Definition df-h0v 30223

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 30421. (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 30177	. 2 class 0ℎ
2		cva 30173	. . . . 5 class +ℎ
3		csm 30174	. . . . 5 class ·ℎ
4	2, 3	cop 4635	. . . 4 class ⟨ +ℎ , ·ℎ ⟩
5		cno 30176	. . . 4 class normℎ
6	4, 5	cop 4635	. . 3 class ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩
7		cn0v 29841	. . 3 class 0vec
8	6, 7	cfv 6544	. 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  30238  axhvaddid-zf  30239  axhvmul0-zf  30245  axhis4-zf  30250