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Theorem xrsdsreclb 17978
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 17975 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A D B )  =  if ( A  <_  B ,  ( B +e  -e A ) ,  ( A +e  -e
B ) ) )
323adant3 1008 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/= 
B )  ->  ( A D B )  =  if ( A  <_  B ,  ( B +e  -e A ) ,  ( A +e  -e
B ) ) )
43eleq1d 2520 . . 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 2523 . . . . 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 2523 . . . . 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 17977 . . . 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 11232 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  \/  B  <_  A ) )
11103adant3 1008 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/= 
B )  ->  ( A  <_  B  \/  B  <_  A ) )
1211orcanai 904 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/=  B )  /\  -.  A  <_  B )  ->  B  <_  A )
13 necom 2717 . . . . . . . . 9  |-  ( A  =/=  B  <->  B  =/=  A )
14133anbi3i 1181 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  =/= 
B )  <->  ( A  e.  RR*  /\  B  e. 
RR*  /\  B  =/=  A ) )
15 3ancoma 972 . . . . . . . 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 17977 . . . . . . 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 3925 . . 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 17976 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A D B )  =  ( abs `  ( A  -  B
) ) )
25 recn 9476 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
26 recn 9476 . . . . 5  |-  ( B  e.  RR  ->  B  e.  CC )
27 subcl 9713 . . . . 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 13033 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( A  -  B )
)  e.  RR )
3024, 29eqeltrd 2539 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2644   ifcif 3892   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   CCcc 9384   RRcr 9385   RR*cxr 9521    <_ cle 9523    - cmin 9699    -ecxne 11190   +ecxad 11191   abscabs 12834   distcds 14358   RR*scxrs 14549
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 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
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-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-rp 11096  df-xneg 11193  df-xadd 11194  df-fz 11548  df-seq 11917  df-exp 11976  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-plusg 14362  df-mulr 14363  df-tset 14368  df-ple 14369  df-ds 14371  df-xrs 14551
This theorem is referenced by:  xrsxmet  20511  xrsblre  20513  xrsmopn  20514
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