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

Theorem imsval 26693
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 5987 . . . 4 (𝑢 = 𝑈 → (normCV𝑢) = (normCV𝑈))
2 fveq2 5987 . . . 4 (𝑢 = 𝑈 → ( −𝑣𝑢) = ( −𝑣𝑈))
31, 2coeq12d 5100 . . 3 (𝑢 = 𝑈 → ((normCV𝑢) ∘ ( −𝑣𝑢)) = ((normCV𝑈) ∘ ( −𝑣𝑈)))
4 df-ims 26596 . . 3 IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV𝑢) ∘ ( −𝑣𝑢)))
5 fvex 5997 . . . 4 (normCV𝑈) ∈ V
6 fvex 5997 . . . 4 ( −𝑣𝑈) ∈ V
75, 6coex 6886 . . 3 ((normCV𝑈) ∘ ( −𝑣𝑈)) ∈ V
83, 4, 7fvmpt 6075 . 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 5099 . 2 (𝑁𝑀) = ((normCV𝑈) ∘ ( −𝑣𝑈))
138, 9, 123eqtr4g 2573 1 (𝑈 ∈ NrmCVec → 𝐷 = (𝑁𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1938  ccom 4936  cfv 5689  NrmCVeccnv 26579  𝑣 cnsb 26584  normCVcnmcv 26585  IndMetcims 26586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-iota 5653  df-fun 5691  df-fv 5697  df-ims 26596
This theorem is referenced by:  imsdval  26694  imsdf  26697  cnims  26705  hhims  27201  hhssims  27304
  Copyright terms: Public domain W3C validator