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Theorem imsdval 26925
 Description: Value of the induced metric (distance function) of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
imsdval.1 𝑋 = (BaseSet‘𝑈)
imsdval.3 𝑀 = ( −𝑣𝑈)
imsdval.6 𝑁 = (normCV𝑈)
imsdval.8 𝐷 = (IndMet‘𝑈)
Assertion
Ref Expression
imsdval ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝑀𝐵)))

Proof of Theorem imsdval
StepHypRef Expression
1 imsdval.3 . . . . . 6 𝑀 = ( −𝑣𝑈)
2 imsdval.6 . . . . . 6 𝑁 = (normCV𝑈)
3 imsdval.8 . . . . . 6 𝐷 = (IndMet‘𝑈)
41, 2, 3imsval 26924 . . . . 5 (𝑈 ∈ NrmCVec → 𝐷 = (𝑁𝑀))
543ad2ant1 1075 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → 𝐷 = (𝑁𝑀))
65fveq1d 6105 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐷‘⟨𝐴, 𝐵⟩) = ((𝑁𝑀)‘⟨𝐴, 𝐵⟩))
7 imsdval.1 . . . . . 6 𝑋 = (BaseSet‘𝑈)
87, 1nvmf 26884 . . . . 5 (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋)
9 opelxpi 5072 . . . . 5 ((𝐴𝑋𝐵𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
10 fvco3 6185 . . . . 5 ((𝑀:(𝑋 × 𝑋)⟶𝑋 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋)) → ((𝑁𝑀)‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
118, 9, 10syl2an 493 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋)) → ((𝑁𝑀)‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
12113impb 1252 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝑀)‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
136, 12eqtrd 2644 . 2 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐷‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
14 df-ov 6552 . 2 (𝐴𝐷𝐵) = (𝐷‘⟨𝐴, 𝐵⟩)
15 df-ov 6552 . . 3 (𝐴𝑀𝐵) = (𝑀‘⟨𝐴, 𝐵⟩)
1615fveq2i 6106 . 2 (𝑁‘(𝐴𝑀𝐵)) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩))
1713, 14, 163eqtr4g 2669 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝑀𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   × cxp 5036   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  NrmCVeccnv 26823  BaseSetcba 26825   −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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  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-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  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-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-ltxr 9958  df-sub 10147  df-neg 10148  df-grpo 26731  df-gid 26732  df-ginv 26733  df-gdiv 26734  df-ablo 26783  df-vc 26798  df-nv 26831  df-va 26834  df-ba 26835  df-sm 26836  df-0v 26837  df-vs 26838  df-nmcv 26839  df-ims 26840 This theorem is referenced by:  imsdval2  26926  nvnd  26927  vacn  26933  smcnlem  26936  sspimsval  26977  blometi  27042  blocnilem  27043  ubthlem2  27111  minvecolem2  27115  minvecolem4  27120  minvecolem5  27121  minvecolem6  27122  h2hmetdval  27219  hhssmetdval  27519
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