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Theorem maxltsup 11240
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 1001 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  A  e.  RR )
2 simpl2 1002 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  B  e.  RR )
3 maxcl 11232 . . . . 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 1003 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  C  e.  RR )
6 maxle1 11233 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  <_  sup ( { A ,  B } ,  RR ,  <  )
)
763adant3 1018 . . . . 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 8095 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  A  <  C )
11 maxle2 11234 . . . . 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 8095 . . 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 11231 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )
16153adant3 1018 . . . 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 9002 . . . . . . . . . . . 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 1003 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  C  e.  RR )
2119, 20remulcld 8001 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  C )  e.  RR )
2221recnd 7999 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  C )  e.  CC )
23 simpl1 1001 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  A  e.  RR )
2423recnd 7999 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  A  e.  CC )
25 simpl2 1002 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  B  e.  RR )
2625recnd 7999 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  B  e.  CC )
2724, 26addcld 7990 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  +  B )  e.  CC )
2822, 27negsubdi2d 8297 . . . . . . . 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 8000 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  +  B )  e.  RR )
3023, 25resubcld 8351 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  -  B )  e.  RR )
31262timesd 9174 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  B )  =  ( B  +  B
) )
3224, 26, 26pnncand 8320 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  +  B )  -  ( A  -  B ) )  =  ( B  +  B
) )
3331, 32eqtr4d 2223 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  B )  =  ( ( A  +  B )  -  ( A  -  B )
) )
34 2rp 9671 . . . . . . . . . . . 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 9766 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  B )  < 
( 2  x.  C
) )
3833, 37eqbrtrrd 4039 . . . . . . . . 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 8520 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  +  B )  -  ( 2  x.  C ) )  < 
( A  -  B
) )
4028, 39eqbrtrd 4037 . . . . . . 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 8308 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  -  B )  +  ( A  +  B ) )  =  ( A  +  A
) )
42242timesd 9174 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  A )  =  ( A  +  A
) )
4341, 42eqtr4d 2223 . . . . . . . . 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 9766 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  A )  < 
( 2  x.  C
) )
4643, 45eqbrtrd 4037 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  -  B )  +  ( A  +  B ) )  < 
( 2  x.  C
) )
4730, 29, 21ltaddsubd 8515 . . . . . . . 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 8351 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( (
2  x.  C )  -  ( A  +  B ) )  e.  RR )
5130, 50absltd 11196 . . . . . 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 7999 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  -  B )  e.  CC )
5453abscld 11203 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( abs `  ( A  -  B
) )  e.  RR )
5529, 54, 21ltaddsub2d 8516 . . . . 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 8000 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  e.  RR )
5857, 20, 35ltdivmuld 9761 . . . 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 4037 . 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 979    = wceq 1363    e. wcel 2158   {cpr 3605   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   supcsup 6994   RRcr 7823    + caddc 7827    x. cmul 7829    < clt 8005    <_ cle 8006    - cmin 8141   -ucneg 8142    / cdiv 8642   2c2 8983   RR+crp 9666   abscabs 11019
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943  ax-caucvg 7944
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-sup 6996  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-n0 9190  df-z 9267  df-uz 9542  df-rp 9667  df-seqfrec 10459  df-exp 10533  df-cj 10864  df-re 10865  df-im 10866  df-rsqrt 11020  df-abs 11021
This theorem is referenced by:  ltmininf  11256  xrmaxltsup  11279  suplociccreex  14342
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