HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  df-h0v Structured version   Visualization version   GIF version

Definition df-h0v 27955
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 28153. (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 27909 . 2 class 0
2 cva 27905 . . . . 5 class +
3 csm 27906 . . . . 5 class ·
42, 3cop 4216 . . . 4 class ⟨ + , ·
5 cno 27908 . . . 4 class norm
64, 5cop 4216 . . 3 class ⟨⟨ + , · ⟩, norm
7 cn0v 27571 . . 3 class 0vec
86, 7cfv 5926 . 2 class (0vec‘⟨⟨ + , · ⟩, norm⟩)
91, 8wceq 1523 1 wff 0 = (0vec‘⟨⟨ + , · ⟩, norm⟩)
Colors of variables: wff setvar class
This definition is referenced by:  axhv0cl-zf  27970  axhvaddid-zf  27971  axhvmul0-zf  27977  axhis4-zf  27982
  Copyright terms: Public domain W3C validator