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Theorem ruclem3 13520
Description: Lemma for ruc 13530. 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 9418 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  B
)  e.  RR )
54rehalfcld 10576 . . . . . . . . 9  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  RR )
61, 5lenltd 9525 . . . . . . . 8  |-  ( ph  ->  ( M  <_  (
( A  +  B
)  /  2 )  <->  -.  ( ( A  +  B )  /  2
)  <  M )
)
7 ruclem2.8 . . . . . . . . . . 11  |-  ( ph  ->  A  <  B )
8 avglt2 10568 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( ( A  +  B
)  /  2 )  <  B ) )
92, 3, 8syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( A  <  B  <->  ( ( A  +  B
)  /  2 )  <  B ) )
107, 9mpbid 210 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  +  B )  /  2
)  <  B )
11 avglt1 10567 . . . . . . . . . . 11  |-  ( ( ( ( A  +  B )  /  2
)  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  / 
2 )  <  B  <->  ( ( A  +  B
)  /  2 )  <  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ) )
125, 3, 11syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  <  B  <->  ( ( A  +  B
)  /  2 )  <  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ) )
1310, 12mpbid 210 . . . . . . . . 9  |-  ( ph  ->  ( ( A  +  B )  /  2
)  <  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) )
145, 3readdcld 9418 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  +  B
)  e.  RR )
1514rehalfcld 10576 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  RR )
16 lelttr 9470 . . . . . . . . . 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 1218 . . . . . . . . 9  |-  ( ph  ->  ( ( M  <_ 
( ( A  +  B )  /  2
)  /\  ( ( A  +  B )  /  2 )  < 
( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) )  ->  M  <  ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) ) )
1813, 17mpan2d 674 . . . . . . . 8  |-  ( ph  ->  ( M  <_  (
( A  +  B
)  /  2 )  ->  M  <  (
( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ) )
196, 18sylbird 235 . . . . . . 7  |-  ( ph  ->  ( -.  ( ( A  +  B )  /  2 )  < 
M  ->  M  <  ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 ) ) )
2019imp 429 . . . . . 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 13518 . . . . . . . 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 1001 . . . . . . 7  |-  ( ph  ->  X  =  if ( ( ( A  +  B )  /  2
)  <  M ,  A ,  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) ) )
27 iffalse 3804 . . . . . . 7  |-  ( -.  ( ( A  +  B )  /  2
)  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  A ,  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) )  =  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )
2826, 27sylan9eq 2495 . . . . . 6  |-  ( (
ph  /\  -.  (
( A  +  B
)  /  2 )  <  M )  ->  X  =  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) )
2920, 28breqtrrd 4323 . . . . 5  |-  ( (
ph  /\  -.  (
( A  +  B
)  /  2 )  <  M )  ->  M  <  X )
3029ex 434 . . . 4  |-  ( ph  ->  ( -.  ( ( A  +  B )  /  2 )  < 
M  ->  M  <  X ) )
3130con1d 124 . . 3  |-  ( ph  ->  ( -.  M  < 
X  ->  ( ( A  +  B )  /  2 )  < 
M ) )
3225simp3d 1002 . . . . . 6  |-  ( ph  ->  Y  =  if ( ( ( A  +  B )  /  2
)  <  M , 
( ( A  +  B )  /  2
) ,  B ) )
33 iftrue 3802 . . . . . 6  |-  ( ( ( A  +  B
)  /  2 )  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
)  =  ( ( A  +  B )  /  2 ) )
3432, 33sylan9eq 2495 . . . . 5  |-  ( (
ph  /\  ( ( A  +  B )  /  2 )  < 
M )  ->  Y  =  ( ( A  +  B )  / 
2 ) )
35 simpr 461 . . . . 5  |-  ( (
ph  /\  ( ( A  +  B )  /  2 )  < 
M )  ->  (
( A  +  B
)  /  2 )  <  M )
3634, 35eqbrtrd 4317 . . . 4  |-  ( (
ph  /\  ( ( A  +  B )  /  2 )  < 
M )  ->  Y  <  M )
3736ex 434 . . 3  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  <  M  ->  Y  <  M ) )
3831, 37syld 44 . 2  |-  ( ph  ->  ( -.  M  < 
X  ->  Y  <  M ) )
3938orrd 378 1  |-  ( ph  ->  ( M  <  X  \/  Y  <  M ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   [_csb 3293   ifcif 3796   <.cop 3888   class class class wbr 4297    X. cxp 4843   -->wf 5419   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   1stc1st 6580   2ndc2nd 6581   RRcr 9286    + caddc 9290    < clt 9423    <_ cle 9424    / cdiv 9998   NNcn 10327   2c2 10376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-2 10385
This theorem is referenced by:  ruclem12  13528
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