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Theorem remetdval 20482
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 6193 . . 3  |-  ( A D B )  =  ( D `  <. A ,  B >. )
2 remet.1 . . . 4  |-  D  =  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )
32fveq1i 5790 . . 3  |-  ( D `
 <. A ,  B >. )  =  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) `  <. A ,  B >. )
41, 3eqtri 2480 . 2  |-  ( A D B )  =  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) `
 <. A ,  B >. )
5 opelxpi 4969 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
<. A ,  B >.  e.  ( RR  X.  RR ) )
6 fvres 5803 . . . 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 6193 . . . 4  |-  ( A ( abs  o.  -  ) B )  =  ( ( abs  o.  -  ) `  <. A ,  B >. )
9 recn 9473 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
10 recn 9473 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
11 eqid 2451 . . . . . 6  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1211cnmetdval 20466 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A ( abs 
o.  -  ) B
)  =  ( abs `  ( A  -  B
) ) )
139, 10, 12syl2an 477 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A ( abs 
o.  -  ) B
)  =  ( abs `  ( A  -  B
) ) )
148, 13syl5eqr 2506 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( abs  o.  -  ) `  <. A ,  B >. )  =  ( abs `  ( A  -  B )
) )
157, 14eqtrd 2492 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) `
 <. A ,  B >. )  =  ( abs `  ( A  -  B
) ) )
164, 15syl5eq 2504 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A D B )  =  ( abs `  ( A  -  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   <.cop 3981    X. cxp 4936    |` cres 4940    o. ccom 4942   ` cfv 5516  (class class class)co 6190   CCcc 9381   RRcr 9382    - cmin 9696   abscabs 12825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-po 4739  df-so 4740  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-ltxr 9524  df-sub 9698
This theorem is referenced by:  bl2ioo  20485  xrsdsre  20503  reconnlem2  20520  rrxdstprj1  21024  dvlip2  21583  nmcvcn  24225  rrndstprj1  28867  rrndstprj2  28868  rrncmslem  28869  ismrer1  28875
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