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Theorem remetdval 18295
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 5861 . . 3  |-  ( A D B )  =  ( D `  <. A ,  B >. )
2 remet.1 . . . 4  |-  D  =  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )
32fveq1i 5526 . . 3  |-  ( D `
 <. A ,  B >. )  =  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) `  <. A ,  B >. )
41, 3eqtri 2303 . 2  |-  ( A D B )  =  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) `
 <. A ,  B >. )
5 opelxpi 4721 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
6 fvres 5542 . . . 4  |-  ( <. A ,  B >.  e.  ( RR  X.  RR )  ->  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) `  <. A ,  B >. )  =  ( ( abs 
o.  -  ) `  <. A ,  B >. ) )
75, 6syl 15 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) `
 <. A ,  B >. )  =  ( ( abs  o.  -  ) `  <. A ,  B >. ) )
8 df-ov 5861 . . . 4  |-  ( A ( abs  o.  -  ) B )  =  ( ( abs  o.  -  ) `  <. A ,  B >. )
9 recn 8827 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
10 recn 8827 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
11 eqid 2283 . . . . . 6  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1211cnmetdval 18280 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ( abs 
o.  -  ) B
)  =  ( abs `  ( A  -  B
) ) )
139, 10, 12syl2an 463 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A ( abs 
o.  -  ) B
)  =  ( abs `  ( A  -  B
) ) )
148, 13syl5eqr 2329 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs  o.  -  ) `  <. A ,  B >. )  =  ( abs `  ( A  -  B )
) )
157, 14eqtrd 2315 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) `
 <. A ,  B >. )  =  ( abs `  ( A  -  B
) ) )
164, 15syl5eq 2327 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A D B )  =  ( abs `  ( A  -  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643    X. cxp 4687    |` cres 4691    o. ccom 4693   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736    - cmin 9037   abscabs 11719
This theorem is referenced by:  bl2ioo  18298  xrsdsre  18316  reconnlem2  18332  dvlip2  19342  nmcvcn  21268  rrndstprj1  26554  rrndstprj2  26555  rrncmslem  26556  ismrer1  26562
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039
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