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Theorem xrminltinf 11072
Description: Two ways of saying an extended real is greater than the minimum of two others. (Contributed by Jim Kingdon, 19-May-2023.)
Assertion
Ref Expression
xrminltinf  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (inf ( { B ,  C } ,  RR* ,  <  )  <  A  <->  ( B  <  A  \/  C  < 
A ) ) )

Proof of Theorem xrminltinf
StepHypRef Expression
1 xnegcl 9644 . . . 4  |-  ( B  e.  RR*  ->  -e
B  e.  RR* )
213ad2ant2 1004 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -e
B  e.  RR* )
3 xnegcl 9644 . . . 4  |-  ( C  e.  RR*  ->  -e
C  e.  RR* )
433ad2ant3 1005 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -e
C  e.  RR* )
5 xnegcl 9644 . . . 4  |-  ( A  e.  RR*  ->  -e
A  e.  RR* )
653ad2ant1 1003 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -e
A  e.  RR* )
7 xrltmaxsup 11057 . . 3  |-  ( ( 
-e B  e. 
RR*  /\  -e C  e.  RR*  /\  -e
A  e.  RR* )  ->  (  -e A  <  sup ( {  -e
B ,  -e
C } ,  RR* ,  <  )  <->  (  -e
A  <  -e B  \/  -e A  <  -e C ) ) )
82, 4, 6, 7syl3anc 1217 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (  -e A  <  sup ( {  -e B ,  -e C } ,  RR* ,  <  )  <-> 
(  -e A  <  -e B  \/  -e A  <  -e
C ) ) )
9 xrminmax 11065 . . . . 5  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  -> inf ( { B ,  C } ,  RR* ,  <  )  =  -e sup ( {  -e B ,  -e C } ,  RR* ,  <  ) )
1093adant1 1000 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  -> inf ( { B ,  C } ,  RR* ,  <  )  =  -e sup ( {  -e B ,  -e C } ,  RR* ,  <  ) )
11 xnegneg 9645 . . . . . 6  |-  ( A  e.  RR*  ->  -e  -e A  =  A )
1211eqcomd 2146 . . . . 5  |-  ( A  e.  RR*  ->  A  = 
-e  -e
A )
13123ad2ant1 1003 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  A  =  -e  -e
A )
1410, 13breq12d 3949 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (inf ( { B ,  C } ,  RR* ,  <  )  <  A  <->  -e sup ( {  -e
B ,  -e
C } ,  RR* ,  <  )  <  -e  -e A ) )
15 xrmaxcl 11052 . . . . 5  |-  ( ( 
-e B  e. 
RR*  /\  -e C  e.  RR* )  ->  sup ( {  -e B ,  -e C } ,  RR* ,  <  )  e.  RR* )
162, 4, 15syl2anc 409 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  sup ( {  -e B ,  -e C } ,  RR* ,  <  )  e.  RR* )
17 xltneg 9648 . . . 4  |-  ( ( 
-e A  e. 
RR*  /\  sup ( {  -e B ,  -e C } ,  RR* ,  <  )  e. 
RR* )  ->  (  -e A  <  sup ( {  -e B ,  -e C } ,  RR* ,  <  )  <->  -e sup ( {  -e B ,  -e C } ,  RR* ,  <  )  <  -e  -e A ) )
186, 16, 17syl2anc 409 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (  -e A  <  sup ( {  -e B ,  -e C } ,  RR* ,  <  )  <->  -e sup ( {  -e B ,  -e C } ,  RR* ,  <  )  <  -e  -e A ) )
1914, 18bitr4d 190 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (inf ( { B ,  C } ,  RR* ,  <  )  <  A  <->  -e A  <  sup ( {  -e
B ,  -e
C } ,  RR* ,  <  ) ) )
20 simp2 983 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  B  e.  RR* )
21 simp1 982 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  A  e.  RR* )
22 xltneg 9648 . . . 4  |-  ( ( B  e.  RR*  /\  A  e.  RR* )  ->  ( B  <  A  <->  -e A  <  -e B ) )
2320, 21, 22syl2anc 409 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( B  <  A  <->  -e A  <  -e B ) )
24 simp3 984 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  C  e.  RR* )
25 xltneg 9648 . . . 4  |-  ( ( C  e.  RR*  /\  A  e.  RR* )  ->  ( C  <  A  <->  -e A  <  -e C ) )
2624, 21, 25syl2anc 409 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  <  A  <->  -e A  <  -e C ) )
2723, 26orbi12d 783 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( B  <  A  \/  C  <  A )  <-> 
(  -e A  <  -e B  \/  -e A  <  -e
C ) ) )
288, 19, 273bitr4d 219 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (inf ( { B ,  C } ,  RR* ,  <  )  <  A  <->  ( B  <  A  \/  C  < 
A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    \/ wo 698    /\ w3a 963    = wceq 1332    e. wcel 1481   {cpr 3532   class class class wbr 3936   supcsup 6876  infcinf 6877   RR*cxr 7822    < clt 7823    -ecxne 9585
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-xneg 9588  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:  bdbl  12709
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