Definition df-h0v 30906

Description: Define the zero vector of Hilbert space. Note that 0_vec 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 31104. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Assertion

Ref	Expression
df-h0v	⊢ 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)

Detailed syntax breakdown of Definition df-h0v

Step	Hyp	Ref	Expression
1		c0v 30860	. 2 class 0ℎ
2		cva 30856	. . . . 5 class +ℎ
3		csm 30857	. . . . 5 class ·ℎ
4	2, 3	cop 4603	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 30859	. . . 4 class normℎ
6	4, 5	cop 4603	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 30524	. . 3 class 0vec
8	6, 7	cfv 6519	. 2 class (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
9	1, 8	wceq 1540	1 wff 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)

This definition is referenced by:  axhv0cl-zf  30921  axhvaddid-zf  30922  axhvmul0-zf  30928  axhis4-zf  30933