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Theorem maxabslemval 11185
Description: Lemma for maxabs 11186. 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 7912 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
2 simpl 109 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
32recnd 7960 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  CC )
4 simpr 110 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
54recnd 7960 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
63, 5subcld 8242 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  CC )
76abscld 11158 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( A  -  B )
)  e.  RR )
81, 7readdcld 7961 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  e.  RR )
98rehalfcld 9138 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  e.  RR )
10 vex 2738 . . . . 5  |-  x  e. 
_V
1110elpr 3610 . . . 4  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
12 maxabsle 11181 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  <_  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
132, 9, 12lensymd 8053 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  A )
14 breq2 4002 . . . . . . 7  |-  ( x  =  A  ->  (
( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x  <->  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  <  A
) )
1514notbid 667 . . . . . 6  |-  ( x  =  A  ->  ( -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x  <->  -.  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  <  A ) )
1613, 15syl5ibrcom 157 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  =  A  ->  -.  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  <  x
) )
17 maxabsle 11181 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  B  <_  ( (
( B  +  A
)  +  ( abs `  ( B  -  A
) ) )  / 
2 ) )
1817ancoms 268 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  <_  ( (
( B  +  A
)  +  ( abs `  ( B  -  A
) ) )  / 
2 ) )
195, 3addcomd 8082 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  +  A
)  =  ( A  +  B ) )
205, 3abssubd 11170 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( B  -  A )
)  =  ( abs `  ( A  -  B
) ) )
2119, 20oveq12d 5883 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( B  +  A )  +  ( abs `  ( B  -  A ) ) )  =  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) ) )
2221oveq1d 5880 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( B  +  A )  +  ( abs `  ( B  -  A )
) )  /  2
)  =  ( ( ( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
2318, 22breqtrd 4024 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  <_  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
244, 9, 23lensymd 8053 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  B )
25 breq2 4002 . . . . . . 7  |-  ( x  =  B  ->  (
( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x  <->  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  <  B
) )
2625notbid 667 . . . . . 6  |-  ( x  =  B  ->  ( -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x  <->  -.  (
( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  /  2 )  <  B ) )
2724, 26syl5ibrcom 157 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  =  B  ->  -.  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 )  <  x
) )
2816, 27jaod 717 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( x  =  A  \/  x  =  B )  ->  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x )
)
2911, 28biimtrid 152 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( x  e.  { A ,  B }  ->  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x )
)
3029ralrimiv 2547 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A. x  e.  { A ,  B }  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x )
31 prid1g 3693 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  { A ,  B } )
3231ad4antr 494 . . . . . 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 4002 . . . . . . 7  |-  ( z  =  A  ->  (
x  <  z  <->  x  <  A ) )
3433rspcev 2839 . . . . . 6  |-  ( ( A  e.  { A ,  B }  /\  x  <  A )  ->  E. z  e.  { A ,  B } x  <  z )
3532, 34sylancom 420 . . . . 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 3694 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  { A ,  B } )
3736ad4antlr 495 . . . . . 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 4002 . . . . . . 7  |-  ( z  =  B  ->  (
x  <  z  <->  x  <  B ) )
3938rspcev 2839 . . . . . 6  |-  ( ( B  e.  { A ,  B }  /\  x  <  B )  ->  E. z  e.  { A ,  B } x  <  z )
4037, 39sylancom 420 . . . . 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 488 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  /\  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  ->  A  e.  RR )
424ad2antrr 488 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  /\  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  ->  B  e.  RR )
43 simplr 528 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  /\  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  ->  x  e.  RR )
44 simpr 110 . . . . . 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 11184 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  x  e.  RR )  /\  x  <  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
) )  ->  (
x  <  A  \/  x  <  B ) )
4635, 40, 45mpjaodan 798 . . . 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 115 . . 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 2548 . 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 1177 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 104    \/ wo 708    /\ w3a 978    = wceq 1353    e. wcel 2146   A.wral 2453   E.wrex 2454   {cpr 3590   class class class wbr 3998   ` cfv 5208  (class class class)co 5865   RRcr 7785    + caddc 7789    < clt 7966    <_ cle 7967    - cmin 8102    / cdiv 8602   2c2 8943   abscabs 10974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8603  df-inn 8893  df-2 8951  df-3 8952  df-4 8953  df-n0 9150  df-z 9227  df-uz 9502  df-rp 9625  df-seqfrec 10416  df-exp 10490  df-cj 10819  df-re 10820  df-im 10821  df-rsqrt 10975  df-abs 10976
This theorem is referenced by:  maxabs  11186  maxleast  11190
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