Definition df-h0v 31041

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 31239. (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 30995	. 2 class 0ℎ
2		cva 30991	. . . . 5 class +ℎ
3		csm 30992	. . . . 5 class ·ℎ
4	2, 3	cop 4573	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 30994	. . . 4 class normℎ
6	4, 5	cop 4573	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 30659	. . 3 class 0vec
8	6, 7	cfv 6498	. 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  31056  axhvaddid-zf  31057  axhvmul0-zf  31063  axhis4-zf  31068