MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  remetdval Structured version   Unicode version

Theorem remetdval 21586
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 6281 . . 3  |-  ( A D B )  =  ( D `  <. A ,  B >. )
2 remet.1 . . . 4  |-  D  =  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )
32fveq1i 5850 . . 3  |-  ( D `
 <. A ,  B >. )  =  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) `  <. A ,  B >. )
41, 3eqtri 2431 . 2  |-  ( A D B )  =  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) `
 <. A ,  B >. )
5 opelxpi 4855 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
6 fvres 5863 . . . 4  |-  ( <. A ,  B >.  e.  ( RR  X.  RR )  ->  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) `  <. A ,  B >. )  =  ( ( abs 
o.  -  ) `  <. A ,  B >. ) )
75, 6syl 17 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) `
 <. A ,  B >. )  =  ( ( abs  o.  -  ) `  <. A ,  B >. ) )
8 df-ov 6281 . . . 4  |-  ( A ( abs  o.  -  ) B )  =  ( ( abs  o.  -  ) `  <. A ,  B >. )
9 recn 9612 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
10 recn 9612 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
11 eqid 2402 . . . . . 6  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1211cnmetdval 21570 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ( abs 
o.  -  ) B
)  =  ( abs `  ( A  -  B
) ) )
139, 10, 12syl2an 475 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A ( abs 
o.  -  ) B
)  =  ( abs `  ( A  -  B
) ) )
148, 13syl5eqr 2457 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs  o.  -  ) `  <. A ,  B >. )  =  ( abs `  ( A  -  B )
) )
157, 14eqtrd 2443 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) `
 <. A ,  B >. )  =  ( abs `  ( A  -  B
) ) )
164, 15syl5eq 2455 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A D B )  =  ( abs `  ( A  -  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   <.cop 3978    X. cxp 4821    |` cres 4825    o. ccom 4827   ` cfv 5569  (class class class)co 6278   CCcc 9520   RRcr 9521    - cmin 9841   abscabs 13216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-ltxr 9663  df-sub 9843
This theorem is referenced by:  bl2ioo  21589  xrsdsre  21607  reconnlem2  21624  rrxdstprj1  22128  dvlip2  22688  nmcvcn  26019  rrndstprj1  31608  rrndstprj2  31609  rrncmslem  31610  ismrer1  31616
  Copyright terms: Public domain W3C validator