Definition df-h0v 30990

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 31188. (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 30944	. 2 class 0ℎ
2		cva 30940	. . . . 5 class +ℎ
3		csm 30941	. . . . 5 class ·ℎ
4	2, 3	cop 4631	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 30943	. . . 4 class normℎ
6	4, 5	cop 4631	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 30608	. . 3 class 0vec
8	6, 7	cfv 6560	. 2 class (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)
9	1, 8	wceq 1539	1 wff 0ℎ = (0vec‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)

This definition is referenced by:  axhv0cl-zf  31005  axhvaddid-zf  31006  axhvmul0-zf  31012  axhis4-zf  31017