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Theorem maxleast 10925
Description: The maximum of two reals is a least upper bound. Lemma 3.11 of [Geuvers], p. 10. (Contributed by Jim Kingdon, 22-Dec-2021.)
Assertion
Ref Expression
maxleast  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  C  /\  B  <_  C ) )  ->  sup ( { A ,  B } ,  RR ,  <  )  <_  C )

Proof of Theorem maxleast
Dummy variables  f  g  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ioran 724 . . . 4  |-  ( -.  ( C  <  A  \/  C  <  B )  <-> 
( -.  C  < 
A  /\  -.  C  <  B ) )
2 simp3 966 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
3 lttri3 7808 . . . . . . . . 9  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
43adantl 273 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( f  e.  RR  /\  g  e.  RR ) )  ->  ( f  =  g  <->  ( -.  f  <  g  /\  -.  g  <  f ) ) )
5 maxabslemval 10920 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  e.  RR  /\  A. y  e.  { A ,  B }  -.  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  <  y  /\  A. y  e.  RR  (
y  <  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  ->  E. z  e.  { A ,  B } y  <  z
) ) )
6 3anass 949 . . . . . . . . . . 11  |-  ( ( ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  e.  RR  /\  A. y  e.  { A ,  B }  -.  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  <  y  /\  A. y  e.  RR  (
y  <  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  ->  E. z  e.  { A ,  B } y  <  z
) )  <->  ( (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  e.  RR  /\  ( A. y  e.  { A ,  B }  -.  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  <  y  /\  A. y  e.  RR  (
y  <  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  ->  E. z  e.  { A ,  B } y  <  z
) ) ) )
75, 6sylib 121 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  e.  RR  /\  ( A. y  e.  { A ,  B }  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  y  /\  A. y  e.  RR  (
y  <  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  ->  E. z  e.  { A ,  B } y  <  z
) ) ) )
8 breq1 3900 . . . . . . . . . . . . . 14  |-  ( x  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  ->  ( x  <  y  <->  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  y )
)
98notbid 639 . . . . . . . . . . . . 13  |-  ( x  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  ->  ( -.  x  <  y  <->  -.  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  <  y ) )
109ralbidv 2412 . . . . . . . . . . . 12  |-  ( x  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  ->  ( A. y  e.  { A ,  B }  -.  x  <  y  <->  A. y  e.  { A ,  B }  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  y )
)
11 breq2 3901 . . . . . . . . . . . . . 14  |-  ( x  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  ->  ( y  <  x  <->  y  <  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) ) )
1211imbi1d 230 . . . . . . . . . . . . 13  |-  ( x  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  ->  ( (
y  <  x  ->  E. z  e.  { A ,  B } y  < 
z )  <->  ( y  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  ->  E. z  e.  { A ,  B } y  <  z
) ) )
1312ralbidv 2412 . . . . . . . . . . . 12  |-  ( x  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  ->  ( A. y  e.  RR  (
y  <  x  ->  E. z  e.  { A ,  B } y  < 
z )  <->  A. y  e.  RR  ( y  < 
( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  ->  E. z  e.  { A ,  B } y  <  z
) ) )
1410, 13anbi12d 462 . . . . . . . . . . 11  |-  ( x  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  ->  ( ( A. y  e.  { A ,  B }  -.  x  <  y  /\  A. y  e.  RR  ( y  < 
x  ->  E. z  e.  { A ,  B } y  <  z
) )  <->  ( A. y  e.  { A ,  B }  -.  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  <  y  /\  A. y  e.  RR  (
y  <  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  ->  E. z  e.  { A ,  B } y  <  z
) ) ) )
1514rspcev 2761 . . . . . . . . . 10  |-  ( ( ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  e.  RR  /\  ( A. y  e.  { A ,  B }  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  y  /\  A. y  e.  RR  (
y  <  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  ->  E. z  e.  { A ,  B } y  <  z
) ) )  ->  E. x  e.  RR  ( A. y  e.  { A ,  B }  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  { A ,  B } y  < 
z ) ) )
167, 15syl 14 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  E. x  e.  RR  ( A. y  e.  { A ,  B }  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  { A ,  B } y  < 
z ) ) )
17163adant3 984 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  E. x  e.  RR  ( A. y  e.  { A ,  B }  -.  x  <  y  /\  A. y  e.  RR  ( y  <  x  ->  E. z  e.  { A ,  B }
y  <  z )
) )
184, 17suplubti 6853 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  e.  RR  /\  C  <  sup ( { A ,  B } ,  RR ,  <  )
)  ->  E. z  e.  { A ,  B } C  <  z ) )
192, 18mpand 423 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  <  sup ( { A ,  B } ,  RR ,  <  )  ->  E. z  e.  { A ,  B } C  <  z ) )
20 elpri 3518 . . . . . . . . 9  |-  ( z  e.  { A ,  B }  ->  ( z  =  A  \/  z  =  B ) )
2120adantr 272 . . . . . . . 8  |-  ( ( z  e.  { A ,  B }  /\  C  <  z )  ->  (
z  =  A  \/  z  =  B )
)
22 breq2 3901 . . . . . . . . . . 11  |-  ( z  =  A  ->  ( C  <  z  <->  C  <  A ) )
2322biimpcd 158 . . . . . . . . . 10  |-  ( C  <  z  ->  (
z  =  A  ->  C  <  A ) )
2423adantl 273 . . . . . . . . 9  |-  ( ( z  e.  { A ,  B }  /\  C  <  z )  ->  (
z  =  A  ->  C  <  A ) )
25 breq2 3901 . . . . . . . . . . 11  |-  ( z  =  B  ->  ( C  <  z  <->  C  <  B ) )
2625biimpcd 158 . . . . . . . . . 10  |-  ( C  <  z  ->  (
z  =  B  ->  C  <  B ) )
2726adantl 273 . . . . . . . . 9  |-  ( ( z  e.  { A ,  B }  /\  C  <  z )  ->  (
z  =  B  ->  C  <  B ) )
2824, 27orim12d 758 . . . . . . . 8  |-  ( ( z  e.  { A ,  B }  /\  C  <  z )  ->  (
( z  =  A  \/  z  =  B )  ->  ( C  <  A  \/  C  < 
B ) ) )
2921, 28mpd 13 . . . . . . 7  |-  ( ( z  e.  { A ,  B }  /\  C  <  z )  ->  ( C  <  A  \/  C  <  B ) )
3029rexlimiva 2519 . . . . . 6  |-  ( E. z  e.  { A ,  B } C  < 
z  ->  ( C  <  A  \/  C  < 
B ) )
3119, 30syl6 33 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  <  sup ( { A ,  B } ,  RR ,  <  )  ->  ( C  <  A  \/  C  <  B ) ) )
3231con3d 603 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( -.  ( C  <  A  \/  C  <  B )  ->  -.  C  <  sup ( { A ,  B } ,  RR ,  <  ) ) )
331, 32syl5bir 152 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( -.  C  < 
A  /\  -.  C  <  B )  ->  -.  C  <  sup ( { A ,  B } ,  RR ,  <  ) ) )
34 simp1 964 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
3534, 2lenltd 7844 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  C  <->  -.  C  <  A ) )
36 simp2 965 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
3736, 2lenltd 7844 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <_  C  <->  -.  C  <  B ) )
3835, 37anbi12d 462 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <_  C  /\  B  <_  C )  <-> 
( -.  C  < 
A  /\  -.  C  <  B ) ) )
394, 17supclti 6851 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR )
4039, 2lenltd 7844 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( sup ( { A ,  B } ,  RR ,  <  )  <_  C  <->  -.  C  <  sup ( { A ,  B } ,  RR ,  <  ) ) )
4133, 38, 403imtr4d 202 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <_  C  /\  B  <_  C )  ->  sup ( { A ,  B } ,  RR ,  <  )  <_  C
) )
4241imp 123 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <_  C  /\  B  <_  C ) )  ->  sup ( { A ,  B } ,  RR ,  <  )  <_  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    /\ w3a 945    = wceq 1314    e. wcel 1463   A.wral 2391   E.wrex 2392   {cpr 3496   class class class wbr 3897   ` cfv 5091  (class class class)co 5740   supcsup 6835   RRcr 7583    + caddc 7587    < clt 7764    <_ cle 7765    - cmin 7897    / cdiv 8392   2c2 8728   abscabs 10709
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-sup 6837  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-2 8736  df-3 8737  df-4 8738  df-n0 8929  df-z 9006  df-uz 9276  df-rp 9391  df-seqfrec 10159  df-exp 10233  df-cj 10554  df-re 10555  df-im 10556  df-rsqrt 10710  df-abs 10711
This theorem is referenced by:  maxleastb  10926  dfabsmax  10929
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