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Theorem maxabslemlub 11009
Description: Lemma for maxabs 11011. 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 7817 . . . . . 6  |-  ( ph  ->  ( A  +  B
)  e.  RR )
63recnd 7816 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
74recnd 7816 . . . . . . . 8  |-  ( ph  ->  B  e.  CC )
86, 7subcld 8095 . . . . . . 7  |-  ( ph  ->  ( A  -  B
)  e.  CC )
98abscld 10983 . . . . . 6  |-  ( ph  ->  ( abs `  ( A  -  B )
)  e.  RR )
105, 9readdcld 7817 . . . . 5  |-  ( ph  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  e.  RR )
1110rehalfcld 8988 . . . 4  |-  ( ph  ->  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  e.  RR )
12 axltwlin 7854 . . . 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 1217 . . 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 274 . . . . 5  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  C  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )
163adantr 274 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  A  e.  RR )
174adantr 274 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  B  e.  RR )
1816, 17resubcld 8165 . . . . . . . 8  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( A  -  B )  e.  RR )
19 2re 8812 . . . . . . . . . . . . . 14  |-  2  e.  RR
2019a1i 9 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  2  e.  RR )
2120, 16remulcld 7818 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( 2  x.  A )  e.  RR )
2221recnd 7816 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( 2  x.  A )  e.  CC )
236adantr 274 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  A  e.  CC )
247adantr 274 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  B  e.  CC )
2522, 23, 24subsub4d 8126 . . . . . . . . . 10  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
( 2  x.  A
)  -  A )  -  B )  =  ( ( 2  x.  A )  -  ( A  +  B )
) )
26 2cnd 8815 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  2  e.  CC )
2726, 23mulsubfacd 8202 . . . . . . . . . . . . 13  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
2  x.  A )  -  A )  =  ( ( 2  -  1 )  x.  A
) )
28 2m1e1 8860 . . . . . . . . . . . . . 14  |-  ( 2  -  1 )  =  1
2928oveq1i 5790 . . . . . . . . . . . . 13  |-  ( ( 2  -  1 )  x.  A )  =  ( 1  x.  A
)
3027, 29eqtrdi 2189 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
2  x.  A )  -  A )  =  ( 1  x.  A
) )
3123mulid2d 7806 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( 1  x.  A )  =  A )
3230, 31eqtrd 2173 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
2  x.  A )  -  A )  =  A )
3332oveq1d 5795 . . . . . . . . . 10  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
( 2  x.  A
)  -  A )  -  B )  =  ( A  -  B
) )
3425, 33eqtr3d 2175 . . . . . . . . 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 274 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  e.  RR )
37 2rp 9473 . . . . . . . . . . . . 13  |-  2  e.  RR+
3837a1i 9 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  2  e.  RR+ )
3916, 36, 38ltmuldiv2d 9560 . . . . . . . . . . 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 274 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( A  +  B )  e.  RR )
429adantr 274 . . . . . . . . . . 11  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( abs `  ( A  -  B
) )  e.  RR )
4321, 41, 42ltsubadd2d 8327 . . . . . . . . . 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 3958 . . . . . . . 8  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( A  -  B )  <  ( abs `  ( A  -  B ) ) )
46 ltabs 10889 . . . . . . . 8  |-  ( ( ( A  -  B
)  e.  RR  /\  ( A  -  B
)  <  ( abs `  ( A  -  B
) ) )  -> 
( A  -  B
)  <  0 )
4718, 45, 46syl2anc 409 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( A  -  B )  <  0
)
4816, 17sublt0d 8354 . . . . . . 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 11008 . . . . 5  |-  ( (
ph  /\  A  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 ) )  ->  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  =  B )
5115, 50breqtrd 3960 . . . 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 778 . 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 698    e. wcel 1481   class class class wbr 3935   ` cfv 5129  (class class class)co 5780   CCcc 7640   RRcr 7641   0cc0 7642   1c1 7643    + caddc 7645    x. cmul 7647    < clt 7822    - cmin 7955    / cdiv 8454   2c2 8793   RR+crp 9468   abscabs 10799
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4049  ax-sep 4052  ax-nul 4060  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-setind 4458  ax-iinf 4508  ax-cnex 7733  ax-resscn 7734  ax-1cn 7735  ax-1re 7736  ax-icn 7737  ax-addcl 7738  ax-addrcl 7739  ax-mulcl 7740  ax-mulrcl 7741  ax-addcom 7742  ax-mulcom 7743  ax-addass 7744  ax-mulass 7745  ax-distr 7746  ax-i2m1 7747  ax-0lt1 7748  ax-1rid 7749  ax-0id 7750  ax-rnegex 7751  ax-precex 7752  ax-cnre 7753  ax-pre-ltirr 7754  ax-pre-ltwlin 7755  ax-pre-lttrn 7756  ax-pre-apti 7757  ax-pre-ltadd 7758  ax-pre-mulgt0 7759  ax-pre-mulext 7760  ax-arch 7761  ax-caucvg 7762
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-if 3478  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-int 3778  df-iun 3821  df-br 3936  df-opab 3996  df-mpt 3997  df-tr 4033  df-id 4221  df-po 4224  df-iso 4225  df-iord 4294  df-on 4296  df-ilim 4297  df-suc 4299  df-iom 4511  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-f1 5134  df-fo 5135  df-f1o 5136  df-fv 5137  df-riota 5736  df-ov 5783  df-oprab 5784  df-mpo 5785  df-1st 6044  df-2nd 6045  df-recs 6208  df-frec 6294  df-pnf 7824  df-mnf 7825  df-xr 7826  df-ltxr 7827  df-le 7828  df-sub 7957  df-neg 7958  df-reap 8359  df-ap 8366  df-div 8455  df-inn 8743  df-2 8801  df-3 8802  df-4 8803  df-n0 9000  df-z 9077  df-uz 9349  df-rp 9469  df-seqfrec 10248  df-exp 10322  df-cj 10644  df-re 10645  df-im 10646  df-rsqrt 10800  df-abs 10801
This theorem is referenced by:  maxabslemval  11010  maxleastlt  11017
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