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Theorem imsdval 26750
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 26749 . . . . 5 (𝑈 ∈ NrmCVec → 𝐷 = (𝑁𝑀))
543ad2ant1 1074 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → 𝐷 = (𝑁𝑀))
65fveq1d 6090 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐷‘⟨𝐴, 𝐵⟩) = ((𝑁𝑀)‘⟨𝐴, 𝐵⟩))
7 imsdval.1 . . . . . 6 𝑋 = (BaseSet‘𝑈)
87, 1nvmf 26699 . . . . 5 (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋)
9 opelxpi 5062 . . . . 5 ((𝐴𝑋𝐵𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
10 fvco3 6170 . . . . 5 ((𝑀:(𝑋 × 𝑋)⟶𝑋 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋)) → ((𝑁𝑀)‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
118, 9, 10syl2an 492 . . . 4 ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋)) → ((𝑁𝑀)‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
12113impb 1251 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝑀)‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
136, 12eqtrd 2643 . 2 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐷‘⟨𝐴, 𝐵⟩) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩)))
14 df-ov 6530 . 2 (𝐴𝐷𝐵) = (𝐷‘⟨𝐴, 𝐵⟩)
15 df-ov 6530 . . 3 (𝐴𝑀𝐵) = (𝑀‘⟨𝐴, 𝐵⟩)
1615fveq2i 6091 . 2 (𝑁‘(𝐴𝑀𝐵)) = (𝑁‘(𝑀‘⟨𝐴, 𝐵⟩))
1713, 14, 163eqtr4g 2668 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝑀𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  cop 4130   × cxp 5026  ccom 5032  wf 5786  cfv 5790  (class class class)co 6527  NrmCVeccnv 26635  BaseSetcba 26637  𝑣 cnsb 26640  normCVcnmcv 26641  IndMetcims 26642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-po 4949  df-so 4950  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7037  df-2nd 7038  df-er 7607  df-en 7820  df-dom 7821  df-sdom 7822  df-pnf 9933  df-mnf 9934  df-ltxr 9936  df-sub 10120  df-neg 10121  df-grpo 26525  df-gid 26526  df-ginv 26527  df-gdiv 26528  df-ablo 26580  df-vc 26595  df-nv 26643  df-va 26646  df-ba 26647  df-sm 26648  df-0v 26649  df-vs 26650  df-nmcv 26651  df-ims 26652
This theorem is referenced by:  imsdval2  26751  nvnd  26752  nvelbl  26757  vacn  26762  smcnlem  26765  sspimsval  26811  blometi  26876  blocnilem  26877  ubthlem2  26945  minvecolem2  26949  minvecolem4  26954  minvecolem5  26955  minvecolem6  26956  h2hmetdval  27053  hhssmetdval  27353
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