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Theorem imsval 29044
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 6776 . . . 4 (𝑢 = 𝑈 → (normCV𝑢) = (normCV𝑈))
2 fveq2 6776 . . . 4 (𝑢 = 𝑈 → ( −𝑣𝑢) = ( −𝑣𝑈))
31, 2coeq12d 5775 . . 3 (𝑢 = 𝑈 → ((normCV𝑢) ∘ ( −𝑣𝑢)) = ((normCV𝑈) ∘ ( −𝑣𝑈)))
4 df-ims 28960 . . 3 IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV𝑢) ∘ ( −𝑣𝑢)))
5 fvex 6789 . . . 4 (normCV𝑈) ∈ V
6 fvex 6789 . . . 4 ( −𝑣𝑈) ∈ V
75, 6coex 7777 . . 3 ((normCV𝑈) ∘ ( −𝑣𝑈)) ∈ V
83, 4, 7fvmpt 6877 . 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 5774 . 2 (𝑁𝑀) = ((normCV𝑈) ∘ ( −𝑣𝑈))
138, 9, 123eqtr4g 2803 1 (𝑈 ∈ NrmCVec → 𝐷 = (𝑁𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  ccom 5595  cfv 6435  NrmCVeccnv 28943  𝑣 cnsb 28948  normCVcnmcv 28949  IndMetcims 28950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5225  ax-nul 5232  ax-pow 5290  ax-pr 5354  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-br 5077  df-opab 5139  df-mpt 5160  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-iota 6393  df-fun 6437  df-fv 6443  df-ims 28960
This theorem is referenced by:  imsdval  29045  imsdf  29048  cnims  29052  hhims  29531  hhssims  29633
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