MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imsval Structured version   Visualization version   GIF version
Theorem imsval 29938
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 6892 . . . 4 (𝑒 = π‘ˆ β†’ (normCVβ€˜π‘’) = (normCVβ€˜π‘ˆ))
2 fveq2 6892 . . . 4 (𝑒 = π‘ˆ β†’ ( βˆ’π‘£ β€˜π‘’) = ( βˆ’π‘£ β€˜π‘ˆ))
31, 2coeq12d 5865 . . 3 (𝑒 = π‘ˆ β†’ ((normCVβ€˜π‘’) ∘ ( βˆ’π‘£ β€˜π‘’)) = ((normCVβ€˜π‘ˆ) ∘ ( βˆ’π‘£ β€˜π‘ˆ)))
4 df-ims 29854 . . 3 IndMet = (𝑒 ∈ NrmCVec ↦ ((normCVβ€˜π‘’) ∘ ( βˆ’π‘£ β€˜π‘’)))
5 fvex 6905 . . . 4 (normCVβ€˜π‘ˆ) ∈ V
6 fvex 6905 . . . 4 ( βˆ’π‘£ β€˜π‘ˆ) ∈ V
75, 6coex 7921 . . 3 ((normCVβ€˜π‘ˆ) ∘ ( βˆ’π‘£ β€˜π‘ˆ)) ∈ V
83, 4, 7fvmpt 6999 . 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 5864 . 2 (𝑁 ∘ 𝑀) = ((normCVβ€˜π‘ˆ) ∘ ( βˆ’π‘£ β€˜π‘ˆ))
138, 9, 123eqtr4g 2798 1 (π‘ˆ ∈ NrmCVec β†’ 𝐷 = (𝑁 ∘ 𝑀))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107   ∘ ccom 5681  β€˜cfv 6544  NrmCVeccnv 29837   βˆ’π‘£ cnsb 29842  normCVcnmcv 29843  IndMetcims 29844
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-ims 29854
This theorem is referenced by:  imsdval  29939  imsdf  29942  cnims  29946  hhims  30425  hhssims  30527
  Copyright terms: Public domain W3C validator