Definition df-h0v 31045

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 31243. (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 30999	. 2 class 0ℎ
2		cva 30995	. . . . 5 class +ℎ
3		csm 30996	. . . . 5 class ·ℎ
4	2, 3	cop 4586	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 30998	. . . 4 class normℎ
6	4, 5	cop 4586	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 30663	. . 3 class 0vec
8	6, 7	cfv 6492	. 2 class (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
9	1, 8	wceq 1541	1 wff 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)

This definition is referenced by:  axhv0cl-zf  31060  axhvaddid-zf  31061  axhvmul0-zf  31067  axhis4-zf  31072