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Theorem imsdval 25719
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 25718 . . . . 5  |-  ( U  e.  NrmCVec  ->  D  =  ( N  o.  M ) )
543ad2ant1 1017 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  D  =  ( N  o.  M ) )
65fveq1d 5874 . . 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 25668 . . . . 5  |-  ( U  e.  NrmCVec  ->  M : ( X  X.  X ) --> X )
9 opelxpi 5040 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  -> 
<. A ,  B >.  e.  ( X  X.  X
) )
10 fvco3 5950 . . . . 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 1192 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N  o.  M
) `  <. A ,  B >. )  =  ( N `  ( M `
 <. A ,  B >. ) ) )
136, 12eqtrd 2498 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( D `  <. A ,  B >. )  =  ( N `  ( M `
 <. A ,  B >. ) ) )
14 df-ov 6299 . 2  |-  ( A D B )  =  ( D `  <. A ,  B >. )
15 df-ov 6299 . . 3  |-  ( A M B )  =  ( M `  <. A ,  B >. )
1615fveq2i 5875 . 2  |-  ( N `
 ( A M B ) )  =  ( N `  ( M `  <. A ,  B >. ) )
1713, 14, 163eqtr4g 2523 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 973    = wceq 1395    e. wcel 1819   <.cop 4038    X. cxp 5006    o. ccom 5012   -->wf 5590   ` cfv 5594  (class class class)co 6296   NrmCVeccnv 25604   BaseSetcba 25606   -vcnsb 25609   normCVcnmcv 25610   IndMetcims 25611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-ltxr 9650  df-sub 9826  df-neg 9827  df-grpo 25320  df-gid 25321  df-ginv 25322  df-gdiv 25323  df-ablo 25411  df-vc 25566  df-nv 25612  df-va 25615  df-ba 25616  df-sm 25617  df-0v 25618  df-vs 25619  df-nmcv 25620  df-ims 25621
This theorem is referenced by:  imsdval2  25720  nvnd  25721  nvelbl  25726  vacn  25731  smcnlem  25734  sspimsval  25780  blometi  25845  blocnilem  25846  ubthlem2  25914  minvecolem2  25918  minvecolem4  25923  minvecolem5  25924  minvecolem6  25925  h2hmetdval  26022  hhssmetdval  26321
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