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Theorem imsval 28229
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 6493 . . . 4 (𝑢 = 𝑈 → (normCV𝑢) = (normCV𝑈))
2 fveq2 6493 . . . 4 (𝑢 = 𝑈 → ( −𝑣𝑢) = ( −𝑣𝑈))
31, 2coeq12d 5578 . . 3 (𝑢 = 𝑈 → ((normCV𝑢) ∘ ( −𝑣𝑢)) = ((normCV𝑈) ∘ ( −𝑣𝑈)))
4 df-ims 28145 . . 3 IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV𝑢) ∘ ( −𝑣𝑢)))
5 fvex 6506 . . . 4 (normCV𝑈) ∈ V
6 fvex 6506 . . . 4 ( −𝑣𝑈) ∈ V
75, 6coex 7444 . . 3 ((normCV𝑈) ∘ ( −𝑣𝑈)) ∈ V
83, 4, 7fvmpt 6589 . 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 5577 . 2 (𝑁𝑀) = ((normCV𝑈) ∘ ( −𝑣𝑈))
138, 9, 123eqtr4g 2833 1 (𝑈 ∈ NrmCVec → 𝐷 = (𝑁𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2048  ccom 5404  cfv 6182  NrmCVeccnv 28128  𝑣 cnsb 28133  normCVcnmcv 28134  IndMetcims 28135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-iota 6146  df-fun 6184  df-fv 6190  df-ims 28145
This theorem is referenced by:  imsdval  28230  imsdf  28233  cnims  28237  hhims  28718  hhssims  28821
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