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Theorem maxltsup 10958
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 969 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  A  e.  RR )
2 simpl2 970 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  B  e.  RR )
3 maxcl 10950 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR )
41, 2, 3syl2anc 408 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR )
5 simpl3 971 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  C  e.  RR )
6 maxle1 10951 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  <_  sup ( { A ,  B } ,  RR ,  <  )
)
763adant3 986 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  <_  sup ( { A ,  B } ,  RR ,  <  ) )
87adantr 274 . . . 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 7855 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  A  <  C )
11 maxle2 10952 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  <_  sup ( { A ,  B } ,  RR ,  <  )
)
121, 2, 11syl2anc 408 . . . 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 7855 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  B  <  C )
1410, 13jca 304 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  ( A  <  C  /\  B  < 
C ) )
15 maxabs 10949 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )
16153adant3 986 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  =  ( ( ( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
1716adantr 274 . . 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 8758 . . . . . . . . . . . 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 971 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  C  e.  RR )
2119, 20remulcld 7764 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  C )  e.  RR )
2221recnd 7762 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  C )  e.  CC )
23 simpl1 969 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  A  e.  RR )
2423recnd 7762 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  A  e.  CC )
25 simpl2 970 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  B  e.  RR )
2625recnd 7762 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  B  e.  CC )
2724, 26addcld 7753 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  +  B )  e.  CC )
2822, 27negsubdi2d 8057 . . . . . . . 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 7763 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  +  B )  e.  RR )
3023, 25resubcld 8111 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  -  B )  e.  RR )
31262timesd 8930 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  B )  =  ( B  +  B
) )
3224, 26, 26pnncand 8080 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  +  B )  -  ( A  -  B ) )  =  ( B  +  B
) )
3331, 32eqtr4d 2153 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  B )  =  ( ( A  +  B )  -  ( A  -  B )
) )
34 2rp 9414 . . . . . . . . . . . 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 506 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  B  <  C )
3725, 20, 35, 36ltmul2dd 9508 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  B )  < 
( 2  x.  C
) )
3833, 37eqbrtrrd 3922 . . . . . . . . 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 8280 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  +  B )  -  ( 2  x.  C ) )  < 
( A  -  B
) )
4028, 39eqbrtrd 3920 . . . . . . 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 8068 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  -  B )  +  ( A  +  B ) )  =  ( A  +  A
) )
42242timesd 8930 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  A )  =  ( A  +  A
) )
4341, 42eqtr4d 2153 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  -  B )  +  ( A  +  B ) )  =  ( 2  x.  A
) )
44 simprl 505 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  A  <  C )
4523, 20, 35, 44ltmul2dd 9508 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  A )  < 
( 2  x.  C
) )
4643, 45eqbrtrd 3920 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  -  B )  +  ( A  +  B ) )  < 
( 2  x.  C
) )
4730, 29, 21ltaddsubd 8275 . . . . . . . 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 304 . . . . . 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 8111 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( (
2  x.  C )  -  ( A  +  B ) )  e.  RR )
5130, 50absltd 10914 . . . . . 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 7762 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  -  B )  e.  CC )
5453abscld 10921 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( abs `  ( A  -  B
) )  e.  RR )
5529, 54, 21ltaddsub2d 8276 . . . . 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 7763 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  e.  RR )
5857, 20, 35ltdivmuld 9503 . . . 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 3920 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  sup ( { A ,  B } ,  RR ,  <  )  <  C )
6114, 60impbida 570 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 947    = wceq 1316    e. wcel 1465   {cpr 3498   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   supcsup 6837   RRcr 7587    + caddc 7591    x. cmul 7593    < clt 7768    <_ cle 7769    - cmin 7901   -ucneg 7902    / cdiv 8400   2c2 8739   RR+crp 9409   abscabs 10737
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-sup 6839  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-n0 8946  df-z 9023  df-uz 9295  df-rp 9410  df-seqfrec 10187  df-exp 10261  df-cj 10582  df-re 10583  df-im 10584  df-rsqrt 10738  df-abs 10739
This theorem is referenced by:  ltmininf  10974  xrmaxltsup  10995  suplociccreex  12698
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