Definition df-h0v 31058

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 31256. (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 31012	. 2 class 0ℎ
2		cva 31008	. . . . 5 class +ℎ
3		csm 31009	. . . . 5 class ·ℎ
4	2, 3	cop 4588	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 31011	. . . 4 class normℎ
6	4, 5	cop 4588	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 30676	. . 3 class 0vec
8	6, 7	cfv 6500	. 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  31073  axhvaddid-zf  31074  axhvmul0-zf  31080  axhis4-zf  31085