co 4884	. . 3 class (0 .h A)
7		c0v 10423	. . 3 class 0h
8	6, 7	wceq 1298	. 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 10525  hvmul0or 10526  hvsubid 10527  hi01 10595  h1de2ctlem 11111  spansneleq 11126  h1datomi 11137

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