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Theorem maxleast 10611
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 704 . . . 4  |-  ( -.  ( C  <  A  \/  C  <  B )  <-> 
( -.  C  < 
A  /\  -.  C  <  B ) )
2 simp3 945 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
3 lttri3 7544 . . . . . . . . 9  |-  ( ( f  e.  RR  /\  g  e.  RR )  ->  ( f  =  g  <-> 
( -.  f  < 
g  /\  -.  g  <  f ) ) )
43adantl 271 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( f  e.  RR  /\  g  e.  RR ) )  ->  ( f  =  g  <->  ( -.  f  <  g  /\  -.  g  <  f ) ) )
5 maxabslemval 10606 . . . . . . . . . . 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 928 . . . . . . . . . . 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 120 . . . . . . . . . 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 3840 . . . . . . . . . . . . . 14  |-  ( x  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  ->  ( x  <  y  <->  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  y )
)
98notbid 627 . . . . . . . . . . . . 13  |-  ( x  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  ->  ( -.  x  <  y  <->  -.  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  <  y ) )
109ralbidv 2380 . . . . . . . . . . . 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 3841 . . . . . . . . . . . . . 14  |-  ( x  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  ->  ( y  <  x  <->  y  <  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) ) )
1211imbi1d 229 . . . . . . . . . . . . 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 2380 . . . . . . . . . . . 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 457 . . . . . . . . . . 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 2722 . . . . . . . . . 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 963 . . . . . . . 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 6674 . . . . . . 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 420 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  <  sup ( { A ,  B } ,  RR ,  <  )  ->  E. z  e.  { A ,  B } C  <  z ) )
20 elpri 3464 . . . . . . . . 9  |-  ( z  e.  { A ,  B }  ->  ( z  =  A  \/  z  =  B ) )
2120adantr 270 . . . . . . . 8  |-  ( ( z  e.  { A ,  B }  /\  C  <  z )  ->  (
z  =  A  \/  z  =  B )
)
22 breq2 3841 . . . . . . . . . . 11  |-  ( z  =  A  ->  ( C  <  z  <->  C  <  A ) )
2322biimpcd 157 . . . . . . . . . 10  |-  ( C  <  z  ->  (
z  =  A  ->  C  <  A ) )
2423adantl 271 . . . . . . . . 9  |-  ( ( z  e.  { A ,  B }  /\  C  <  z )  ->  (
z  =  A  ->  C  <  A ) )
25 breq2 3841 . . . . . . . . . . 11  |-  ( z  =  B  ->  ( C  <  z  <->  C  <  B ) )
2625biimpcd 157 . . . . . . . . . 10  |-  ( C  <  z  ->  (
z  =  B  ->  C  <  B ) )
2726adantl 271 . . . . . . . . 9  |-  ( ( z  e.  { A ,  B }  /\  C  <  z )  ->  (
z  =  B  ->  C  <  B ) )
2824, 27orim12d 735 . . . . . . . 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 2484 . . . . . 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 596 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( -.  ( C  <  A  \/  C  <  B )  ->  -.  C  <  sup ( { A ,  B } ,  RR ,  <  ) ) )
331, 32syl5bir 151 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( -.  C  < 
A  /\  -.  C  <  B )  ->  -.  C  <  sup ( { A ,  B } ,  RR ,  <  ) ) )
34 simp1 943 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
3534, 2lenltd 7580 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  C  <->  -.  C  <  A ) )
36 simp2 944 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
3736, 2lenltd 7580 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <_  C  <->  -.  C  <  B ) )
3835, 37anbi12d 457 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <_  C  /\  B  <_  C )  <-> 
( -.  C  < 
A  /\  -.  C  <  B ) ) )
394, 17supclti 6672 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR )
4039, 2lenltd 7580 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( sup ( { A ,  B } ,  RR ,  <  )  <_  C  <->  -.  C  <  sup ( { A ,  B } ,  RR ,  <  ) ) )
4133, 38, 403imtr4d 201 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <_  C  /\  B  <_  C )  ->  sup ( { A ,  B } ,  RR ,  <  )  <_  C
) )
4241imp 122 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 102    <-> wb 103    \/ wo 664    /\ w3a 924    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   {cpr 3442   class class class wbr 3837   ` cfv 5002  (class class class)co 5634   supcsup 6656   RRcr 7328    + caddc 7332    < clt 7501    <_ cle 7502    - cmin 7632    / cdiv 8113   2c2 8444   abscabs 10395
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-sup 6658  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-rp 9104  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397
This theorem is referenced by:  maxleastb  10612  dfabsmax  10615
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