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Theorem maxleim 10534
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 7509 . . . 4  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
21adantl 271 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
f  e.  RR  /\  g  e.  RR )
)  ->  ( f  =  g  <->  ( -.  f  <  g  /\  -.  g  <  f ) ) )
3 simplr 497 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  B  e.  RR )
4 prid2g 3530 . . . 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 496 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  A  e.  RR )
76ad2antrr 472 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  A  e.  RR )
83ad2antrr 472 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  B  e.  RR )
9 simpllr 501 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  A  <_  B )
107, 8, 9lensymd 7549 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  -.  B  <  A )
11 breq2 3824 . . . . . . 7  |-  ( y  =  A  ->  ( B  <  y  <->  B  <  A ) )
1211notbid 625 . . . . . 6  |-  ( y  =  A  ->  ( -.  B  <  y  <->  -.  B  <  A ) )
1312adantl 271 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  ( -.  B  <  y  <->  -.  B  <  A ) )
1410, 13mpbird 165 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  A )  ->  -.  B  <  y )
153ad2antrr 472 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  B  e.  RR )
1615ltnrd 7540 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  -.  B  <  B )
17 breq2 3824 . . . . . . 7  |-  ( y  =  B  ->  ( B  <  y  <->  B  <  B ) )
1817notbid 625 . . . . . 6  |-  ( y  =  B  ->  ( -.  B  <  y  <->  -.  B  <  B ) )
1918adantl 271 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  ( -.  B  <  y  <->  -.  B  <  B ) )
2016, 19mpbird 165 . . . 4  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  /\  y  =  B )  ->  -.  B  <  y )
21 elpri 3454 . . . . 5  |-  ( y  e.  { A ,  B }  ->  ( y  =  A  \/  y  =  B ) )
2221adantl 271 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  ->  (
y  =  A  \/  y  =  B )
)
2314, 20, 22mpjaodan 745 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  y  e.  { A ,  B } )  ->  -.  B  <  y )
242, 3, 5, 23supmaxti 6643 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  sup ( { A ,  B } ,  RR ,  <  )  =  B )
2524ex 113 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 102    <-> wb 103    \/ wo 662    = wceq 1287    e. wcel 1436   {cpr 3432   class class class wbr 3820   supcsup 6621   RRcr 7293    < clt 7466    <_ cle 7467
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-cnex 7380  ax-resscn 7381  ax-pre-ltirr 7401  ax-pre-apti 7404
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-xp 4417  df-cnv 4419  df-iota 4946  df-riota 5569  df-sup 6623  df-pnf 7468  df-mnf 7469  df-xr 7470  df-ltxr 7471  df-le 7472
This theorem is referenced by:  maxleb  10545
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