Definition df-h0v 30254

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 30452. (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 30208	. 2 class 0ℎ
2		cva 30204	. . . . 5 class +ℎ
3		csm 30205	. . . . 5 class ·ℎ
4	2, 3	cop 4635	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 30207	. . . 4 class normℎ
6	4, 5	cop 4635	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 29872	. . 3 class 0vec
8	6, 7	cfv 6544	. 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  30269  axhvaddid-zf  30270  axhvmul0-zf  30276  axhis4-zf  30281