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Theorem imsdval 21257
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 21256 . . . . 5  |-  ( U  e.  NrmCVec  ->  D  =  ( N  o.  M ) )
543ad2ant1 976 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  D  =  ( N  o.  M ) )
65fveq1d 5529 . . 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 21206 . . . . 5  |-  ( U  e.  NrmCVec  ->  M : ( X  X.  X ) --> X )
9 opelxpi 4723 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  -> 
<. A ,  B >.  e.  ( X  X.  X
) )
10 fvco3 5598 . . . . 5  |-  ( ( M : ( X  X.  X ) --> X  /\  <. A ,  B >.  e.  ( X  X.  X ) )  -> 
( ( N  o.  M ) `  <. A ,  B >. )  =  ( N `  ( M `  <. A ,  B >. ) ) )
118, 9, 10syl2an 463 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( N  o.  M ) `  <. A ,  B >. )  =  ( N `
 ( M `  <. A ,  B >. ) ) )
12113impb 1147 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N  o.  M
) `  <. A ,  B >. )  =  ( N `  ( M `
 <. A ,  B >. ) ) )
136, 12eqtrd 2317 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( D `  <. A ,  B >. )  =  ( N `  ( M `
 <. A ,  B >. ) ) )
14 df-ov 5863 . 2  |-  ( A D B )  =  ( D `  <. A ,  B >. )
15 df-ov 5863 . . 3  |-  ( A M B )  =  ( M `  <. A ,  B >. )
1615fveq2i 5530 . 2  |-  ( N `
 ( A M B ) )  =  ( N `  ( M `  <. A ,  B >. ) )
1713, 14, 163eqtr4g 2342 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( N `  ( A M B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686   <.cop 3645    X. cxp 4689    o. ccom 4695   -->wf 5253   ` cfv 5257  (class class class)co 5860   NrmCVeccnv 21142   BaseSetcba 21144   -vcnsb 21147   normCVcnmcv 21148   IndMetcims 21149
This theorem is referenced by:  imsdval2  21258  nvnd  21259  nvelbl  21264  vacn  21269  smcnlem  21272  sspimsval  21318  blometi  21383  blocnilem  21384  ubthlem2  21452  minvecolem2  21456  minvecolem4  21461  minvecolem5  21462  minvecolem6  21463  h2hmetdval  21560  hhssmetdval  21857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-ltxr 8874  df-sub 9041  df-neg 9042  df-grpo 20860  df-gid 20861  df-ginv 20862  df-gdiv 20863  df-ablo 20951  df-vc 21104  df-nv 21150  df-va 21153  df-ba 21154  df-sm 21155  df-0v 21156  df-vs 21157  df-nmcv 21158  df-ims 21159
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