Definition df-h0v 29341

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 29539. (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 29295	. 2 class 0ℎ
2		cva 29291	. . . . 5 class +ℎ
3		csm 29292	. . . . 5 class ·ℎ
4	2, 3	cop 4573	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 29294	. . . 4 class normℎ
6	4, 5	cop 4573	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 28959	. . 3 class 0vec
8	6, 7	cfv 6432	. 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  29356  axhvaddid-zf  29357  axhvmul0-zf  29363  axhis4-zf  29368