Definition df-h0v 30994

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 31192. (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 30948	. 2 class 0ℎ
2		cva 30944	. . . . 5 class +ℎ
3		csm 30945	. . . . 5 class ·ℎ
4	2, 3	cop 4584	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 30947	. . . 4 class normℎ
6	4, 5	cop 4584	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 30612	. . 3 class 0vec
8	6, 7	cfv 6490	. 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  31009  axhvaddid-zf  31010  axhvmul0-zf  31016  axhis4-zf  31021