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Theorem ruclem3 13977
Description: Lemma for ruc 13987. 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 9640 . . . . . . . . . 10  |-  ( ph  ->  ( A  +  B
)  e.  RR )
54rehalfcld 10806 . . . . . . . . 9  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  RR )
61, 5lenltd 9748 . . . . . . . 8  |-  ( ph  ->  ( M  <_  (
( A  +  B
)  /  2 )  <->  -.  ( ( A  +  B )  /  2
)  <  M )
)
7 ruclem2.8 . . . . . . . . . . 11  |-  ( ph  ->  A  <  B )
8 avglt2 10798 . . . . . . . . . . . 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 10797 . . . . . . . . . . 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 9640 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  +  B
)  e.  RR )
1514rehalfcld 10806 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  RR )
16 lelttr 9692 . . . . . . . . . 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 1228 . . . . . . . . 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 13975 . . . . . . . 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 1009 . . . . . . 7  |-  ( ph  ->  X  =  if ( ( ( A  +  B )  /  2
)  <  M ,  A ,  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) ) )
27 iffalse 3953 . . . . . . 7  |-  ( -.  ( ( A  +  B )  /  2
)  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  A ,  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) )  =  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )
2826, 27sylan9eq 2518 . . . . . 6  |-  ( (
ph  /\  -.  (
( A  +  B
)  /  2 )  <  M )  ->  X  =  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) )
2920, 28breqtrrd 4482 . . . . 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 1010 . . . . . 6  |-  ( ph  ->  Y  =  if ( ( ( A  +  B )  /  2
)  <  M , 
( ( A  +  B )  /  2
) ,  B ) )
33 iftrue 3950 . . . . . 6  |-  ( ( ( A  +  B
)  /  2 )  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
)  =  ( ( A  +  B )  /  2 ) )
3432, 33sylan9eq 2518 . . . . 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 4476 . . . 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 1395    e. wcel 1819   [_csb 3430   ifcif 3944   <.cop 4038   class class class wbr 4456    X. cxp 5006   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798   RRcr 9508    + caddc 9512    < clt 9645    <_ cle 9646    / cdiv 10227   NNcn 10556   2c2 10606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-2 10615
This theorem is referenced by:  ruclem12  13985
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