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Theorem imsval 26924
 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 6103 . . . 4 (𝑢 = 𝑈 → (normCV𝑢) = (normCV𝑈))
2 fveq2 6103 . . . 4 (𝑢 = 𝑈 → ( −𝑣𝑢) = ( −𝑣𝑈))
31, 2coeq12d 5208 . . 3 (𝑢 = 𝑈 → ((normCV𝑢) ∘ ( −𝑣𝑢)) = ((normCV𝑈) ∘ ( −𝑣𝑈)))
4 df-ims 26840 . . 3 IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV𝑢) ∘ ( −𝑣𝑢)))
5 fvex 6113 . . . 4 (normCV𝑈) ∈ V
6 fvex 6113 . . . 4 ( −𝑣𝑈) ∈ V
75, 6coex 7011 . . 3 ((normCV𝑈) ∘ ( −𝑣𝑈)) ∈ V
83, 4, 7fvmpt 6191 . 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 5207 . 2 (𝑁𝑀) = ((normCV𝑈) ∘ ( −𝑣𝑈))
138, 9, 123eqtr4g 2669 1 (𝑈 ∈ NrmCVec → 𝐷 = (𝑁𝑀))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ∘ ccom 5042  ‘cfv 5804  NrmCVeccnv 26823   −𝑣 cnsb 26828  normCVcnmcv 26829  IndMetcims 26830 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-ims 26840 This theorem is referenced by:  imsdval  26925  imsdf  26928  cnims  26932  hhims  27413  hhssims  27516
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