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Definition df-h0v 28739
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 28937. (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 28693 . 2 class 0
2 cva 28689 . . . . 5 class +
3 csm 28690 . . . . 5 class ·
42, 3cop 4565 . . . 4 class ⟨ + , ·
5 cno 28692 . . . 4 class norm
64, 5cop 4565 . . 3 class ⟨⟨ + , · ⟩, norm
7 cn0v 28357 . . 3 class 0vec
86, 7cfv 6348 . 2 class (0vec‘⟨⟨ + , · ⟩, norm⟩)
91, 8wceq 1530 1 wff 0 = (0vec‘⟨⟨ + , · ⟩, norm⟩)
Colors of variables: wff setvar class
This definition is referenced by:  axhv0cl-zf  28754  axhvaddid-zf  28755  axhvmul0-zf  28761  axhis4-zf  28766
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