ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  maxabslemlub Unicode version

Theorem maxabslemlub 10819
Description: Lemma for maxabs 10821. A least upper bound for  { A ,  B }. (Contributed by Jim Kingdon, 20-Dec-2021.)
Hypotheses
Ref Expression
maxabslemlub.a  |-  ( ph  ->  A  e.  RR )
maxabslemlub.b  |-  ( ph  ->  B  e.  RR )
maxabslemlub.c  |-  ( ph  ->  C  e.  RR )
maxabslemlub.clt  |-  ( ph  ->  C  <  ( ( ( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
Assertion
Ref Expression
maxabslemlub  |-  ( ph  ->  ( C  <  A  \/  C  <  B ) )

Proof of Theorem maxabslemlub
StepHypRef Expression
1 maxabslemlub.clt . . 3  |-  ( ph  ->  C  <  ( ( ( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
2 maxabslemlub.c . . . 4  |-  ( ph  ->  C  e.  RR )
3 maxabslemlub.a . . . . . . 7  |-  ( ph  ->  A  e.  RR )
4 maxabslemlub.b . . . . . . 7  |-  ( ph  ->  B  e.  RR )
53, 4readdcld 7667 . . . . . 6  |-  ( ph  ->  ( A  +  B
)  e.  RR )
63recnd 7666 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
74recnd 7666 . . . . . . . 8  |-  ( ph  ->  B  e.  CC )
86, 7subcld 7944 . . . . . . 7  |-  ( ph  ->  ( A  -  B
)  e.  CC )
98abscld 10793 . . . . . 6  |-  ( ph  ->  ( abs `  ( A  -  B )
)  e.  RR )
105, 9readdcld 7667 . . . . 5  |-  ( ph  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  e.  RR )
1110rehalfcld 8818 . . . 4  |-  ( ph  ->  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  e.  RR )
12 axltwlin 7704 . . . 4  |-  ( ( C  e.  RR  /\  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  e.  RR  /\  A  e.  RR )  ->  ( C  <  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  ->  ( C  < 
A  \/  A  < 
( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) ) ) )
132, 11, 3, 12syl3anc 1184 . . 3  |-  ( ph  ->  ( C  <  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  ->  ( C  < 
A  \/  A  < 
( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) ) ) )
141, 13mpd 13 . 2  |-  ( ph  ->  ( C  <  A  \/  A  <  ( ( ( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) ) )
151adantr 272 . . . . 5  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  C  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )
163adantr 272 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  A  e.  RR )
174adantr 272 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  B  e.  RR )
1816, 17resubcld 8010 . . . . . . . 8  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( A  -  B )  e.  RR )
19 2re 8648 . . . . . . . . . . . . . 14  |-  2  e.  RR
2019a1i 9 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  2  e.  RR )
2120, 16remulcld 7668 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( 2  x.  A )  e.  RR )
2221recnd 7666 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( 2  x.  A )  e.  CC )
236adantr 272 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  A  e.  CC )
247adantr 272 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  B  e.  CC )
2522, 23, 24subsub4d 7975 . . . . . . . . . 10  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
( 2  x.  A
)  -  A )  -  B )  =  ( ( 2  x.  A )  -  ( A  +  B )
) )
26 2cnd 8651 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  2  e.  CC )
2726, 23mulsubfacd 8047 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
2  x.  A )  -  A )  =  ( ( 2  -  1 )  x.  A
) )
28 2m1e1 8696 . . . . . . . . . . . . . 14  |-  ( 2  -  1 )  =  1
2928oveq1i 5716 . . . . . . . . . . . . 13  |-  ( ( 2  -  1 )  x.  A )  =  ( 1  x.  A
)
3027, 29syl6eq 2148 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
2  x.  A )  -  A )  =  ( 1  x.  A
) )
3123mulid2d 7656 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( 1  x.  A )  =  A )
3230, 31eqtrd 2132 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
2  x.  A )  -  A )  =  A )
3332oveq1d 5721 . . . . . . . . . 10  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
( 2  x.  A
)  -  A )  -  B )  =  ( A  -  B
) )
3425, 33eqtr3d 2134 . . . . . . . . 9  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
2  x.  A )  -  ( A  +  B ) )  =  ( A  -  B
) )
35 simpr 109 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )
3610adantr 272 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  e.  RR )
37 2rp 9296 . . . . . . . . . . . . 13  |-  2  e.  RR+
3837a1i 9 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  2  e.  RR+ )
3916, 36, 38ltmuldiv2d 9379 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
2  x.  A )  <  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  <->  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) ) )
4035, 39mpbird 166 . . . . . . . . . 10  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( 2  x.  A )  < 
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) ) )
415adantr 272 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( A  +  B )  e.  RR )
429adantr 272 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( abs `  ( A  -  B
) )  e.  RR )
4321, 41, 42ltsubadd2d 8171 . . . . . . . . . 10  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
( 2  x.  A
)  -  ( A  +  B ) )  <  ( abs `  ( A  -  B )
)  <->  ( 2  x.  A )  <  (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) ) ) )
4440, 43mpbird 166 . . . . . . . . 9  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
2  x.  A )  -  ( A  +  B ) )  < 
( abs `  ( A  -  B )
) )
4534, 44eqbrtrrd 3897 . . . . . . . 8  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( A  -  B )  <  ( abs `  ( A  -  B ) ) )
46 ltabs 10699 . . . . . . . 8  |-  ( ( ( A  -  B
)  e.  RR  /\  ( A  -  B
)  <  ( abs `  ( A  -  B
) ) )  -> 
( A  -  B
)  <  0 )
4718, 45, 46syl2anc 406 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( A  -  B )  <  0
)
4816, 17sublt0d 8198 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( ( A  -  B )  <  0  <->  A  <  B ) )
4947, 48mpbid 146 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  A  <  B )
5016, 17, 49maxabslemab 10818 . . . . 5  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  =  B )
5115, 50breqtrd 3899 . . . 4  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  C  <  B )
5251ex 114 . . 3  |-  ( ph  ->  ( A  <  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  ->  C  <  B
) )
5352orim2d 743 . 2  |-  ( ph  ->  ( ( C  < 
A  \/  A  < 
( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  ->  ( C  <  A  \/  C  <  B ) ) )
5414, 53mpd 13 1  |-  ( ph  ->  ( C  <  A  \/  C  <  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 670    e. wcel 1448   class class class wbr 3875   ` cfv 5059  (class class class)co 5706   CCcc 7498   RRcr 7499   0cc0 7500   1c1 7501    + caddc 7503    x. cmul 7505    < clt 7672    - cmin 7804    / cdiv 8293   2c2 8629   RR+crp 9291   abscabs 10609
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 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-rp 9292  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611
This theorem is referenced by:  maxabslemval  10820  maxleastlt  10827
  Copyright terms: Public domain W3C validator