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Mirrors > Home > HSE Home > Th. List > ax-hvmul0 | Structured version Visualization version GIF version |
Description: Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 30714 and hvsubval 30704). (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax-hvmul0 | ⊢ (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | chba 30607 | . . 3 class ℋ | |
3 | 1, 2 | wcel 2098 | . 2 wff 𝐴 ∈ ℋ |
4 | cc0 11105 | . . . 4 class 0 | |
5 | csm 30609 | . . . 4 class ·ℎ | |
6 | 4, 1, 5 | co 7401 | . . 3 class (0 ·ℎ 𝐴) |
7 | c0v 30612 | . . 3 class 0ℎ | |
8 | 6, 7 | wceq 1533 | . 2 wff (0 ·ℎ 𝐴) = 0ℎ |
9 | 3, 8 | wi 4 | 1 wff (𝐴 ∈ ℋ → (0 ·ℎ 𝐴) = 0ℎ) |
Colors of variables: wff setvar class |
This axiom is referenced by: hvmul0 30712 hvmul0or 30713 hvsubid 30714 hi01 30784 h1de2ctlem 31243 spansneleq 31258 h1datomi 31269 |
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