Definition df-h0v 31331

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 31529. (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 31285	. 2 class 0ℎ
2		cva 31281	. . . . 5 class +ℎ
3		csm 31282	. . . . 5 class ·ℎ
4	2, 3	cop 4595	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 31284	. . . 4 class normℎ
6	4, 5	cop 4595	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 30949	. . 3 class 0vec
8	6, 7	cfv 6536	. 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  31346  axhvaddid-zf  31347  axhvmul0-zf  31353  axhis4-zf  31358