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Theorem maxleim 11246
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 8067 . . . 4  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
21adantl 277 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
f  e.  RR  /\  g  e.  RR )
)  ->  ( f  =  g  <->  ( -.  f  <  g  /\  -.  g  <  f ) ) )
3 simplr 528 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  B  e.  RR )
4 prid2g 3712 . . . 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 527 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  A  e.  RR )
76ad2antrr 488 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  A  e.  RR )
83ad2antrr 488 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  B  e.  RR )
9 simpllr 534 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  A  <_  B )
107, 8, 9lensymd 8109 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  -.  B  <  A )
11 breq2 4022 . . . . . . 7  |-  ( y  =  A  ->  ( B  <  y  <->  B  <  A ) )
1211notbid 668 . . . . . 6  |-  ( y  =  A  ->  ( -.  B  <  y  <->  -.  B  <  A ) )
1312adantl 277 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  ( -.  B  <  y  <->  -.  B  <  A ) )
1410, 13mpbird 167 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  -.  B  <  y )
153ad2antrr 488 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  B  e.  RR )
1615ltnrd 8099 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  -.  B  <  B )
17 breq2 4022 . . . . . . 7  |-  ( y  =  B  ->  ( B  <  y  <->  B  <  B ) )
1817notbid 668 . . . . . 6  |-  ( y  =  B  ->  ( -.  B  <  y  <->  -.  B  <  B ) )
1918adantl 277 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  ( -.  B  <  y  <->  -.  B  <  B ) )
2016, 19mpbird 167 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  -.  B  <  y )
21 elpri 3630 . . . . 5  |-  ( y  e.  { A ,  B }  ->  ( y  =  A  \/  y  =  B ) )
2221adantl 277 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  ->  (
y  =  A  \/  y  =  B )
)
2314, 20, 22mpjaodan 799 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  ->  -.  B  <  y )
242, 3, 5, 23supmaxti 7033 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  sup ( { A ,  B } ,  RR ,  <  )  =  B )
2524ex 115 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 104    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2160   {cpr 3608   class class class wbr 4018   supcsup 7011   RRcr 7840    < clt 8022    <_ cle 8023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-pre-ltirr 7953  ax-pre-apti 7956
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-cnv 4652  df-iota 5196  df-riota 5852  df-sup 7013  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028
This theorem is referenced by:  maxleb  11257  xrmaxiflemab  11287
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