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Definition df-h0v 28178
Description: Define the zero vector of Hilbert space. Note that 0vec is considered a primitive in the Hilbert space axioms below, and we don't use this definition outside of this section. It is proved from the axioms as theorem hh0v 28376. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
Assertion
Ref Expression
df-h0v 0 = (0vec‘⟨⟨ + , · ⟩, norm⟩)

Detailed syntax breakdown of Definition df-h0v
StepHypRef Expression
1 c0v 28132 . 2 class 0
2 cva 28128 . . . . 5 class +
3 csm 28129 . . . . 5 class ·
42, 3cop 4387 . . . 4 class ⟨ + , ·
5 cno 28131 . . . 4 class norm
64, 5cop 4387 . . 3 class ⟨⟨ + , · ⟩, norm
7 cn0v 27794 . . 3 class 0vec
86, 7cfv 6111 . 2 class (0vec‘⟨⟨ + , · ⟩, norm⟩)
91, 8wceq 1637 1 wff 0 = (0vec‘⟨⟨ + , · ⟩, norm⟩)
Colors of variables: wff setvar class
This definition is referenced by:  axhv0cl-zf  28193  axhvaddid-zf  28194  axhvmul0-zf  28200  axhis4-zf  28205
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