Definition df-h0v 30999

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 31197. (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 30953	. 2 class 0ℎ
2		cva 30949	. . . . 5 class +ℎ
3		csm 30950	. . . . 5 class ·ℎ
4	2, 3	cop 4637	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 30952	. . . 4 class normℎ
6	4, 5	cop 4637	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 30617	. . 3 class 0vec
8	6, 7	cfv 6563	. 2 class (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
9	1, 8	wceq 1537	1 wff 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)

This definition is referenced by:  axhv0cl-zf  31014  axhvaddid-zf  31015  axhvmul0-zf  31021  axhis4-zf  31026