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Theorem remetdval 12635
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 5745 . . 3  |-  ( A D B )  =  ( D `  <. A ,  B >. )
2 remet.1 . . . 4  |-  D  =  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )
32fveq1i 5390 . . 3  |-  ( D `
 <. A ,  B >. )  =  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) `  <. A ,  B >. )
41, 3eqtri 2138 . 2  |-  ( A D B )  =  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) `
 <. A ,  B >. )
5 opelxpi 4541 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
6 fvres 5413 . . . 4  |-  ( <. A ,  B >.  e.  ( RR  X.  RR )  ->  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) `  <. A ,  B >. )  =  ( ( abs 
o.  -  ) `  <. A ,  B >. ) )
75, 6syl 14 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) `
 <. A ,  B >. )  =  ( ( abs  o.  -  ) `  <. A ,  B >. ) )
8 df-ov 5745 . . . 4  |-  ( A ( abs  o.  -  ) B )  =  ( ( abs  o.  -  ) `  <. A ,  B >. )
9 recn 7721 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
10 recn 7721 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
11 eqid 2117 . . . . . 6  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1211cnmetdval 12625 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ( abs 
o.  -  ) B
)  =  ( abs `  ( A  -  B
) ) )
139, 10, 12syl2an 287 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A ( abs 
o.  -  ) B
)  =  ( abs `  ( A  -  B
) ) )
148, 13syl5eqr 2164 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs  o.  -  ) `  <. A ,  B >. )  =  ( abs `  ( A  -  B )
) )
157, 14eqtrd 2150 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) `
 <. A ,  B >. )  =  ( abs `  ( A  -  B
) ) )
164, 15syl5eq 2162 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A D B )  =  ( abs `  ( A  -  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   <.cop 3500    X. cxp 4507    |` cres 4511    o. ccom 4513   ` cfv 5093  (class class class)co 5742   CCcc 7586   RRcr 7587    - cmin 7901   abscabs 10737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-resscn 7680  ax-1cn 7681  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-sub 7903
This theorem is referenced by:  bl2ioo  12638
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