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Theorem maxabslemval 11011
Description: Lemma for maxabs 11012. 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 7769 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
2 simpl 108 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
32recnd 7817 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  CC )
4 simpr 109 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
54recnd 7817 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
63, 5subcld 8096 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  CC )
76abscld 10984 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( A  -  B )
)  e.  RR )
81, 7readdcld 7818 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  e.  RR )
98rehalfcld 8989 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  e.  RR )
10 vex 2692 . . . . 5  |-  x  e. 
_V
1110elpr 3552 . . . 4  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
12 maxabsle 11007 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  <_  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
132, 9, 12lensymd 7907 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  A )
14 breq2 3940 . . . . . . 7  |-  ( x  =  A  ->  (
( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x  <->  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  <  A
) )
1514notbid 657 . . . . . 6  |-  ( x  =  A  ->  ( -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x  <->  -.  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  <  A ) )
1613, 15syl5ibrcom 156 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  =  A  ->  -.  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  <  x
) )
17 maxabsle 11007 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  B  <_  ( (
( B  +  A
)  +  ( abs `  ( B  -  A
) ) )  / 
2 ) )
1817ancoms 266 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  <_  ( (
( B  +  A
)  +  ( abs `  ( B  -  A
) ) )  / 
2 ) )
195, 3addcomd 7936 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  +  A
)  =  ( A  +  B ) )
205, 3abssubd 10996 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( B  -  A )
)  =  ( abs `  ( A  -  B
) ) )
2119, 20oveq12d 5799 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( B  +  A )  +  ( abs `  ( B  -  A ) ) )  =  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) ) )
2221oveq1d 5796 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( B  +  A )  +  ( abs `  ( B  -  A )
) )  /  2
)  =  ( ( ( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
2318, 22breqtrd 3961 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  <_  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
244, 9, 23lensymd 7907 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  B )
25 breq2 3940 . . . . . . 7  |-  ( x  =  B  ->  (
( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x  <->  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  <  B
) )
2625notbid 657 . . . . . 6  |-  ( x  =  B  ->  ( -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x  <->  -.  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  <  B ) )
2724, 26syl5ibrcom 156 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  =  B  ->  -.  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  <  x
) )
2816, 27jaod 707 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( x  =  A  \/  x  =  B )  ->  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x )
)
2911, 28syl5bi 151 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  { A ,  B }  ->  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x )
)
3029ralrimiv 2507 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A. x  e.  { A ,  B }  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x )
31 prid1g 3634 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  { A ,  B } )
3231ad4antr 486 . . . . . 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 3940 . . . . . . 7  |-  ( z  =  A  ->  (
x  <  z  <->  x  <  A ) )
3433rspcev 2792 . . . . . 6  |-  ( ( A  e.  { A ,  B }  /\  x  <  A )  ->  E. z  e.  { A ,  B } x  <  z )
3532, 34sylancom 417 . . . . 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 3635 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  { A ,  B } )
3736ad4antlr 487 . . . . . 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 3940 . . . . . . 7  |-  ( z  =  B  ->  (
x  <  z  <->  x  <  B ) )
3938rspcev 2792 . . . . . 6  |-  ( ( B  e.  { A ,  B }  /\  x  <  B )  ->  E. z  e.  { A ,  B } x  <  z )
4037, 39sylancom 417 . . . . 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 480 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  /\  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  ->  A  e.  RR )
424ad2antrr 480 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  /\  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  ->  B  e.  RR )
43 simplr 520 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  /\  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  ->  x  e.  RR )
44 simpr 109 . . . . . 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 11010 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  /\  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  ->  (
x  <  A  \/  x  <  B ) )
4635, 40, 45mpjaodan 788 . . . 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 114 . . 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 2508 . 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 1162 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 103    \/ wo 698    /\ w3a 963    = wceq 1332    e. wcel 1481   A.wral 2417   E.wrex 2418   {cpr 3532   class class class wbr 3936   ` cfv 5130  (class class class)co 5781   RRcr 7642    + caddc 7646    < clt 7823    <_ cle 7824    - cmin 7956    / cdiv 8455   2c2 8794   abscabs 10800
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 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761  ax-arch 7762  ax-caucvg 7763
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 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-ilim 4298  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-frec 6295  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456  df-inn 8744  df-2 8802  df-3 8803  df-4 8804  df-n0 9001  df-z 9078  df-uz 9350  df-rp 9470  df-seqfrec 10249  df-exp 10323  df-cj 10645  df-re 10646  df-im 10647  df-rsqrt 10801  df-abs 10802
This theorem is referenced by:  maxabs  11012  maxleast  11016
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