Definition df-h0v 31263

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 31461. (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 31217	. 2 class 0ℎ
2		cva 31213	. . . . 5 class +ℎ
3		csm 31214	. . . . 5 class ·ℎ
4	2, 3	cop 4600	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 31216	. . . 4 class normℎ
6	4, 5	cop 4600	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 30881	. . 3 class 0vec
8	6, 7	cfv 6537	. 2 class (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
9	1, 8	wceq 1567	1 wff 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)

This definition is referenced by:  axhv0cl-zf  31278  axhvaddid-zf  31279  axhvmul0-zf  31285  axhis4-zf  31290