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Theorem imsdval 24099
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 24098 . . . . 5  |-  ( U  e.  NrmCVec  ->  D  =  ( N  o.  M ) )
543ad2ant1 1009 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  D  =  ( N  o.  M ) )
65fveq1d 5714 . . 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 24048 . . . . 5  |-  ( U  e.  NrmCVec  ->  M : ( X  X.  X ) --> X )
9 opelxpi 4892 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  -> 
<. A ,  B >.  e.  ( X  X.  X
) )
10 fvco3 5789 . . . . 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 1183 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N  o.  M
) `  <. A ,  B >. )  =  ( N `  ( M `
 <. A ,  B >. ) ) )
136, 12eqtrd 2475 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( D `  <. A ,  B >. )  =  ( N `  ( M `
 <. A ,  B >. ) ) )
14 df-ov 6115 . 2  |-  ( A D B )  =  ( D `  <. A ,  B >. )
15 df-ov 6115 . . 3  |-  ( A M B )  =  ( M `  <. A ,  B >. )
1615fveq2i 5715 . 2  |-  ( N `
 ( A M B ) )  =  ( N `  ( M `  <. A ,  B >. ) )
1713, 14, 163eqtr4g 2500 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 965    = wceq 1369    e. wcel 1756   <.cop 3904    X. cxp 4859    o. ccom 4865   -->wf 5435   ` cfv 5439  (class class class)co 6112   NrmCVeccnv 23984   BaseSetcba 23986   -vcnsb 23989   normCVcnmcv 23990   IndMetcims 23991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-po 4662  df-so 4663  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-ltxr 9444  df-sub 9618  df-neg 9619  df-grpo 23700  df-gid 23701  df-ginv 23702  df-gdiv 23703  df-ablo 23791  df-vc 23946  df-nv 23992  df-va 23995  df-ba 23996  df-sm 23997  df-0v 23998  df-vs 23999  df-nmcv 24000  df-ims 24001
This theorem is referenced by:  imsdval2  24100  nvnd  24101  nvelbl  24106  vacn  24111  smcnlem  24114  sspimsval  24160  blometi  24225  blocnilem  24226  ubthlem2  24294  minvecolem2  24298  minvecolem4  24303  minvecolem5  24304  minvecolem6  24305  h2hmetdval  24402  hhssmetdval  24701
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