Axiom ax-hvmul0 8801

Description: Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubidt 8816 and hvsubvalt 8807).

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 8727	. . 3 class H~
3	1, 2	wcel 955	. 2 wff A e. H~
4		cc0 5206	. . . 4 class 0
5		csm 8729	. . . 4 class .h
6	4, 1, 5	co 3948	. . 3 class (0 .h A)
7		c0v 8730	. . 3 class 0h
8	6, 7	wceq 953	. 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:  hvmul0t 8814  hvmul0ort 8815  hvsubidt 8816  hi01t 8883  h1de2ctlem 9394  spansneleq 9409  spansneleqOLD 9410  h1datom 9421

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