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Theorem ruclem3 12752
Description: Lemma for ruc 12762. 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 9041 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  B
)  e.  RR )
54rehalfcld 10139 . . . . . . . . 9  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  RR )
61, 5lenltd 9144 . . . . . . . 8  |-  ( ph  ->  ( M  <_  (
( A  +  B
)  /  2 )  <->  -.  ( ( A  +  B )  /  2
)  <  M )
)
7 ruclem2.8 . . . . . . . . . . 11  |-  ( ph  ->  A  <  B )
8 avglt2 10131 . . . . . . . . . . . 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 10130 . . . . . . . . . . 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 9041 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  +  B
)  e.  RR )
1514rehalfcld 10139 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  RR )
16 lelttr 9091 . . . . . . . . . 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 12750 . . . . . . . 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 3682 . . . . . . 7  |-  ( -.  ( ( A  +  B )  /  2
)  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  A ,  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) )  =  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )
2826, 27sylan9eq 2432 . . . . . 6  |-  ( (
ph  /\  -.  (
( A  +  B
)  /  2 )  <  M )  ->  X  =  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) )
2920, 28breqtrrd 4172 . . . . 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 3681 . . . . . 6  |-  ( ( ( A  +  B
)  /  2 )  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
)  =  ( ( A  +  B )  /  2 ) )
3432, 33sylan9eq 2432 . . . . 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 4166 . . . 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 1717   [_csb 3187   ifcif 3675   <.cop 3753   class class class wbr 4146    X. cxp 4809   -->wf 5383   ` cfv 5387  (class class class)co 6013    e. cmpt2 6015   1stc1st 6279   2ndc2nd 6280   RRcr 8915    + caddc 8919    < clt 9046    <_ cle 9047    / cdiv 9602   NNcn 9925   2c2 9974
This theorem is referenced by:  ruclem12  12760
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-2 9983
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