Definition df-h0v 29233

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 29431. (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 29187	. 2 class 0ℎ
2		cva 29183	. . . . 5 class +ℎ
3		csm 29184	. . . . 5 class ·ℎ
4	2, 3	cop 4564	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 29186	. . . 4 class normℎ
6	4, 5	cop 4564	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 28851	. . 3 class 0vec
8	6, 7	cfv 6418	. 2 class (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
9	1, 8	wceq 1539	1 wff 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)

This definition is referenced by:  axhv0cl-zf  29248  axhvaddid-zf  29249  axhvmul0-zf  29255  axhis4-zf  29260