MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ruclem3 Unicode version

Theorem ruclem3 12527
Description: Lemma for ruc 12537. 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 8878 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  B
)  e.  RR )
54rehalfcld 9974 . . . . . . . . 9  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  RR )
61, 5lenltd 8981 . . . . . . . 8  |-  ( ph  ->  ( M  <_  (
( A  +  B
)  /  2 )  <->  -.  ( ( A  +  B )  /  2
)  <  M )
)
7 ruclem2.8 . . . . . . . . . . 11  |-  ( ph  ->  A  <  B )
8 avglt2 9966 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( ( A  +  B
)  /  2 )  <  B ) )
92, 3, 8syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( A  <  B  <->  ( ( A  +  B
)  /  2 )  <  B ) )
107, 9mpbid 201 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  +  B )  /  2
)  <  B )
11 avglt1 9965 . . . . . . . . . . 11  |-  ( ( ( ( A  +  B )  /  2
)  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  / 
2 )  <  B  <->  ( ( A  +  B
)  /  2 )  <  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ) )
125, 3, 11syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  <  B  <->  ( ( A  +  B
)  /  2 )  <  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ) )
1310, 12mpbid 201 . . . . . . . . 9  |-  ( ph  ->  ( ( A  +  B )  /  2
)  <  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) )
145, 3readdcld 8878 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  +  B
)  e.  RR )
1514rehalfcld 9974 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  RR )
16 lelttr 8928 . . . . . . . . . 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 1182 . . . . . . . . 9  |-  ( ph  ->  ( ( M  <_ 
( ( A  +  B )  /  2
)  /\  ( ( A  +  B )  /  2 )  < 
( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) )  ->  M  <  ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) ) )
1813, 17mpan2d 655 . . . . . . . 8  |-  ( ph  ->  ( M  <_  (
( A  +  B
)  /  2 )  ->  M  <  (
( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ) )
196, 18sylbird 226 . . . . . . 7  |-  ( ph  ->  ( -.  ( ( A  +  B )  /  2 )  < 
M  ->  M  <  ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ) )
2019imp 418 . . . . . 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 12525 . . . . . . . 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 968 . . . . . . 7  |-  ( ph  ->  X  =  if ( ( ( A  +  B )  /  2
)  <  M ,  A ,  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) ) )
27 iffalse 3585 . . . . . . 7  |-  ( -.  ( ( A  +  B )  /  2
)  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  A ,  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) )  =  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )
2826, 27sylan9eq 2348 . . . . . 6  |-  ( (
ph  /\  -.  (
( A  +  B
)  /  2 )  <  M )  ->  X  =  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) )
2920, 28breqtrrd 4065 . . . . 5  |-  ( (
ph  /\  -.  (
( A  +  B
)  /  2 )  <  M )  ->  M  <  X )
3029ex 423 . . . 4  |-  ( ph  ->  ( -.  ( ( A  +  B )  /  2 )  < 
M  ->  M  <  X ) )
3130con1d 116 . . 3  |-  ( ph  ->  ( -.  M  < 
X  ->  ( ( A  +  B )  /  2 )  < 
M ) )
3225simp3d 969 . . . . . 6  |-  ( ph  ->  Y  =  if ( ( ( A  +  B )  /  2
)  <  M , 
( ( A  +  B )  /  2
) ,  B ) )
33 iftrue 3584 . . . . . 6  |-  ( ( ( A  +  B
)  /  2 )  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
)  =  ( ( A  +  B )  /  2 ) )
3432, 33sylan9eq 2348 . . . . 5  |-  ( (
ph  /\  ( ( A  +  B )  /  2 )  < 
M )  ->  Y  =  ( ( A  +  B )  / 
2 ) )
35 simpr 447 . . . . 5  |-  ( (
ph  /\  ( ( A  +  B )  /  2 )  < 
M )  ->  (
( A  +  B
)  /  2 )  <  M )
3634, 35eqbrtrd 4059 . . . 4  |-  ( (
ph  /\  ( ( A  +  B )  /  2 )  < 
M )  ->  Y  <  M )
3736ex 423 . . 3  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  <  M  ->  Y  <  M ) )
3831, 37syld 40 . 2  |-  ( ph  ->  ( -.  M  < 
X  ->  Y  <  M ) )
3938orrd 367 1  |-  ( ph  ->  ( M  <  X  \/  Y  <  M ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   [_csb 3094   ifcif 3578   <.cop 3656   class class class wbr 4039    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   RRcr 8752    + caddc 8756    < clt 8883    <_ cle 8884    / cdiv 9439   NNcn 9762   2c2 9811
This theorem is referenced by:  ruclem12  12535
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820
  Copyright terms: Public domain W3C validator