Definition df-h0v 31059

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 31257. (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 31013	. 2 class 0ℎ
2		cva 31009	. . . . 5 class +ℎ
3		csm 31010	. . . . 5 class ·ℎ
4	2, 3	cop 4574	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 31012	. . . 4 class normℎ
6	4, 5	cop 4574	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 30677	. . 3 class 0vec
8	6, 7	cfv 6493	. 2 class (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
9	1, 8	wceq 1542	1 wff 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)

This definition is referenced by:  axhv0cl-zf  31074  axhvaddid-zf  31075  axhvmul0-zf  31081  axhis4-zf  31086