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Theorem xrltmininf 11063
Description: Two ways of saying an extended real is less than the minimum of two others. (Contributed by NM, 7-Feb-2007.) (Revised by Jim Kingdon, 3-May-2023.)
Assertion
Ref Expression
xrltmininf  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  < inf ( { B ,  C } ,  RR* ,  <  )  <->  ( A  <  B  /\  A  < 
C ) ) )

Proof of Theorem xrltmininf
StepHypRef Expression
1 xrminmax 11058 . . . 4  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  -> inf ( { B ,  C } ,  RR* ,  <  )  =  -e sup ( {  -e B ,  -e C } ,  RR* ,  <  ) )
213adant1 999 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  -> inf ( { B ,  C } ,  RR* ,  <  )  =  -e sup ( {  -e B ,  -e C } ,  RR* ,  <  ) )
32breq2d 3944 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  < inf ( { B ,  C } ,  RR* ,  <  )  <->  A  <  -e sup ( { 
-e B ,  -e C } ,  RR* ,  <  ) ) )
4 simp2 982 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  B  e.  RR* )
54xnegcld 9661 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -e
B  e.  RR* )
6 simp3 983 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  C  e.  RR* )
76xnegcld 9661 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -e
C  e.  RR* )
8 simp1 981 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  A  e.  RR* )
98xnegcld 9661 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -e
A  e.  RR* )
10 xrmaxltsup 11051 . . . 4  |-  ( ( 
-e B  e. 
RR*  /\  -e C  e.  RR*  /\  -e
A  e.  RR* )  ->  ( sup ( { 
-e B ,  -e C } ,  RR* ,  <  )  <  -e A  <->  (  -e
B  <  -e A  /\  -e C  <  -e A ) ) )
115, 7, 9, 10syl3anc 1216 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( sup ( {  -e
B ,  -e
C } ,  RR* ,  <  )  <  -e
A  <->  (  -e
B  <  -e A  /\  -e C  <  -e A ) ) )
12 xrmaxcl 11045 . . . . . . 7  |-  ( ( 
-e B  e. 
RR*  /\  -e C  e.  RR* )  ->  sup ( {  -e B ,  -e C } ,  RR* ,  <  )  e.  RR* )
135, 7, 12syl2anc 408 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  sup ( {  -e B ,  -e C } ,  RR* ,  <  )  e.  RR* )
1413xnegcld 9661 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -e sup ( {  -e
B ,  -e
C } ,  RR* ,  <  )  e.  RR* )
15 xltneg 9642 . . . . 5  |-  ( ( A  e.  RR*  /\  -e sup ( {  -e
B ,  -e
C } ,  RR* ,  <  )  e.  RR* )  ->  ( A  <  -e sup ( {  -e B ,  -e C } ,  RR* ,  <  )  <->  -e  -e sup ( {  -e
B ,  -e
C } ,  RR* ,  <  )  <  -e
A ) )
168, 14, 15syl2anc 408 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <  -e sup ( {  -e B ,  -e C } ,  RR* ,  <  )  <->  -e  -e sup ( {  -e
B ,  -e
C } ,  RR* ,  <  )  <  -e
A ) )
17 xnegneg 9639 . . . . . 6  |-  ( sup ( {  -e
B ,  -e
C } ,  RR* ,  <  )  e.  RR*  -> 
-e  -e sup ( {  -e
B ,  -e
C } ,  RR* ,  <  )  =  sup ( {  -e B ,  -e C } ,  RR* ,  <  ) )
1813, 17syl 14 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  -e  -e sup ( { 
-e B ,  -e C } ,  RR* ,  <  )  =  sup ( {  -e
B ,  -e
C } ,  RR* ,  <  ) )
1918breq1d 3942 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (  -e  -e sup ( {  -e
B ,  -e
C } ,  RR* ,  <  )  <  -e
A  <->  sup ( {  -e
B ,  -e
C } ,  RR* ,  <  )  <  -e
A ) )
2016, 19bitrd 187 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <  -e sup ( {  -e B ,  -e C } ,  RR* ,  <  )  <->  sup ( {  -e B ,  -e C } ,  RR* ,  <  )  <  -e A ) )
21 xltneg 9642 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  -e B  <  -e A ) )
22213adant3 1001 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <  B  <->  -e B  <  -e A ) )
23 xltneg 9642 . . . . 5  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  ( A  <  C  <->  -e C  <  -e A ) )
24233adant2 1000 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <  C  <->  -e C  <  -e A ) )
2522, 24anbi12d 464 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <  B  /\  A  <  C )  <-> 
(  -e B  <  -e A  /\  -e C  <  -e
A ) ) )
2611, 20, 253bitr4d 219 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <  -e sup ( {  -e B ,  -e C } ,  RR* ,  <  )  <->  ( A  <  B  /\  A  < 
C ) ) )
273, 26bitrd 187 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  < inf ( { B ,  C } ,  RR* ,  <  )  <->  ( A  <  B  /\  A  < 
C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   {cpr 3528   class class class wbr 3932   supcsup 6872  infcinf 6873   RR*cxr 7818    < clt 7819    -ecxne 9579
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4046  ax-sep 4049  ax-nul 4057  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455  ax-iinf 4505  ax-cnex 7730  ax-resscn 7731  ax-1cn 7732  ax-1re 7733  ax-icn 7734  ax-addcl 7735  ax-addrcl 7736  ax-mulcl 7737  ax-mulrcl 7738  ax-addcom 7739  ax-mulcom 7740  ax-addass 7741  ax-mulass 7742  ax-distr 7743  ax-i2m1 7744  ax-0lt1 7745  ax-1rid 7746  ax-0id 7747  ax-rnegex 7748  ax-precex 7749  ax-cnre 7750  ax-pre-ltirr 7751  ax-pre-ltwlin 7752  ax-pre-lttrn 7753  ax-pre-apti 7754  ax-pre-ltadd 7755  ax-pre-mulgt0 7756  ax-pre-mulext 7757  ax-arch 7758  ax-caucvg 7759
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-int 3775  df-iun 3818  df-br 3933  df-opab 3993  df-mpt 3994  df-tr 4030  df-id 4218  df-po 4221  df-iso 4222  df-iord 4291  df-on 4293  df-ilim 4294  df-suc 4296  df-iom 4508  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-f1 5131  df-fo 5132  df-f1o 5133  df-fv 5134  df-isom 5135  df-riota 5733  df-ov 5780  df-oprab 5781  df-mpo 5782  df-1st 6041  df-2nd 6042  df-recs 6205  df-frec 6291  df-sup 6874  df-inf 6875  df-pnf 7821  df-mnf 7822  df-xr 7823  df-ltxr 7824  df-le 7825  df-sub 7954  df-neg 7955  df-reap 8356  df-ap 8363  df-div 8452  df-inn 8740  df-2 8798  df-3 8799  df-4 8800  df-n0 8997  df-z 9074  df-uz 9346  df-rp 9464  df-xneg 9582  df-seqfrec 10243  df-exp 10317  df-cj 10638  df-re 10639  df-im 10640  df-rsqrt 10794  df-abs 10795
This theorem is referenced by:  xrminrpcl  11067  iooinsup  11070  blininf  12619  bdxmet  12696  bdmopn  12699
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