Definition df-h0v 30218

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 30416. (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 30172	. 2 class 0ℎ
2		cva 30168	. . . . 5 class +ℎ
3		csm 30169	. . . . 5 class ·ℎ
4	2, 3	cop 4634	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 30171	. . . 4 class normℎ
6	4, 5	cop 4634	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 29836	. . 3 class 0vec
8	6, 7	cfv 6543	. 2 class (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
9	1, 8	wceq 1541	1 wff 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)

This definition is referenced by:  axhv0cl-zf  30233  axhvaddid-zf  30234  axhvmul0-zf  30240  axhis4-zf  30245