Definition df-h0v 27211

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 27409. (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 27165	. 2 class 0ℎ
2		cva 27161	. . . . 5 class +ℎ
3		csm 27162	. . . . 5 class ·ℎ
4	2, 3	cop 4131	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 27164	. . . 4 class normℎ
6	4, 5	cop 4131	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 26827	. . 3 class 0vec
8	6, 7	cfv 5804	. 2 class (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
9	1, 8	wceq 1475	1 wff 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)

This definition is referenced by:  axhv0cl-zf  27226  axhvaddid-zf  27227  axhvmul0-zf  27233  axhis4-zf  27238