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Theorem maxltsup 10766
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 949 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  A  e.  RR )
2 simpl2 950 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  B  e.  RR )
3 maxcl 10758 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR )
41, 2, 3syl2anc 404 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR )
5 simpl3 951 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  C  e.  RR )
6 maxle1 10759 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  <_  sup ( { A ,  B } ,  RR ,  <  )
)
763adant3 966 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  <_  sup ( { A ,  B } ,  RR ,  <  ) )
87adantr 271 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  A  <_  sup ( { A ,  B } ,  RR ,  <  ) )
9 simpr 109 . . . 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 7705 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  A  <  C )
11 maxle2 10760 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  <_  sup ( { A ,  B } ,  RR ,  <  )
)
121, 2, 11syl2anc 404 . . . 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 7705 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  B  <  C )
1410, 13jca 301 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  ( A  <  C  /\  B  < 
C ) )
15 maxabs 10757 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )
16153adant3 966 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  =  ( ( ( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
1716adantr 271 . . 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 8590 . . . . . . . . . . . 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 951 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  C  e.  RR )
2119, 20remulcld 7615 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  C )  e.  RR )
2221recnd 7613 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  C )  e.  CC )
23 simpl1 949 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  A  e.  RR )
2423recnd 7613 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  A  e.  CC )
25 simpl2 950 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  B  e.  RR )
2625recnd 7613 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  B  e.  CC )
2724, 26addcld 7604 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  +  B )  e.  CC )
2822, 27negsubdi2d 7906 . . . . . . . 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 7614 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  +  B )  e.  RR )
3023, 25resubcld 7956 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  -  B )  e.  RR )
31262timesd 8756 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  B )  =  ( B  +  B
) )
3224, 26, 26pnncand 7929 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  +  B )  -  ( A  -  B ) )  =  ( B  +  B
) )
3331, 32eqtr4d 2130 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  B )  =  ( ( A  +  B )  -  ( A  -  B )
) )
34 2rp 9238 . . . . . . . . . . . 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 500 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  B  <  C )
3725, 20, 35, 36ltmul2dd 9329 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  B )  < 
( 2  x.  C
) )
3833, 37eqbrtrrd 3889 . . . . . . . . 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 8124 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  +  B )  -  ( 2  x.  C ) )  < 
( A  -  B
) )
4028, 39eqbrtrd 3887 . . . . . . 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 7917 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  -  B )  +  ( A  +  B ) )  =  ( A  +  A
) )
42242timesd 8756 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  A )  =  ( A  +  A
) )
4341, 42eqtr4d 2130 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  -  B )  +  ( A  +  B ) )  =  ( 2  x.  A
) )
44 simprl 499 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  A  <  C )
4523, 20, 35, 44ltmul2dd 9329 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  A )  < 
( 2  x.  C
) )
4643, 45eqbrtrd 3887 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  -  B )  +  ( A  +  B ) )  < 
( 2  x.  C
) )
4730, 29, 21ltaddsubd 8119 . . . . . . . 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 146 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  -  B )  <  (
( 2  x.  C
)  -  ( A  +  B ) ) )
4940, 48jca 301 . . . . . 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 7956 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( (
2  x.  C )  -  ( A  +  B ) )  e.  RR )
5130, 50absltd 10722 . . . . . 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 166 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( abs `  ( A  -  B
) )  <  (
( 2  x.  C
)  -  ( A  +  B ) ) )
5330recnd 7613 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  -  B )  e.  CC )
5453abscld 10729 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( abs `  ( A  -  B
) )  e.  RR )
5529, 54, 21ltaddsub2d 8120 . . . . 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 166 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  <  (
2  x.  C ) )
5729, 54readdcld 7614 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  e.  RR )
5857, 20, 35ltdivmuld 9324 . . . 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 166 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  <  C
)
6017, 59eqbrtrd 3887 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  sup ( { A ,  B } ,  RR ,  <  )  <  C )
6114, 60impbida 564 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 103    <-> wb 104    /\ w3a 927    = wceq 1296    e. wcel 1445   {cpr 3467   class class class wbr 3867   ` cfv 5049  (class class class)co 5690   supcsup 6757   RRcr 7446    + caddc 7450    x. cmul 7452    < clt 7619    <_ cle 7620    - cmin 7750   -ucneg 7751    / cdiv 8236   2c2 8571   RR+crp 9233   abscabs 10545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560  ax-arch 7561  ax-caucvg 7562
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-ilim 4220  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-frec 6194  df-sup 6759  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-2 8579  df-3 8580  df-4 8581  df-n0 8772  df-z 8849  df-uz 9119  df-rp 9234  df-seqfrec 10001  df-exp 10070  df-cj 10391  df-re 10392  df-im 10393  df-rsqrt 10546  df-abs 10547
This theorem is referenced by:  ltmininf  10781  xrmaxltsup  10801
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