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Theorem imsdval 25254
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  =  ( -v `  U
)
imsdval.6  |-  N  =  ( normCV `  U )
imsdval.8  |-  D  =  ( IndMet `  U )
Assertion
Ref Expression
imsdval  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( N `  ( A M B ) ) )

Proof of Theorem imsdval
StepHypRef Expression
1 imsdval.3 . . . . . 6  |-  M  =  ( -v `  U
)
2 imsdval.6 . . . . . 6  |-  N  =  ( normCV `  U )
3 imsdval.8 . . . . . 6  |-  D  =  ( IndMet `  U )
41, 2, 3imsval 25253 . . . . 5  |-  ( U  e.  NrmCVec  ->  D  =  ( N  o.  M ) )
543ad2ant1 1012 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  D  =  ( N  o.  M ) )
65fveq1d 5859 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( D `  <. A ,  B >. )  =  ( ( N  o.  M
) `  <. A ,  B >. ) )
7 imsdval.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
87, 1nvmf 25203 . . . . 5  |-  ( U  e.  NrmCVec  ->  M : ( X  X.  X ) --> X )
9 opelxpi 5023 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  -> 
<. A ,  B >.  e.  ( X  X.  X
) )
10 fvco3 5935 . . . . 5  |-  ( ( M : ( X  X.  X ) --> X  /\  <. A ,  B >.  e.  ( X  X.  X ) )  -> 
( ( N  o.  M ) `  <. A ,  B >. )  =  ( N `  ( M `  <. A ,  B >. ) ) )
118, 9, 10syl2an 477 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( N  o.  M ) `  <. A ,  B >. )  =  ( N `
 ( M `  <. A ,  B >. ) ) )
12113impb 1187 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N  o.  M
) `  <. A ,  B >. )  =  ( N `  ( M `
 <. A ,  B >. ) ) )
136, 12eqtrd 2501 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( D `  <. A ,  B >. )  =  ( N `  ( M `
 <. A ,  B >. ) ) )
14 df-ov 6278 . 2  |-  ( A D B )  =  ( D `  <. A ,  B >. )
15 df-ov 6278 . . 3  |-  ( A M B )  =  ( M `  <. A ,  B >. )
1615fveq2i 5860 . 2  |-  ( N `
 ( A M B ) )  =  ( N `  ( M `  <. A ,  B >. ) )
1713, 14, 163eqtr4g 2526 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( N `  ( A M B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   <.cop 4026    X. cxp 4990    o. ccom 4996   -->wf 5575   ` cfv 5579  (class class class)co 6275   NrmCVeccnv 25139   BaseSetcba 25141   -vcnsb 25144   normCVcnmcv 25145   IndMetcims 25146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9796  df-neg 9797  df-grpo 24855  df-gid 24856  df-ginv 24857  df-gdiv 24858  df-ablo 24946  df-vc 25101  df-nv 25147  df-va 25150  df-ba 25151  df-sm 25152  df-0v 25153  df-vs 25154  df-nmcv 25155  df-ims 25156
This theorem is referenced by:  imsdval2  25255  nvnd  25256  nvelbl  25261  vacn  25266  smcnlem  25269  sspimsval  25315  blometi  25380  blocnilem  25381  ubthlem2  25449  minvecolem2  25453  minvecolem4  25458  minvecolem5  25459  minvecolem6  25460  h2hmetdval  25557  hhssmetdval  25856
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