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Theorem imsval 30508
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 6897 . . . 4 (𝑒 = π‘ˆ β†’ (normCVβ€˜π‘’) = (normCVβ€˜π‘ˆ))
2 fveq2 6897 . . . 4 (𝑒 = π‘ˆ β†’ ( βˆ’π‘£ β€˜π‘’) = ( βˆ’π‘£ β€˜π‘ˆ))
31, 2coeq12d 5867 . . 3 (𝑒 = π‘ˆ β†’ ((normCVβ€˜π‘’) ∘ ( βˆ’π‘£ β€˜π‘’)) = ((normCVβ€˜π‘ˆ) ∘ ( βˆ’π‘£ β€˜π‘ˆ)))
4 df-ims 30424 . . 3 IndMet = (𝑒 ∈ NrmCVec ↦ ((normCVβ€˜π‘’) ∘ ( βˆ’π‘£ β€˜π‘’)))
5 fvex 6910 . . . 4 (normCVβ€˜π‘ˆ) ∈ V
6 fvex 6910 . . . 4 ( βˆ’π‘£ β€˜π‘ˆ) ∈ V
75, 6coex 7938 . . 3 ((normCVβ€˜π‘ˆ) ∘ ( βˆ’π‘£ β€˜π‘ˆ)) ∈ V
83, 4, 7fvmpt 7005 . 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 5866 . 2 (𝑁 ∘ 𝑀) = ((normCVβ€˜π‘ˆ) ∘ ( βˆ’π‘£ β€˜π‘ˆ))
138, 9, 123eqtr4g 2793 1 (π‘ˆ ∈ NrmCVec β†’ 𝐷 = (𝑁 ∘ 𝑀))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099   ∘ ccom 5682  β€˜cfv 6548  NrmCVeccnv 30407   βˆ’π‘£ cnsb 30412  normCVcnmcv 30413  IndMetcims 30414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fv 6556  df-ims 30424
This theorem is referenced by:  imsdval  30509  imsdf  30512  cnims  30516  hhims  30995  hhssims  31097
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