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Theorem imsval 30433
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 = ( βˆ’π‘£ β€˜π‘ˆ)
imsval.6 𝑁 = (normCVβ€˜π‘ˆ)
imsval.8 𝐷 = (IndMetβ€˜π‘ˆ)
Assertion
Ref Expression
imsval (π‘ˆ ∈ NrmCVec β†’ 𝐷 = (𝑁 ∘ 𝑀))
Proof of Theorem imsval
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6882 . . . 4 (𝑒 = π‘ˆ β†’ (normCVβ€˜π‘’) = (normCVβ€˜π‘ˆ))
2 fveq2 6882 . . . 4 (𝑒 = π‘ˆ β†’ ( βˆ’π‘£ β€˜π‘’) = ( βˆ’π‘£ β€˜π‘ˆ))
31, 2coeq12d 5855 . . 3 (𝑒 = π‘ˆ β†’ ((normCVβ€˜π‘’) ∘ ( βˆ’π‘£ β€˜π‘’)) = ((normCVβ€˜π‘ˆ) ∘ ( βˆ’π‘£ β€˜π‘ˆ)))
4 df-ims 30349 . . 3 IndMet = (𝑒 ∈ NrmCVec ↦ ((normCVβ€˜π‘’) ∘ ( βˆ’π‘£ β€˜π‘’)))
5 fvex 6895 . . . 4 (normCVβ€˜π‘ˆ) ∈ V
6 fvex 6895 . . . 4 ( βˆ’π‘£ β€˜π‘ˆ) ∈ V
75, 6coex 7915 . . 3 ((normCVβ€˜π‘ˆ) ∘ ( βˆ’π‘£ β€˜π‘ˆ)) ∈ V
83, 4, 7fvmpt 6989 . 2 (π‘ˆ ∈ NrmCVec β†’ (IndMetβ€˜π‘ˆ) = ((normCVβ€˜π‘ˆ) ∘ ( βˆ’π‘£ β€˜π‘ˆ)))
9 imsval.8 . 2 𝐷 = (IndMetβ€˜π‘ˆ)
10 imsval.6 . . 3 𝑁 = (normCVβ€˜π‘ˆ)
11 imsval.3 . . 3 𝑀 = ( βˆ’π‘£ β€˜π‘ˆ)
1210, 11coeq12i 5854 . 2 (𝑁 ∘ 𝑀) = ((normCVβ€˜π‘ˆ) ∘ ( βˆ’π‘£ β€˜π‘ˆ))
138, 9, 123eqtr4g 2789 1 (π‘ˆ ∈ NrmCVec β†’ 𝐷 = (𝑁 ∘ 𝑀))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098   ∘ ccom 5671  β€˜cfv 6534  NrmCVeccnv 30332   βˆ’π‘£ cnsb 30337  normCVcnmcv 30338  IndMetcims 30339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fv 6542  df-ims 30349
This theorem is referenced by:  imsdval  30434  imsdf  30437  cnims  30441  hhims  30920  hhssims  31022
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