Definition df-h0v 30899

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 31097. (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 30853	. 2 class 0ℎ
2		cva 30849	. . . . 5 class +ℎ
3		csm 30850	. . . . 5 class ·ℎ
4	2, 3	cop 4595	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 30852	. . . 4 class normℎ
6	4, 5	cop 4595	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 30517	. . 3 class 0vec
8	6, 7	cfv 6511	. 2 class (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
9	1, 8	wceq 1540	1 wff 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)

This definition is referenced by:  axhv0cl-zf  30914  axhvaddid-zf  30915  axhvmul0-zf  30921  axhis4-zf  30926