Metamath Proof Explorer


Theorem imsval

Description: Value of the induced metric of a normed complex vector space. (Contributed by NM, 11-Sep-2007) (Revised by Mario Carneiro, 16-Nov-2013) (New usage is discouraged.)

Ref Expression
Hypotheses imsval.3 M = - v U
imsval.6 N = norm CV U
imsval.8 D = IndMet U
Assertion imsval U NrmCVec D = N M

Proof

Step Hyp Ref Expression
1 imsval.3 M = - v U
2 imsval.6 N = norm CV U
3 imsval.8 D = IndMet U
4 fveq2 u = U norm CV u = norm CV U
5 fveq2 u = U - v u = - v U
6 4 5 coeq12d u = U norm CV u - v u = norm CV U - v U
7 df-ims IndMet = u NrmCVec norm CV u - v u
8 fvex norm CV U V
9 fvex - v U V
10 8 9 coex norm CV U - v U V
11 6 7 10 fvmpt U NrmCVec IndMet U = norm CV U - v U
12 2 1 coeq12i N M = norm CV U - v U
13 11 3 12 3eqtr4g U NrmCVec D = N M