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Theorem maxltsup 11246
Description: Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 10-Feb-2022.)
Assertion
Ref Expression
maxltsup  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( sup ( { A ,  B } ,  RR ,  <  )  <  C  <->  ( A  <  C  /\  B  < 
C ) ) )

Proof of Theorem maxltsup
StepHypRef Expression
1 simpl1 1002 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  A  e.  RR )
2 simpl2 1003 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  B  e.  RR )
3 maxcl 11238 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR )
41, 2, 3syl2anc 411 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR )
5 simpl3 1004 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  C  e.  RR )
6 maxle1 11239 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  <_  sup ( { A ,  B } ,  RR ,  <  )
)
763adant3 1019 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  <_  sup ( { A ,  B } ,  RR ,  <  ) )
87adantr 276 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  A  <_  sup ( { A ,  B } ,  RR ,  <  ) )
9 simpr 110 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  sup ( { A ,  B } ,  RR ,  <  )  <  C )
101, 4, 5, 8, 9lelttrd 8101 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  A  <  C )
11 maxle2 11240 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  <_  sup ( { A ,  B } ,  RR ,  <  )
)
121, 2, 11syl2anc 411 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  B  <_  sup ( { A ,  B } ,  RR ,  <  ) )
132, 4, 5, 12, 9lelttrd 8101 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  B  <  C )
1410, 13jca 306 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  ( A  <  C  /\  B  < 
C ) )
15 maxabs 11237 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )
16153adant3 1019 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  =  ( ( ( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
1716adantr 276 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  sup ( { A ,  B } ,  RR ,  <  )  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )
18 2re 9008 . . . . . . . . . . . 12  |-  2  e.  RR
1918a1i 9 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  2  e.  RR )
20 simpl3 1004 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  C  e.  RR )
2119, 20remulcld 8007 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  C )  e.  RR )
2221recnd 8005 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  C )  e.  CC )
23 simpl1 1002 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  A  e.  RR )
2423recnd 8005 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  A  e.  CC )
25 simpl2 1003 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  B  e.  RR )
2625recnd 8005 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  B  e.  CC )
2724, 26addcld 7996 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  +  B )  e.  CC )
2822, 27negsubdi2d 8303 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  -u ( ( 2  x.  C )  -  ( A  +  B ) )  =  ( ( A  +  B )  -  (
2  x.  C ) ) )
2923, 25readdcld 8006 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  +  B )  e.  RR )
3023, 25resubcld 8357 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  -  B )  e.  RR )
31262timesd 9180 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  B )  =  ( B  +  B
) )
3224, 26, 26pnncand 8326 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  +  B )  -  ( A  -  B ) )  =  ( B  +  B
) )
3331, 32eqtr4d 2225 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  B )  =  ( ( A  +  B )  -  ( A  -  B )
) )
34 2rp 9677 . . . . . . . . . . . 12  |-  2  e.  RR+
3534a1i 9 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  2  e.  RR+ )
36 simprr 531 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  B  <  C )
3725, 20, 35, 36ltmul2dd 9772 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  B )  < 
( 2  x.  C
) )
3833, 37eqbrtrrd 4042 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  +  B )  -  ( A  -  B ) )  < 
( 2  x.  C
) )
3929, 30, 21, 38ltsub23d 8526 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  +  B )  -  ( 2  x.  C ) )  < 
( A  -  B
) )
4028, 39eqbrtrd 4040 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  -u ( ( 2  x.  C )  -  ( A  +  B ) )  < 
( A  -  B
) )
4124, 26, 24nppcan3d 8314 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  -  B )  +  ( A  +  B ) )  =  ( A  +  A
) )
42242timesd 9180 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  A )  =  ( A  +  A
) )
4341, 42eqtr4d 2225 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  -  B )  +  ( A  +  B ) )  =  ( 2  x.  A
) )
44 simprl 529 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  A  <  C )
4523, 20, 35, 44ltmul2dd 9772 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  A )  < 
( 2  x.  C
) )
4643, 45eqbrtrd 4040 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  -  B )  +  ( A  +  B ) )  < 
( 2  x.  C
) )
4730, 29, 21ltaddsubd 8521 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( (
( A  -  B
)  +  ( A  +  B ) )  <  ( 2  x.  C )  <->  ( A  -  B )  <  (
( 2  x.  C
)  -  ( A  +  B ) ) ) )
4846, 47mpbid 147 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  -  B )  <  (
( 2  x.  C
)  -  ( A  +  B ) ) )
4940, 48jca 306 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( -u (
( 2  x.  C
)  -  ( A  +  B ) )  <  ( A  -  B )  /\  ( A  -  B )  <  ( ( 2  x.  C )  -  ( A  +  B )
) ) )
5021, 29resubcld 8357 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( (
2  x.  C )  -  ( A  +  B ) )  e.  RR )
5130, 50absltd 11202 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( abs `  ( A  -  B ) )  < 
( ( 2  x.  C )  -  ( A  +  B )
)  <->  ( -u (
( 2  x.  C
)  -  ( A  +  B ) )  <  ( A  -  B )  /\  ( A  -  B )  <  ( ( 2  x.  C )  -  ( A  +  B )
) ) ) )
5249, 51mpbird 167 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( abs `  ( A  -  B
) )  <  (
( 2  x.  C
)  -  ( A  +  B ) ) )
5330recnd 8005 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  -  B )  e.  CC )
5453abscld 11209 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( abs `  ( A  -  B
) )  e.  RR )
5529, 54, 21ltaddsub2d 8522 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  < 
( 2  x.  C
)  <->  ( abs `  ( A  -  B )
)  <  ( (
2  x.  C )  -  ( A  +  B ) ) ) )
5652, 55mpbird 167 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  <  (
2  x.  C ) )
5729, 54readdcld 8006 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  e.  RR )
5857, 20, 35ltdivmuld 9767 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  <  C  <->  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  <  (
2  x.  C ) ) )
5956, 58mpbird 167 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  <  C
)
6017, 59eqbrtrd 4040 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  sup ( { A ,  B } ,  RR ,  <  )  <  C )
6114, 60impbida 596 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( sup ( { A ,  B } ,  RR ,  <  )  <  C  <->  ( A  <  C  /\  B  < 
C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160   {cpr 3608   class class class wbr 4018   ` cfv 5231  (class class class)co 5891   supcsup 7000   RRcr 7829    + caddc 7833    x. cmul 7835    < clt 8011    <_ cle 8012    - cmin 8147   -ucneg 8148    / cdiv 8648   2c2 8989   RR+crp 9672   abscabs 11025
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-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-mulrcl 7929  ax-addcom 7930  ax-mulcom 7931  ax-addass 7932  ax-mulass 7933  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-1rid 7937  ax-0id 7938  ax-rnegex 7939  ax-precex 7940  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946  ax-pre-mulgt0 7947  ax-pre-mulext 7948  ax-arch 7949  ax-caucvg 7950
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  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-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-sup 7002  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-reap 8551  df-ap 8558  df-div 8649  df-inn 8939  df-2 8997  df-3 8998  df-4 8999  df-n0 9196  df-z 9273  df-uz 9548  df-rp 9673  df-seqfrec 10465  df-exp 10539  df-cj 10870  df-re 10871  df-im 10872  df-rsqrt 11026  df-abs 11027
This theorem is referenced by:  ltmininf  11262  xrmaxltsup  11285  suplociccreex  14505
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