Definition df-h0v 31002

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 31200. (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 30956	. 2 class 0ℎ
2		cva 30952	. . . . 5 class +ℎ
3		csm 30953	. . . . 5 class ·ℎ
4	2, 3	cop 4654	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 30955	. . . 4 class normℎ
6	4, 5	cop 4654	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 30620	. . 3 class 0vec
8	6, 7	cfv 6573	. 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  31017  axhvaddid-zf  31018  axhvmul0-zf  31024  axhis4-zf  31029