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

Theorem xrsdsreclb 16737
Description: The metric of the extended real number structure is only real when both arguments are real. (Contributed by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
xrsds.d  |-  D  =  ( dist `  RR* s
)
Assertion
Ref Expression
xrsdsreclb  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/= 
B )  ->  (
( A D B )  e.  RR  <->  ( A  e.  RR  /\  B  e.  RR ) ) )

Proof of Theorem xrsdsreclb
StepHypRef Expression
1 xrsds.d . . . . . 6  |-  D  =  ( dist `  RR* s
)
21xrsdsval 16734 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A D B )  =  if ( A  <_  B ,  ( B + e  - e A ) ,  ( A + e  - e B ) ) )
323adant3 977 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/= 
B )  ->  ( A D B )  =  if ( A  <_  B ,  ( B + e  - e A ) ,  ( A + e  - e B ) ) )
43eleq1d 2501 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/= 
B )  ->  (
( A D B )  e.  RR  <->  if ( A  <_  B ,  ( B + e  - e A ) ,  ( A + e  - e B ) )  e.  RR ) )
5 eleq1 2495 . . . . 5  |-  ( ( B + e  - e A )  =  if ( A  <_  B ,  ( B + e  - e A ) ,  ( A + e  - e B ) )  ->  ( ( B + e  - e A )  e.  RR  <->  if ( A  <_  B ,  ( B + e  - e A ) ,  ( A + e  - e B ) )  e.  RR ) )
65imbi1d 309 . . . 4  |-  ( ( B + e  - e A )  =  if ( A  <_  B ,  ( B + e  - e A ) ,  ( A + e  - e B ) )  ->  ( (
( B + e  - e A )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR ) )  <->  ( if ( A  <_  B , 
( B + e  - e A ) ,  ( A + e  - e B ) )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR ) ) ) )
7 eleq1 2495 . . . . 5  |-  ( ( A + e  - e B )  =  if ( A  <_  B ,  ( B + e  - e A ) ,  ( A + e  - e B ) )  ->  ( ( A + e  - e B )  e.  RR  <->  if ( A  <_  B ,  ( B + e  - e A ) ,  ( A + e  - e B ) )  e.  RR ) )
87imbi1d 309 . . . 4  |-  ( ( A + e  - e B )  =  if ( A  <_  B ,  ( B + e  - e A ) ,  ( A + e  - e B ) )  ->  ( (
( A + e  - e B )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR ) )  <->  ( if ( A  <_  B , 
( B + e  - e A ) ,  ( A + e  - e B ) )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR ) ) ) )
91xrsdsreclblem 16736 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/=  B )  /\  A  <_  B )  ->  (
( B + e  - e A )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR ) ) )
10 xrletri 10736 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  \/  B  <_  A ) )
11103adant3 977 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/= 
B )  ->  ( A  <_  B  \/  B  <_  A ) )
1211orcanai 880 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/=  B )  /\  -.  A  <_  B )  ->  B  <_  A )
13 necom 2679 . . . . . . . . 9  |-  ( A  =/=  B  <->  B  =/=  A )
14133anbi3i 1146 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/= 
B )  <->  ( A  e.  RR*  /\  B  e. 
RR*  /\  B  =/=  A ) )
15 3ancoma 943 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  B  =/= 
A )  <->  ( B  e.  RR*  /\  A  e. 
RR*  /\  B  =/=  A ) )
1614, 15bitri 241 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/= 
B )  <->  ( B  e.  RR*  /\  A  e. 
RR*  /\  B  =/=  A ) )
171xrsdsreclblem 16736 . . . . . . 7  |-  ( ( ( B  e.  RR*  /\  A  e.  RR*  /\  B  =/=  A )  /\  B  <_  A )  ->  (
( A + e  - e B )  e.  RR  ->  ( B  e.  RR  /\  A  e.  RR ) ) )
1816, 17sylanb 459 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/=  B )  /\  B  <_  A )  ->  (
( A + e  - e B )  e.  RR  ->  ( B  e.  RR  /\  A  e.  RR ) ) )
19 ancom 438 . . . . . 6  |-  ( ( B  e.  RR  /\  A  e.  RR )  <->  ( A  e.  RR  /\  B  e.  RR )
)
2018, 19syl6ib 218 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/=  B )  /\  B  <_  A )  ->  (
( A + e  - e B )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR ) ) )
2112, 20syldan 457 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/=  B )  /\  -.  A  <_  B )  -> 
( ( A + e  - e B )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR ) ) )
226, 8, 9, 21ifbothda 3761 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/= 
B )  ->  ( if ( A  <_  B ,  ( B + e  - e A ) ,  ( A + e  - e B ) )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR )
) )
234, 22sylbid 207 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/= 
B )  ->  (
( A D B )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR )
) )
241xrsdsreval 16735 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A D B )  =  ( abs `  ( A  -  B
) ) )
25 recn 9072 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
26 recn 9072 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
27 subcl 9297 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
2825, 26, 27syl2an 464 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  CC )
2928abscld 12230 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( A  -  B )
)  e.  RR )
3024, 29eqeltrd 2509 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A D B )  e.  RR )
3123, 30impbid1 195 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/= 
B )  ->  (
( A D B )  e.  RR  <->  ( A  e.  RR  /\  B  e.  RR ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   ifcif 3731   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   RRcr 8981   RR*cxr 9111    <_ cle 9113    - cmin 9283    - ecxne 10699   + ecxad 10700   abscabs 12031   distcds 13530   RR* scxrs 13714
This theorem is referenced by:  xrsxmet  18832  xrsblre  18834  xrsmopn  18835
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-rp 10605  df-xneg 10702  df-xadd 10703  df-fz 11036  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-plusg 13534  df-mulr 13535  df-tset 13540  df-ple 13541  df-ds 13543  df-xrs 13718
  Copyright terms: Public domain W3C validator