Definition df-h0v 31572

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 31770. (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 31526	. 2 class 0ℎ
2		cva 31522	. . . . 5 class +ℎ
3		csm 31523	. . . . 5 class ·ℎ
4	2, 3	cop 4590	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 31525	. . . 4 class normℎ
6	4, 5	cop 4590	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 31190	. . 3 class 0vec
8	6, 7	cfv 6538	. 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  31587  axhvaddid-zf  31588  axhvmul0-zf  31594  axhis4-zf  31599