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Theorem remetdval 18812
Description: Value of the distance function of the metric space of real numbers. (Contributed by NM, 16-May-2007.)
Hypothesis
Ref Expression
remet.1  |-  D  =  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )
Assertion
Ref Expression
remetdval  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A D B )  =  ( abs `  ( A  -  B
) ) )

Proof of Theorem remetdval
StepHypRef Expression
1 df-ov 6076 . . 3  |-  ( A D B )  =  ( D `  <. A ,  B >. )
2 remet.1 . . . 4  |-  D  =  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )
32fveq1i 5721 . . 3  |-  ( D `
 <. A ,  B >. )  =  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) `  <. A ,  B >. )
41, 3eqtri 2455 . 2  |-  ( A D B )  =  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) `
 <. A ,  B >. )
5 opelxpi 4902 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
6 fvres 5737 . . . 4  |-  ( <. A ,  B >.  e.  ( RR  X.  RR )  ->  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) `  <. A ,  B >. )  =  ( ( abs 
o.  -  ) `  <. A ,  B >. ) )
75, 6syl 16 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) `
 <. A ,  B >. )  =  ( ( abs  o.  -  ) `  <. A ,  B >. ) )
8 df-ov 6076 . . . 4  |-  ( A ( abs  o.  -  ) B )  =  ( ( abs  o.  -  ) `  <. A ,  B >. )
9 recn 9072 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
10 recn 9072 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
11 eqid 2435 . . . . . 6  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1211cnmetdval 18797 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ( abs 
o.  -  ) B
)  =  ( abs `  ( A  -  B
) ) )
139, 10, 12syl2an 464 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A ( abs 
o.  -  ) B
)  =  ( abs `  ( A  -  B
) ) )
148, 13syl5eqr 2481 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs  o.  -  ) `  <. A ,  B >. )  =  ( abs `  ( A  -  B )
) )
157, 14eqtrd 2467 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) `
 <. A ,  B >. )  =  ( abs `  ( A  -  B
) ) )
164, 15syl5eq 2479 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A D B )  =  ( abs `  ( A  -  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3809    X. cxp 4868    |` cres 4872    o. ccom 4874   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981    - cmin 9283   abscabs 12031
This theorem is referenced by:  bl2ioo  18815  xrsdsre  18833  reconnlem2  18850  dvlip2  19871  nmcvcn  22183  rrndstprj1  26530  rrndstprj2  26531  rrncmslem  26532  ismrer1  26538
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-ltxr 9117  df-sub 9285
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