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Theorem minclpr 11039
Description: The minimum of two real numbers is one of those numbers if and only if dichotomy ( A  <_  B  \/  B  <_  A) holds. For example, this can be combined with zletric 9121 if one is dealing with integers, but real number dichotomy in general does not follow from our axioms. (Contributed by Jim Kingdon, 23-May-2023.)
Assertion
Ref Expression
minclpr  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (inf ( { A ,  B } ,  RR ,  <  )  e.  { A ,  B }  <->  ( A  <_  B  \/  B  <_  A ) ) )

Proof of Theorem minclpr
StepHypRef Expression
1 renegcl 8046 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
2 renegcl 8046 . . . . 5  |-  ( B  e.  RR  ->  -u B  e.  RR )
3 maxcl 11013 . . . . 5  |-  ( (
-u A  e.  RR  /\  -u B  e.  RR )  ->  sup ( { -u A ,  -u B } ,  RR ,  <  )  e.  RR )
41, 2, 3syl2an 287 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { -u A ,  -u B } ,  RR ,  <  )  e.  RR )
5 elprg 3551 . . . 4  |-  ( sup ( { -u A ,  -u B } ,  RR ,  <  )  e.  RR  ->  ( sup ( { -u A ,  -u B } ,  RR ,  <  )  e.  { -u A ,  -u B } 
<->  ( sup ( {
-u A ,  -u B } ,  RR ,  <  )  =  -u A  \/  sup ( { -u A ,  -u B } ,  RR ,  <  )  =  -u B ) ) )
64, 5syl 14 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( sup ( {
-u A ,  -u B } ,  RR ,  <  )  e.  { -u A ,  -u B }  <->  ( sup ( { -u A ,  -u B } ,  RR ,  <  )  =  -u A  \/  sup ( { -u A ,  -u B } ,  RR ,  <  )  =  -u B ) ) )
7 maxclpr 11025 . . . 4  |-  ( (
-u A  e.  RR  /\  -u B  e.  RR )  ->  ( sup ( { -u A ,  -u B } ,  RR ,  <  )  e.  { -u A ,  -u B }  <->  (
-u A  <_  -u B  \/  -u B  <_  -u A
) ) )
81, 2, 7syl2an 287 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( sup ( {
-u A ,  -u B } ,  RR ,  <  )  e.  { -u A ,  -u B }  <->  (
-u A  <_  -u B  \/  -u B  <_  -u A
) ) )
94recnd 7817 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { -u A ,  -u B } ,  RR ,  <  )  e.  CC )
101adantr 274 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u A  e.  RR )
1110recnd 7817 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u A  e.  CC )
129, 11neg11ad 8092 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u sup ( { -u A ,  -u B } ,  RR ,  <  )  =  -u -u A  <->  sup ( { -u A ,  -u B } ,  RR ,  <  )  = 
-u A ) )
13 minmax 11032 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> inf ( { A ,  B } ,  RR ,  <  )  =  -u sup ( { -u A ,  -u B } ,  RR ,  <  ) )
1413eqcomd 2146 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u sup ( { -u A ,  -u B } ,  RR ,  <  )  = inf ( { A ,  B } ,  RR ,  <  ) )
15 recn 7776 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
1615adantr 274 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  CC )
1716negnegd 8087 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u -u A  =  A )
1814, 17eqeq12d 2155 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u sup ( { -u A ,  -u B } ,  RR ,  <  )  =  -u -u A  <-> inf ( { A ,  B } ,  RR ,  <  )  =  A ) )
1912, 18bitr3d 189 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( sup ( {
-u A ,  -u B } ,  RR ,  <  )  =  -u A  <-> inf ( { A ,  B } ,  RR ,  <  )  =  A ) )
202adantl 275 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u B  e.  RR )
2120recnd 7817 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u B  e.  CC )
229, 21neg11ad 8092 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u sup ( { -u A ,  -u B } ,  RR ,  <  )  =  -u -u B  <->  sup ( { -u A ,  -u B } ,  RR ,  <  )  = 
-u B ) )
23 recn 7776 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  CC )
2423adantl 275 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
2524negnegd 8087 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> 
-u -u B  =  B )
2614, 25eqeq12d 2155 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u sup ( { -u A ,  -u B } ,  RR ,  <  )  =  -u -u B  <-> inf ( { A ,  B } ,  RR ,  <  )  =  B ) )
2722, 26bitr3d 189 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( sup ( {
-u A ,  -u B } ,  RR ,  <  )  =  -u B  <-> inf ( { A ,  B } ,  RR ,  <  )  =  B ) )
2819, 27orbi12d 783 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( sup ( { -u A ,  -u B } ,  RR ,  <  )  =  -u A  \/  sup ( { -u A ,  -u B } ,  RR ,  <  )  =  -u B )  <->  (inf ( { A ,  B } ,  RR ,  <  )  =  A  \/ inf ( { A ,  B } ,  RR ,  <  )  =  B ) ) )
296, 8, 283bitr3rd 218 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( (inf ( { A ,  B } ,  RR ,  <  )  =  A  \/ inf ( { A ,  B } ,  RR ,  <  )  =  B )  <->  ( -u A  <_ 
-u B  \/  -u B  <_ 
-u A ) ) )
30 mincl 11033 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  -> inf ( { A ,  B } ,  RR ,  <  )  e.  RR )
31 elprg 3551 . . 3  |-  (inf ( { A ,  B } ,  RR ,  <  )  e.  RR  ->  (inf ( { A ,  B } ,  RR ,  <  )  e.  { A ,  B }  <->  (inf ( { A ,  B } ,  RR ,  <  )  =  A  \/ inf ( { A ,  B } ,  RR ,  <  )  =  B ) ) )
3230, 31syl 14 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (inf ( { A ,  B } ,  RR ,  <  )  e.  { A ,  B }  <->  (inf ( { A ,  B } ,  RR ,  <  )  =  A  \/ inf ( { A ,  B } ,  RR ,  <  )  =  B ) ) )
33 orcom 718 . . 3  |-  ( ( B  <_  A  \/  A  <_  B )  <->  ( A  <_  B  \/  B  <_  A ) )
34 simpr 109 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
35 simpl 108 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
3634, 35lenegd 8309 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  <_  A  <->  -u A  <_  -u B ) )
37 leneg 8250 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  -u B  <_  -u A ) )
3836, 37orbi12d 783 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( B  <_  A  \/  A  <_  B )  <->  ( -u A  <_ 
-u B  \/  -u B  <_ 
-u A ) ) )
3933, 38bitr3id 193 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  <_  B  \/  B  <_  A )  <->  ( -u A  <_ 
-u B  \/  -u B  <_ 
-u A ) ) )
4029, 32, 393bitr4d 219 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  (inf ( { A ,  B } ,  RR ,  <  )  e.  { A ,  B }  <->  ( A  <_  B  \/  B  <_  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698    = wceq 1332    e. wcel 1481   {cpr 3532   class class class wbr 3936   supcsup 6876  infcinf 6877   CCcc 7641   RRcr 7642    < clt 7823    <_ cle 7824   -ucneg 7957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761  ax-arch 7762  ax-caucvg 7763
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-ilim 4298  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-isom 5139  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-frec 6295  df-sup 6878  df-inf 6879  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456  df-inn 8744  df-2 8802  df-3 8803  df-4 8804  df-n0 9001  df-z 9078  df-uz 9350  df-rp 9470  df-seqfrec 10249  df-exp 10323  df-cj 10645  df-re 10646  df-im 10647  df-rsqrt 10801  df-abs 10802
This theorem is referenced by:  qtopbas  12728
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