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Theorem xrsdsre 20362
Description: The metric on the extended reals coincides with the usual metric on the reals. (Contributed by Mario Carneiro, 4-Sep-2015.)
Hypothesis
Ref Expression
xrsxmet.1  |-  D  =  ( dist `  RR*s
)
Assertion
Ref Expression
xrsdsre  |-  ( D  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )

Proof of Theorem xrsdsre
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xrsxmet.1 . . . . 5  |-  D  =  ( dist `  RR*s
)
21xrsdsreval 17833 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x D y )  =  ( abs `  ( x  -  y
) ) )
3 ovres 6225 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x ( D  |`  ( RR  X.  RR ) ) y )  =  ( x D y ) )
4 eqid 2438 . . . . 5  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
54remetdval 20341 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) y )  =  ( abs `  (
x  -  y ) ) )
62, 3, 53eqtr4d 2480 . . 3  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x ( D  |`  ( RR  X.  RR ) ) y )  =  ( x ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) y ) )
76rgen2a 2777 . 2  |-  A. x  e.  RR  A. y  e.  RR  ( x ( D  |`  ( RR  X.  RR ) ) y )  =  ( x ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) y )
81xrsxmet 20361 . . . . 5  |-  D  e.  ( *Met `  RR* )
9 xmetf 19879 . . . . 5  |-  ( D  e.  ( *Met ` 
RR* )  ->  D : ( RR*  X.  RR* )
--> RR* )
10 ffn 5554 . . . . 5  |-  ( D : ( RR*  X.  RR* )
--> RR*  ->  D  Fn  ( RR*  X.  RR* )
)
118, 9, 10mp2b 10 . . . 4  |-  D  Fn  ( RR*  X.  RR* )
12 rexpssxrxp 9420 . . . 4  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
13 fnssres 5519 . . . 4  |-  ( ( D  Fn  ( RR*  X. 
RR* )  /\  ( RR  X.  RR )  C_  ( RR*  X.  RR* )
)  ->  ( D  |`  ( RR  X.  RR ) )  Fn  ( RR  X.  RR ) )
1411, 12, 13mp2an 672 . . 3  |-  ( D  |`  ( RR  X.  RR ) )  Fn  ( RR  X.  RR )
15 cnmet 20326 . . . . 5  |-  ( abs 
o.  -  )  e.  ( Met `  CC )
16 metf 19880 . . . . 5  |-  ( ( abs  o.  -  )  e.  ( Met `  CC )  ->  ( abs  o.  -  ) : ( CC  X.  CC ) --> RR )
17 ffn 5554 . . . . 5  |-  ( ( abs  o.  -  ) : ( CC  X.  CC ) --> RR  ->  ( abs  o.  -  )  Fn  ( CC  X.  CC ) )
1815, 16, 17mp2b 10 . . . 4  |-  ( abs 
o.  -  )  Fn  ( CC  X.  CC )
19 ax-resscn 9331 . . . . 5  |-  RR  C_  CC
20 xpss12 4940 . . . . 5  |-  ( ( RR  C_  CC  /\  RR  C_  CC )  ->  ( RR  X.  RR )  C_  ( CC  X.  CC ) )
2119, 19, 20mp2an 672 . . . 4  |-  ( RR 
X.  RR )  C_  ( CC  X.  CC )
22 fnssres 5519 . . . 4  |-  ( ( ( abs  o.  -  )  Fn  ( CC  X.  CC )  /\  ( RR  X.  RR )  C_  ( CC  X.  CC ) )  ->  (
( abs  o.  -  )  |`  ( RR  X.  RR ) )  Fn  ( RR  X.  RR ) )
2318, 21, 22mp2an 672 . . 3  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  Fn  ( RR  X.  RR )
24 eqfnov2 6192 . . 3  |-  ( ( ( D  |`  ( RR  X.  RR ) )  Fn  ( RR  X.  RR )  /\  (
( abs  o.  -  )  |`  ( RR  X.  RR ) )  Fn  ( RR  X.  RR ) )  ->  ( ( D  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  <->  A. x  e.  RR  A. y  e.  RR  ( x ( D  |`  ( RR  X.  RR ) ) y )  =  ( x ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) y ) ) )
2514, 23, 24mp2an 672 . 2  |-  ( ( D  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  <->  A. x  e.  RR  A. y  e.  RR  (
x ( D  |`  ( RR  X.  RR ) ) y )  =  ( x ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) y ) )
267, 25mpbir 209 1  |-  ( D  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710    C_ wss 3323    X. cxp 4833    |` cres 4837    o. ccom 4839    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   RR*cxr 9409    - cmin 9587   abscabs 12715   distcds 14239   RR*scxrs 14430   *Metcxmt 17776   Metcme 17777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-rp 10984  df-xneg 11081  df-xadd 11082  df-icc 11299  df-fz 11430  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-plusg 14243  df-mulr 14244  df-tset 14249  df-ple 14250  df-ds 14252  df-xrs 14432  df-xmet 17785  df-met 17786
This theorem is referenced by:  xrsmopn  20364  metdscn2  20408
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