MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imsval Structured version   Visualization version   GIF version

Theorem imsval 28468
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.)
Hypotheses
Ref Expression
imsval.3 𝑀 = ( −𝑣𝑈)
imsval.6 𝑁 = (normCV𝑈)
imsval.8 𝐷 = (IndMet‘𝑈)
Assertion
Ref Expression
imsval (𝑈 ∈ NrmCVec → 𝐷 = (𝑁𝑀))

Proof of Theorem imsval
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6645 . . . 4 (𝑢 = 𝑈 → (normCV𝑢) = (normCV𝑈))
2 fveq2 6645 . . . 4 (𝑢 = 𝑈 → ( −𝑣𝑢) = ( −𝑣𝑈))
31, 2coeq12d 5699 . . 3 (𝑢 = 𝑈 → ((normCV𝑢) ∘ ( −𝑣𝑢)) = ((normCV𝑈) ∘ ( −𝑣𝑈)))
4 df-ims 28384 . . 3 IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV𝑢) ∘ ( −𝑣𝑢)))
5 fvex 6658 . . . 4 (normCV𝑈) ∈ V
6 fvex 6658 . . . 4 ( −𝑣𝑈) ∈ V
75, 6coex 7617 . . 3 ((normCV𝑈) ∘ ( −𝑣𝑈)) ∈ V
83, 4, 7fvmpt 6745 . 2 (𝑈 ∈ NrmCVec → (IndMet‘𝑈) = ((normCV𝑈) ∘ ( −𝑣𝑈)))
9 imsval.8 . 2 𝐷 = (IndMet‘𝑈)
10 imsval.6 . . 3 𝑁 = (normCV𝑈)
11 imsval.3 . . 3 𝑀 = ( −𝑣𝑈)
1210, 11coeq12i 5698 . 2 (𝑁𝑀) = ((normCV𝑈) ∘ ( −𝑣𝑈))
138, 9, 123eqtr4g 2858 1 (𝑈 ∈ NrmCVec → 𝐷 = (𝑁𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2111  ccom 5523  cfv 6324  NrmCVeccnv 28367  𝑣 cnsb 28372  normCVcnmcv 28373  IndMetcims 28374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fv 6332  df-ims 28384
This theorem is referenced by:  imsdval  28469  imsdf  28472  cnims  28476  hhims  28955  hhssims  29057
  Copyright terms: Public domain W3C validator