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Theorem imsdval 26330
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 26329 . . . . 5  |-  ( U  e.  NrmCVec  ->  D  =  ( N  o.  M ) )
543ad2ant1 1030 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  D  =  ( N  o.  M ) )
65fveq1d 5872 . . 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 26279 . . . . 5  |-  ( U  e.  NrmCVec  ->  M : ( X  X.  X ) --> X )
9 opelxpi 4869 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  -> 
<. A ,  B >.  e.  ( X  X.  X
) )
10 fvco3 5947 . . . . 5  |-  ( ( M : ( X  X.  X ) --> X  /\  <. A ,  B >.  e.  ( X  X.  X ) )  -> 
( ( N  o.  M ) `  <. A ,  B >. )  =  ( N `  ( M `  <. A ,  B >. ) ) )
118, 9, 10syl2an 480 . . . 4  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( N  o.  M ) `  <. A ,  B >. )  =  ( N `
 ( M `  <. A ,  B >. ) ) )
12113impb 1205 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
( N  o.  M
) `  <. A ,  B >. )  =  ( N `  ( M `
 <. A ,  B >. ) ) )
136, 12eqtrd 2487 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( D `  <. A ,  B >. )  =  ( N `  ( M `
 <. A ,  B >. ) ) )
14 df-ov 6298 . 2  |-  ( A D B )  =  ( D `  <. A ,  B >. )
15 df-ov 6298 . . 3  |-  ( A M B )  =  ( M `  <. A ,  B >. )
1615fveq2i 5873 . 2  |-  ( N `
 ( A M B ) )  =  ( N `  ( M `  <. A ,  B >. ) )
1713, 14, 163eqtr4g 2512 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 371    /\ w3a 986    = wceq 1446    e. wcel 1889   <.cop 3976    X. cxp 4835    o. ccom 4841   -->wf 5581   ` cfv 5585  (class class class)co 6295   NrmCVeccnv 26215   BaseSetcba 26217   -vcnsb 26220   normCVcnmcv 26221   IndMetcims 26222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-po 4758  df-so 4759  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-ltxr 9685  df-sub 9867  df-neg 9868  df-grpo 25931  df-gid 25932  df-ginv 25933  df-gdiv 25934  df-ablo 26022  df-vc 26177  df-nv 26223  df-va 26226  df-ba 26227  df-sm 26228  df-0v 26229  df-vs 26230  df-nmcv 26231  df-ims 26232
This theorem is referenced by:  imsdval2  26331  nvnd  26332  nvelbl  26337  vacn  26342  smcnlem  26345  sspimsval  26391  blometi  26456  blocnilem  26457  ubthlem2  26525  minvecolem2  26529  minvecolem4  26534  minvecolem5  26535  minvecolem6  26536  minvecolem2OLD  26539  minvecolem4OLD  26544  minvecolem5OLD  26545  minvecolem6OLD  26546  h2hmetdval  26643  hhssmetdval  26941
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