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Theorem maxabslemval 10295
Description: Lemma for maxabs 10296. Value of the supremum. (Contributed by Jim Kingdon, 22-Dec-2021.)
Assertion
Ref Expression
maxabslemval  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  e.  RR  /\  A. x  e.  { A ,  B }  -.  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  <  x  /\  A. x  e.  RR  (
x  <  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  ->  E. z  e.  { A ,  B } x  <  z ) ) )
Distinct variable groups:    x, A, z   
x, B, z

Proof of Theorem maxabslemval
StepHypRef Expression
1 readdcl 7213 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
2 simpl 107 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
32recnd 7261 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  CC )
4 simpr 108 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
54recnd 7261 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
63, 5subcld 7538 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  CC )
76abscld 10268 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( A  -  B )
)  e.  RR )
81, 7readdcld 7262 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  e.  RR )
98rehalfcld 8396 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  e.  RR )
10 vex 2613 . . . . 5  |-  x  e. 
_V
1110elpr 3437 . . . 4  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
12 maxabsle 10291 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  <_  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
132, 9, 12lensymd 7350 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  A )
14 breq2 3809 . . . . . . 7  |-  ( x  =  A  ->  (
( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x  <->  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  <  A
) )
1514notbid 625 . . . . . 6  |-  ( x  =  A  ->  ( -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x  <->  -.  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  <  A ) )
1613, 15syl5ibrcom 155 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  =  A  ->  -.  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  <  x
) )
17 maxabsle 10291 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  B  <_  ( (
( B  +  A
)  +  ( abs `  ( B  -  A
) ) )  / 
2 ) )
1817ancoms 264 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  <_  ( (
( B  +  A
)  +  ( abs `  ( B  -  A
) ) )  / 
2 ) )
195, 3addcomd 7378 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  +  A
)  =  ( A  +  B ) )
205, 3abssubd 10280 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( B  -  A )
)  =  ( abs `  ( A  -  B
) ) )
2119, 20oveq12d 5581 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( B  +  A )  +  ( abs `  ( B  -  A ) ) )  =  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) ) )
2221oveq1d 5578 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( B  +  A )  +  ( abs `  ( B  -  A )
) )  /  2
)  =  ( ( ( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
2318, 22breqtrd 3829 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  <_  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
244, 9, 23lensymd 7350 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  B )
25 breq2 3809 . . . . . . 7  |-  ( x  =  B  ->  (
( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x  <->  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  <  B
) )
2625notbid 625 . . . . . 6  |-  ( x  =  B  ->  ( -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x  <->  -.  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  <  B ) )
2724, 26syl5ibrcom 155 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  =  B  ->  -.  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  <  x
) )
2816, 27jaod 670 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( x  =  A  \/  x  =  B )  ->  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x )
)
2911, 28syl5bi 150 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  { A ,  B }  ->  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x )
)
3029ralrimiv 2438 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A. x  e.  { A ,  B }  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x )
31 prid1g 3514 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  { A ,  B } )
3231ad4antr 478 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  /\  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  /\  x  <  A )  ->  A  e.  { A ,  B } )
33 breq2 3809 . . . . . . 7  |-  ( z  =  A  ->  (
x  <  z  <->  x  <  A ) )
3433rspcev 2710 . . . . . 6  |-  ( ( A  e.  { A ,  B }  /\  x  <  A )  ->  E. z  e.  { A ,  B } x  <  z )
3532, 34sylancom 411 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  /\  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  /\  x  <  A )  ->  E. z  e.  { A ,  B } x  <  z )
36 prid2g 3515 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  { A ,  B } )
3736ad4antlr 479 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  /\  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  /\  x  <  B )  ->  B  e.  { A ,  B } )
38 breq2 3809 . . . . . . 7  |-  ( z  =  B  ->  (
x  <  z  <->  x  <  B ) )
3938rspcev 2710 . . . . . 6  |-  ( ( B  e.  { A ,  B }  /\  x  <  B )  ->  E. z  e.  { A ,  B } x  <  z )
4037, 39sylancom 411 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  /\  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  /\  x  <  B )  ->  E. z  e.  { A ,  B } x  <  z )
412ad2antrr 472 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  /\  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  ->  A  e.  RR )
424ad2antrr 472 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  /\  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  ->  B  e.  RR )
43 simplr 497 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  /\  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  ->  x  e.  RR )
44 simpr 108 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  /\  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  ->  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )
4541, 42, 43, 44maxabslemlub 10294 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  /\  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  ->  (
x  <  A  \/  x  <  B ) )
4635, 40, 45mpjaodan 745 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  /\  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  ->  E. z  e.  { A ,  B } x  <  z )
4746ex 113 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  ->  ( x  < 
( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  ->  E. z  e.  { A ,  B } x  <  z ) )
4847ralrimiva 2439 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A. x  e.  RR  ( x  <  ( ( ( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  ->  E. z  e.  { A ,  B } x  <  z ) )
499, 30, 483jca 1119 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  e.  RR  /\  A. x  e.  { A ,  B }  -.  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  <  x  /\  A. x  e.  RR  (
x  <  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  ->  E. z  e.  { A ,  B } x  <  z ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    \/ wo 662    /\ w3a 920    = wceq 1285    e. wcel 1434   A.wral 2353   E.wrex 2354   {cpr 3417   class class class wbr 3805   ` cfv 4952  (class class class)co 5563   RRcr 7094    + caddc 7098    < clt 7267    <_ cle 7268    - cmin 7398    / cdiv 7879   2c2 8208   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:  maxabs  10296  maxleast  10300
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