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Theorem xrsdsreclb 18226
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 18223 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A D B )  =  if ( A  <_  B ,  ( B +e  -e A ) ,  ( A +e  -e
B ) ) )
323adant3 1011 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/= 
B )  ->  ( A D B )  =  if ( A  <_  B ,  ( B +e  -e A ) ,  ( A +e  -e
B ) ) )
43eleq1d 2529 . . 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 2532 . . . . 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 317 . . . 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 2532 . . . . 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 317 . . . 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 18225 . . . 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 11346 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  \/  B  <_  A ) )
11103adant3 1011 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/= 
B )  ->  ( A  <_  B  \/  B  <_  A ) )
1211orcanai 906 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/=  B )  /\  -.  A  <_  B )  ->  B  <_  A )
13 necom 2729 . . . . . . . . 9  |-  ( A  =/=  B  <->  B  =/=  A )
14133anbi3i 1184 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/= 
B )  <->  ( A  e.  RR*  /\  B  e. 
RR*  /\  B  =/=  A ) )
15 3ancoma 975 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  B  =/= 
A )  <->  ( B  e.  RR*  /\  A  e. 
RR*  /\  B  =/=  A ) )
1614, 15bitri 249 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/= 
B )  <->  ( B  e.  RR*  /\  A  e. 
RR*  /\  B  =/=  A ) )
171xrsdsreclblem 18225 . . . . . . 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 472 . . . . . 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 450 . . . . . 6  |-  ( ( B  e.  RR  /\  A  e.  RR )  <->  ( A  e.  RR  /\  B  e.  RR )
)
2018, 19syl6ib 226 . . . . 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 470 . . . 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 3967 . . 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 215 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/= 
B )  ->  (
( A D B )  e.  RR  ->  ( A  e.  RR  /\  B  e.  RR )
) )
241xrsdsreval 18224 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A D B )  =  ( abs `  ( A  -  B
) ) )
25 recn 9571 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
26 recn 9571 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
27 subcl 9808 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
2825, 26, 27syl2an 477 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  CC )
2928abscld 13216 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( A  -  B )
)  e.  RR )
3024, 29eqeltrd 2548 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A D B )  e.  RR )
3123, 30impbid1 203 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/= 
B )  ->  (
( A D B )  e.  RR  <->  ( A  e.  RR  /\  B  e.  RR ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   ifcif 3932   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   CCcc 9479   RRcr 9480   RR*cxr 9616    <_ cle 9618    - cmin 9794    -ecxne 11304   +ecxad 11305   abscabs 13017   distcds 14553   RR*scxrs 14744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-rp 11210  df-xneg 11307  df-xadd 11308  df-fz 11662  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-plusg 14557  df-mulr 14558  df-tset 14563  df-ple 14564  df-ds 14566  df-xrs 14746
This theorem is referenced by:  xrsxmet  21042  xrsblre  21044  xrsmopn  21045
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