Definition df-ims 30620

Description: Define the induced metric on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
df-ims	⊢ IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)))

Detailed syntax breakdown of Definition df-ims

Step	Hyp	Ref	Expression
1		cims 30610	. 2 class IndMet
2		vu	. . 3 setvar 𝑢
3		cnv 30603	. . 3 class NrmCVec
4	2	cv 1539	. . . . 5 class 𝑢
5		cnmcv 30609	. . . . 5 class normCV
6	4, 5	cfv 6561	. . . 4 class (normCV‘𝑢)
7		cnsb 30608	. . . . 5 class −𝑣
8	4, 7	cfv 6561	. . . 4 class ( −𝑣 ‘𝑢)
9	6, 8	ccom 5689	. . 3 class ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢))
10	2, 3, 9	cmpt 5225	. 2 class (𝑢 ∈ NrmCVec ↦ ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)))
11	1, 10	wceq 1540	1 wff IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)))

This definition is referenced by:  imsval  30704