ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  maxleim Unicode version

Theorem maxleim 10863
Description: Value of maximum when we know which number is larger. (Contributed by Jim Kingdon, 21-Dec-2021.)
Assertion
Ref Expression
maxleim  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  sup ( { A ,  B } ,  RR ,  <  )  =  B ) )

Proof of Theorem maxleim
Dummy variables  f  g  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lttri3 7761 . . . 4  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
21adantl 273 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
f  e.  RR  /\  g  e.  RR )
)  ->  ( f  =  g  <->  ( -.  f  <  g  /\  -.  g  <  f ) ) )
3 simplr 502 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  B  e.  RR )
4 prid2g 3592 . . . 4  |-  ( B  e.  RR  ->  B  e.  { A ,  B } )
53, 4syl 14 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  B  e.  { A ,  B }
)
6 simpll 501 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  A  e.  RR )
76ad2antrr 477 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  A  e.  RR )
83ad2antrr 477 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  B  e.  RR )
9 simpllr 506 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  A  <_  B )
107, 8, 9lensymd 7801 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  -.  B  <  A )
11 breq2 3897 . . . . . . 7  |-  ( y  =  A  ->  ( B  <  y  <->  B  <  A ) )
1211notbid 639 . . . . . 6  |-  ( y  =  A  ->  ( -.  B  <  y  <->  -.  B  <  A ) )
1312adantl 273 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  ( -.  B  <  y  <->  -.  B  <  A ) )
1410, 13mpbird 166 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  -.  B  <  y )
153ad2antrr 477 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  B  e.  RR )
1615ltnrd 7792 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  -.  B  <  B )
17 breq2 3897 . . . . . . 7  |-  ( y  =  B  ->  ( B  <  y  <->  B  <  B ) )
1817notbid 639 . . . . . 6  |-  ( y  =  B  ->  ( -.  B  <  y  <->  -.  B  <  B ) )
1918adantl 273 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  ( -.  B  <  y  <->  -.  B  <  B ) )
2016, 19mpbird 166 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  -.  B  <  y )
21 elpri 3514 . . . . 5  |-  ( y  e.  { A ,  B }  ->  ( y  =  A  \/  y  =  B ) )
2221adantl 273 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  ->  (
y  =  A  \/  y  =  B )
)
2314, 20, 22mpjaodan 770 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  ->  -.  B  <  y )
242, 3, 5, 23supmaxti 6841 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  sup ( { A ,  B } ,  RR ,  <  )  =  B )
2524ex 114 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  ->  sup ( { A ,  B } ,  RR ,  <  )  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    = wceq 1312    e. wcel 1461   {cpr 3492   class class class wbr 3893   supcsup 6819   RRcr 7540    < clt 7718    <_ cle 7719
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-pre-ltirr 7651  ax-pre-apti 7654
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-xp 4503  df-cnv 4505  df-iota 5044  df-riota 5682  df-sup 6821  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724
This theorem is referenced by:  maxleb  10874  xrmaxiflemab  10902
  Copyright terms: Public domain W3C validator