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Theorem maxabslemlub 10294
Description: Lemma for maxabs 10296. 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 7262 . . . . . 6  |-  ( ph  ->  ( A  +  B
)  e.  RR )
63recnd 7261 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
74recnd 7261 . . . . . . . 8  |-  ( ph  ->  B  e.  CC )
86, 7subcld 7538 . . . . . . 7  |-  ( ph  ->  ( A  -  B
)  e.  CC )
98abscld 10268 . . . . . 6  |-  ( ph  ->  ( abs `  ( A  -  B )
)  e.  RR )
105, 9readdcld 7262 . . . . 5  |-  ( ph  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  e.  RR )
1110rehalfcld 8396 . . . 4  |-  ( ph  ->  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  e.  RR )
12 axltwlin 7299 . . . 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 1170 . . 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 270 . . . . 5  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  C  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )
163adantr 270 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  A  e.  RR )
174adantr 270 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  B  e.  RR )
1816, 17resubcld 7604 . . . . . . . 8  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( A  -  B )  e.  RR )
19 2re 8228 . . . . . . . . . . . . . 14  |-  2  e.  RR
2019a1i 9 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  2  e.  RR )
2120, 16remulcld 7263 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( 2  x.  A )  e.  RR )
2221recnd 7261 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( 2  x.  A )  e.  CC )
236adantr 270 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  A  e.  CC )
247adantr 270 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  B  e.  CC )
2522, 23, 24subsub4d 7569 . . . . . . . . . 10  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
( 2  x.  A
)  -  A )  -  B )  =  ( ( 2  x.  A )  -  ( A  +  B )
) )
26 2cnd 8231 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  2  e.  CC )
2726, 23mulsubfacd 7641 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
2  x.  A )  -  A )  =  ( ( 2  -  1 )  x.  A
) )
28 2m1e1 8275 . . . . . . . . . . . . . 14  |-  ( 2  -  1 )  =  1
2928oveq1i 5573 . . . . . . . . . . . . 13  |-  ( ( 2  -  1 )  x.  A )  =  ( 1  x.  A
)
3027, 29syl6eq 2131 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
2  x.  A )  -  A )  =  ( 1  x.  A
) )
3123mulid2d 7251 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( 1  x.  A )  =  A )
3230, 31eqtrd 2115 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
2  x.  A )  -  A )  =  A )
3332oveq1d 5578 . . . . . . . . . 10  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
( 2  x.  A
)  -  A )  -  B )  =  ( A  -  B
) )
3425, 33eqtr3d 2117 . . . . . . . . 9  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
2  x.  A )  -  ( A  +  B ) )  =  ( A  -  B
) )
35 simpr 108 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )
3610adantr 270 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  e.  RR )
37 2rp 8872 . . . . . . . . . . . . 13  |-  2  e.  RR+
3837a1i 9 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  2  e.  RR+ )
3916, 36, 38ltmuldiv2d 8955 . . . . . . . . . . 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 165 . . . . . . . . . 10  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( 2  x.  A )  < 
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) ) )
415adantr 270 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( A  +  B )  e.  RR )
429adantr 270 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( abs `  ( A  -  B
) )  e.  RR )
4321, 41, 42ltsubadd2d 7762 . . . . . . . . . 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 165 . . . . . . . . 9  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
2  x.  A )  -  ( A  +  B ) )  < 
( abs `  ( A  -  B )
) )
4534, 44eqbrtrrd 3827 . . . . . . . 8  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( A  -  B )  <  ( abs `  ( A  -  B ) ) )
46 ltabs 10174 . . . . . . . 8  |-  ( ( ( A  -  B
)  e.  RR  /\  ( A  -  B
)  <  ( abs `  ( A  -  B
) ) )  -> 
( A  -  B
)  <  0 )
4718, 45, 46syl2anc 403 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( A  -  B )  <  0
)
4816, 17sublt0d 7789 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( ( A  -  B )  <  0  <->  A  <  B ) )
4947, 48mpbid 145 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  A  <  B )
5016, 17, 49maxabslemab 10293 . . . . 5  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  =  B )
5115, 50breqtrd 3829 . . . 4  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  C  <  B )
5251ex 113 . . 3  |-  ( ph  ->  ( A  <  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  ->  C  <  B
) )
5352orim2d 735 . 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 102    \/ wo 662    e. wcel 1434   class class class wbr 3805   ` cfv 4952  (class class class)co 5563   CCcc 7093   RRcr 7094   0cc0 7095   1c1 7096    + caddc 7098    x. cmul 7100    < clt 7267    - cmin 7398    / cdiv 7879   2c2 8208   RR+crp 8867   abscabs 10084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209  ax-caucvg 7210
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-n0 8408  df-z 8485  df-uz 8753  df-rp 8868  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086
This theorem is referenced by:  maxabslemval  10295  maxleastlt  10302
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