Definition df-h0v 28749

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 28947. (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 28703	. 2 class 0ℎ
2		cva 28699	. . . . 5 class +ℎ
3		csm 28700	. . . . 5 class ·ℎ
4	2, 3	cop 4575	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 28702	. . . 4 class normℎ
6	4, 5	cop 4575	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 28367	. . 3 class 0vec
8	6, 7	cfv 6357	. 2 class (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
9	1, 8	wceq 1537	1 wff 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)

This definition is referenced by:  axhv0cl-zf  28764  axhvaddid-zf  28765  axhvmul0-zf  28771  axhis4-zf  28776