Definition df-h0v 31397

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 31595. (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 31351	. 2 class 0ℎ
2		cva 31347	. . . . 5 class +ℎ
3		csm 31348	. . . . 5 class ·ℎ
4	2, 3	cop 4597	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 31350	. . . 4 class normℎ
6	4, 5	cop 4597	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 31015	. . 3 class 0vec
8	6, 7	cfv 6540	. 2 class (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
9	1, 8	wceq 1570	1 wff 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)

This definition is used by:  axhv0cl-zf  31412  axhvaddid-zf  31413  axhvmul0-zf  31419  axhis4-zf  31424