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Theorem maxltsup 11118
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 985 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  A  e.  RR )
2 simpl2 986 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  B  e.  RR )
3 maxcl 11110 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR )
41, 2, 3syl2anc 409 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  sup ( { A ,  B } ,  RR ,  <  )  e.  RR )
5 simpl3 987 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  C  e.  RR )
6 maxle1 11111 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  <_  sup ( { A ,  B } ,  RR ,  <  )
)
763adant3 1002 . . . . 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 8000 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  sup ( { A ,  B } ,  RR ,  <  )  <  C
)  ->  A  <  C )
11 maxle2 11112 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  <_  sup ( { A ,  B } ,  RR ,  <  )
)
121, 2, 11syl2anc 409 . . . 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 8000 . . 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 11109 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  sup ( { A ,  B } ,  RR ,  <  )  =  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )
16153adant3 1002 . . . 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 8903 . . . . . . . . . . . 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 987 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  C  e.  RR )
2119, 20remulcld 7908 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  C )  e.  RR )
2221recnd 7906 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  C )  e.  CC )
23 simpl1 985 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  A  e.  RR )
2423recnd 7906 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  A  e.  CC )
25 simpl2 986 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  B  e.  RR )
2625recnd 7906 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  B  e.  CC )
2724, 26addcld 7897 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  +  B )  e.  CC )
2822, 27negsubdi2d 8202 . . . . . . . 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 7907 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  +  B )  e.  RR )
3023, 25resubcld 8256 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  -  B )  e.  RR )
31262timesd 9075 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  B )  =  ( B  +  B
) )
3224, 26, 26pnncand 8225 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  +  B )  -  ( A  -  B ) )  =  ( B  +  B
) )
3331, 32eqtr4d 2193 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  B )  =  ( ( A  +  B )  -  ( A  -  B )
) )
34 2rp 9565 . . . . . . . . . . . 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 522 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  B  <  C )
3725, 20, 35, 36ltmul2dd 9660 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  B )  < 
( 2  x.  C
) )
3833, 37eqbrtrrd 3988 . . . . . . . . 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 8425 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  +  B )  -  ( 2  x.  C ) )  < 
( A  -  B
) )
4028, 39eqbrtrd 3986 . . . . . . 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 8213 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  -  B )  +  ( A  +  B ) )  =  ( A  +  A
) )
42242timesd 9075 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  A )  =  ( A  +  A
) )
4341, 42eqtr4d 2193 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  -  B )  +  ( A  +  B ) )  =  ( 2  x.  A
) )
44 simprl 521 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  A  <  C )
4523, 20, 35, 44ltmul2dd 9660 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( 2  x.  A )  < 
( 2  x.  C
) )
4643, 45eqbrtrd 3986 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  -  B )  +  ( A  +  B ) )  < 
( 2  x.  C
) )
4730, 29, 21ltaddsubd 8420 . . . . . . . 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 8256 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( (
2  x.  C )  -  ( A  +  B ) )  e.  RR )
5130, 50absltd 11074 . . . . . 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 7906 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( A  -  B )  e.  CC )
5453abscld 11081 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( abs `  ( A  -  B
) )  e.  RR )
5529, 54, 21ltaddsub2d 8421 . . . . 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 7907 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  e.  RR )
5857, 20, 35ltdivmuld 9655 . . . 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 3986 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  /\  ( A  <  C  /\  B  <  C ) )  ->  sup ( { A ,  B } ,  RR ,  <  )  <  C )
6114, 60impbida 586 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 963    = wceq 1335    e. wcel 2128   {cpr 3561   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   supcsup 6926   RRcr 7731    + caddc 7735    x. cmul 7737    < clt 7912    <_ cle 7913    - cmin 8046   -ucneg 8047    / cdiv 8545   2c2 8884   RR+crp 9560   abscabs 10897
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-sup 6928  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-rp 9561  df-seqfrec 10345  df-exp 10419  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899
This theorem is referenced by:  ltmininf  11134  xrmaxltsup  11155  suplociccreex  13013
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