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Theorem maxabslemval 10537
Description: Lemma for maxabs 10538. 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 7412 . . . 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 7460 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  CC )
4 simpr 108 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
54recnd 7460 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
63, 5subcld 7737 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  CC )
76abscld 10510 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( A  -  B )
)  e.  RR )
81, 7readdcld 7461 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  e.  RR )
98rehalfcld 8595 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  e.  RR )
10 vex 2618 . . . . 5  |-  x  e. 
_V
1110elpr 3452 . . . 4  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
12 maxabsle 10533 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  <_  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
132, 9, 12lensymd 7549 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  A )
14 breq2 3824 . . . . . . 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 10533 . . . . . . . . 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 7577 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  +  A
)  =  ( A  +  B ) )
205, 3abssubd 10522 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( B  -  A )
)  =  ( abs `  ( A  -  B
) ) )
2119, 20oveq12d 5631 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( B  +  A )  +  ( abs `  ( B  -  A ) ) )  =  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) ) )
2221oveq1d 5628 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( B  +  A )  +  ( abs `  ( B  -  A )
) )  /  2
)  =  ( ( ( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
2318, 22breqtrd 3844 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  <_  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
244, 9, 23lensymd 7549 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  B )
25 breq2 3824 . . . . . . 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 2441 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A. x  e.  { A ,  B }  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x )
31 prid1g 3529 . . . . . . 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 3824 . . . . . . 7  |-  ( z  =  A  ->  (
x  <  z  <->  x  <  A ) )
3433rspcev 2715 . . . . . 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 3530 . . . . . . 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 3824 . . . . . . 7  |-  ( z  =  B  ->  (
x  <  z  <->  x  <  B ) )
3938rspcev 2715 . . . . . 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 10536 . . . . 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 2442 . 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 1121 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 922    = wceq 1287    e. wcel 1436   A.wral 2355   E.wrex 2356   {cpr 3432   class class class wbr 3820   ` cfv 4981  (class class class)co 5613   RRcr 7293    + caddc 7297    < clt 7466    <_ cle 7467    - cmin 7597    / cdiv 8078   2c2 8407   abscabs 10326
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-iinf 4376  ax-cnex 7380  ax-resscn 7381  ax-1cn 7382  ax-1re 7383  ax-icn 7384  ax-addcl 7385  ax-addrcl 7386  ax-mulcl 7387  ax-mulrcl 7388  ax-addcom 7389  ax-mulcom 7390  ax-addass 7391  ax-mulass 7392  ax-distr 7393  ax-i2m1 7394  ax-0lt1 7395  ax-1rid 7396  ax-0id 7397  ax-rnegex 7398  ax-precex 7399  ax-cnre 7400  ax-pre-ltirr 7401  ax-pre-ltwlin 7402  ax-pre-lttrn 7403  ax-pre-apti 7404  ax-pre-ltadd 7405  ax-pre-mulgt0 7406  ax-pre-mulext 7407  ax-arch 7408  ax-caucvg 7409
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-if 3380  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-id 4094  df-po 4097  df-iso 4098  df-iord 4167  df-on 4169  df-ilim 4170  df-suc 4172  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-riota 5569  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-1st 5868  df-2nd 5869  df-recs 6024  df-frec 6110  df-pnf 7468  df-mnf 7469  df-xr 7470  df-ltxr 7471  df-le 7472  df-sub 7599  df-neg 7600  df-reap 7993  df-ap 8000  df-div 8079  df-inn 8358  df-2 8416  df-3 8417  df-4 8418  df-n0 8607  df-z 8684  df-uz 8952  df-rp 9067  df-iseq 9780  df-iexp 9854  df-cj 10172  df-re 10173  df-im 10174  df-rsqrt 10327  df-abs 10328
This theorem is referenced by:  maxabs  10538  maxleast  10542
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