Definition df-h0v 31480

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 31678. (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 31434	. 2 class 0ℎ
2		cva 31430	. . . . 5 class +ℎ
3		csm 31431	. . . . 5 class ·ℎ
4	2, 3	cop 4590	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 31433	. . . 4 class normℎ
6	4, 5	cop 4590	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 31098	. . 3 class 0vec
8	6, 7	cfv 6534	. 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  31495  axhvaddid-zf  31496  axhvmul0-zf  31502  axhis4-zf  31507