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Theorem ruclem3 12787
Description: Lemma for ruc 12797. The constructed interval  [ X ,  Y ] always excludes  M. (Contributed by Mario Carneiro, 28-May-2014.)
Hypotheses
Ref Expression
ruc.1  |-  ( ph  ->  F : NN --> RR )
ruc.2  |-  ( ph  ->  D  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) )
ruclem1.3  |-  ( ph  ->  A  e.  RR )
ruclem1.4  |-  ( ph  ->  B  e.  RR )
ruclem1.5  |-  ( ph  ->  M  e.  RR )
ruclem1.6  |-  X  =  ( 1st `  ( <. A ,  B >. D M ) )
ruclem1.7  |-  Y  =  ( 2nd `  ( <. A ,  B >. D M ) )
ruclem2.8  |-  ( ph  ->  A  <  B )
Assertion
Ref Expression
ruclem3  |-  ( ph  ->  ( M  <  X  \/  Y  <  M ) )
Distinct variable groups:    x, m, y, A    B, m, x, y    m, F, x, y    m, M, x, y
Allowed substitution hints:    ph( x, y, m)    D( x, y, m)    X( x, y, m)    Y( x, y, m)

Proof of Theorem ruclem3
StepHypRef Expression
1 ruclem1.5 . . . . . . . . 9  |-  ( ph  ->  M  e.  RR )
2 ruclem1.3 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  RR )
3 ruclem1.4 . . . . . . . . . . 11  |-  ( ph  ->  B  e.  RR )
42, 3readdcld 9071 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  B
)  e.  RR )
54rehalfcld 10170 . . . . . . . . 9  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  RR )
61, 5lenltd 9175 . . . . . . . 8  |-  ( ph  ->  ( M  <_  (
( A  +  B
)  /  2 )  <->  -.  ( ( A  +  B )  /  2
)  <  M )
)
7 ruclem2.8 . . . . . . . . . . 11  |-  ( ph  ->  A  <  B )
8 avglt2 10162 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( ( A  +  B
)  /  2 )  <  B ) )
92, 3, 8syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( A  <  B  <->  ( ( A  +  B
)  /  2 )  <  B ) )
107, 9mpbid 202 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  +  B )  /  2
)  <  B )
11 avglt1 10161 . . . . . . . . . . 11  |-  ( ( ( ( A  +  B )  /  2
)  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  / 
2 )  <  B  <->  ( ( A  +  B
)  /  2 )  <  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ) )
125, 3, 11syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  <  B  <->  ( ( A  +  B
)  /  2 )  <  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ) )
1310, 12mpbid 202 . . . . . . . . 9  |-  ( ph  ->  ( ( A  +  B )  /  2
)  <  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) )
145, 3readdcld 9071 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  +  B
)  e.  RR )
1514rehalfcld 10170 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  RR )
16 lelttr 9121 . . . . . . . . . 10  |-  ( ( M  e.  RR  /\  ( ( A  +  B )  /  2
)  e.  RR  /\  ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  RR )  ->  ( ( M  <_  ( ( A  +  B )  / 
2 )  /\  (
( A  +  B
)  /  2 )  <  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )  ->  M  <  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ) )
171, 5, 15, 16syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( ( M  <_ 
( ( A  +  B )  /  2
)  /\  ( ( A  +  B )  /  2 )  < 
( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) )  ->  M  <  ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) ) )
1813, 17mpan2d 656 . . . . . . . 8  |-  ( ph  ->  ( M  <_  (
( A  +  B
)  /  2 )  ->  M  <  (
( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ) )
196, 18sylbird 227 . . . . . . 7  |-  ( ph  ->  ( -.  ( ( A  +  B )  /  2 )  < 
M  ->  M  <  ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ) )
2019imp 419 . . . . . 6  |-  ( (
ph  /\  -.  (
( A  +  B
)  /  2 )  <  M )  ->  M  <  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )
21 ruc.1 . . . . . . . . 9  |-  ( ph  ->  F : NN --> RR )
22 ruc.2 . . . . . . . . 9  |-  ( ph  ->  D  =  ( x  e.  ( RR  X.  RR ) ,  y  e.  RR  |->  [_ ( ( ( 1st `  x )  +  ( 2nd `  x
) )  /  2
)  /  m ]_ if ( m  <  y ,  <. ( 1st `  x
) ,  m >. , 
<. ( ( m  +  ( 2nd `  x ) )  /  2 ) ,  ( 2nd `  x
) >. ) ) )
23 ruclem1.6 . . . . . . . . 9  |-  X  =  ( 1st `  ( <. A ,  B >. D M ) )
24 ruclem1.7 . . . . . . . . 9  |-  Y  =  ( 2nd `  ( <. A ,  B >. D M ) )
2521, 22, 2, 3, 1, 23, 24ruclem1 12785 . . . . . . . 8  |-  ( ph  ->  ( ( <. A ,  B >. D M )  e.  ( RR  X.  RR )  /\  X  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  A , 
( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) )  /\  Y  =  if ( ( ( A  +  B )  /  2 )  < 
M ,  ( ( A  +  B )  /  2 ) ,  B ) ) )
2625simp2d 970 . . . . . . 7  |-  ( ph  ->  X  =  if ( ( ( A  +  B )  /  2
)  <  M ,  A ,  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) ) )
27 iffalse 3706 . . . . . . 7  |-  ( -.  ( ( A  +  B )  /  2
)  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  A ,  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) )  =  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )
2826, 27sylan9eq 2456 . . . . . 6  |-  ( (
ph  /\  -.  (
( A  +  B
)  /  2 )  <  M )  ->  X  =  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) )
2920, 28breqtrrd 4198 . . . . 5  |-  ( (
ph  /\  -.  (
( A  +  B
)  /  2 )  <  M )  ->  M  <  X )
3029ex 424 . . . 4  |-  ( ph  ->  ( -.  ( ( A  +  B )  /  2 )  < 
M  ->  M  <  X ) )
3130con1d 118 . . 3  |-  ( ph  ->  ( -.  M  < 
X  ->  ( ( A  +  B )  /  2 )  < 
M ) )
3225simp3d 971 . . . . . 6  |-  ( ph  ->  Y  =  if ( ( ( A  +  B )  /  2
)  <  M , 
( ( A  +  B )  /  2
) ,  B ) )
33 iftrue 3705 . . . . . 6  |-  ( ( ( A  +  B
)  /  2 )  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
)  =  ( ( A  +  B )  /  2 ) )
3432, 33sylan9eq 2456 . . . . 5  |-  ( (
ph  /\  ( ( A  +  B )  /  2 )  < 
M )  ->  Y  =  ( ( A  +  B )  / 
2 ) )
35 simpr 448 . . . . 5  |-  ( (
ph  /\  ( ( A  +  B )  /  2 )  < 
M )  ->  (
( A  +  B
)  /  2 )  <  M )
3634, 35eqbrtrd 4192 . . . 4  |-  ( (
ph  /\  ( ( A  +  B )  /  2 )  < 
M )  ->  Y  <  M )
3736ex 424 . . 3  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  <  M  ->  Y  <  M ) )
3831, 37syld 42 . 2  |-  ( ph  ->  ( -.  M  < 
X  ->  Y  <  M ) )
3938orrd 368 1  |-  ( ph  ->  ( M  <  X  \/  Y  <  M ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   [_csb 3211   ifcif 3699   <.cop 3777   class class class wbr 4172    X. cxp 4835   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307   RRcr 8945    + caddc 8949    < clt 9076    <_ cle 9077    / cdiv 9633   NNcn 9956   2c2 10005
This theorem is referenced by:  ruclem12  12795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014
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