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Axiom ax-hvmul0 28786
Description: Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 28802 and hvsubval 28792). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
Assertion
Ref Expression
ax-hvmul0 (𝐴 ∈ ℋ → (0 · 𝐴) = 0)

Detailed syntax breakdown of Axiom ax-hvmul0
StepHypRef Expression
1 cA . . 3 class 𝐴
2 chba 28695 . . 3 class
31, 2wcel 2110 . 2 wff 𝐴 ∈ ℋ
4 cc0 10536 . . . 4 class 0
5 csm 28697 . . . 4 class ·
64, 1, 5co 7155 . . 3 class (0 · 𝐴)
7 c0v 28700 . . 3 class 0
86, 7wceq 1533 . 2 wff (0 · 𝐴) = 0
93, 8wi 4 1 wff (𝐴 ∈ ℋ → (0 · 𝐴) = 0)
Colors of variables: wff setvar class
This axiom is referenced by:  hvmul0  28800  hvmul0or  28801  hvsubid  28802  hi01  28872  h1de2ctlem  29331  spansneleq  29346  h1datomi  29357
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