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Theorem maxabslemval 11201
Description: Lemma for maxabs 11202. 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 7928 . . . 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 7976 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  CC )
4 simpr 110 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
54recnd 7976 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
63, 5subcld 8258 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B
)  e.  CC )
76abscld 11174 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( A  -  B )
)  e.  RR )
81, 7readdcld 7977 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  B )  +  ( abs `  ( A  -  B ) ) )  e.  RR )
98rehalfcld 9154 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  e.  RR )
10 vex 2740 . . . . 5  |-  x  e. 
_V
1110elpr 3612 . . . 4  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
12 maxabsle 11197 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  <_  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
132, 9, 12lensymd 8069 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  A )
14 breq2 4004 . . . . . . 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 11197 . . . . . . . . 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 8098 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( B  +  A
)  =  ( A  +  B ) )
205, 3abssubd 11186 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( abs `  ( B  -  A )
)  =  ( abs `  ( A  -  B
) ) )
2119, 20oveq12d 5887 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( B  +  A )  +  ( abs `  ( B  -  A ) ) )  =  ( ( A  +  B )  +  ( abs `  ( A  -  B )
) ) )
2221oveq1d 5884 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( B  +  A )  +  ( abs `  ( B  -  A )
) )  /  2
)  =  ( ( ( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
2318, 22breqtrd 4026 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  <_  ( (
( A  +  B
)  +  ( abs `  ( A  -  B
) ) )  / 
2 ) )
244, 9, 23lensymd 8069 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  B )
25 breq2 4004 . . . . . . 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 2549 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A. x  e.  { A ,  B }  -.  ( ( ( A  +  B )  +  ( abs `  ( A  -  B )
) )  /  2
)  <  x )
31 prid1g 3695 . . . . . . 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 4004 . . . . . . 7  |-  ( z  =  A  ->  (
x  <  z  <->  x  <  A ) )
3433rspcev 2841 . . . . . 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 3696 . . . . . . 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 4004 . . . . . . 7  |-  ( z  =  B  ->  (
x  <  z  <->  x  <  B ) )
3938rspcev 2841 . . . . . 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 11200 . . . . 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 2550 . 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 2148   A.wral 2455   E.wrex 2456   {cpr 3592   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   RRcr 7801    + caddc 7805    < clt 7982    <_ cle 7983    - cmin 8118    / cdiv 8618   2c2 8959   abscabs 10990
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
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 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-rp 9641  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992
This theorem is referenced by:  maxabs  11202  maxleast  11206
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