MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ruclem2 Unicode version

Theorem ruclem2 12510
Description: Lemma for ruc 12521. Ordering property for the input to 
D. (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
ruclem2  |-  ( ph  ->  ( A  <_  X  /\  X  <  Y  /\  Y  <_  B ) )
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 ruclem2
StepHypRef Expression
1 ruclem1.3 . . . . 5  |-  ( ph  ->  A  e.  RR )
21leidd 9339 . . . 4  |-  ( ph  ->  A  <_  A )
3 ruclem1.4 . . . . . . . . 9  |-  ( ph  ->  B  e.  RR )
41, 3readdcld 8862 . . . . . . . 8  |-  ( ph  ->  ( A  +  B
)  e.  RR )
54rehalfcld 9958 . . . . . . 7  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  RR )
65, 3readdcld 8862 . . . . . 6  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  +  B
)  e.  RR )
76rehalfcld 9958 . . . . 5  |-  ( ph  ->  ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  RR )
8 ruclem2.8 . . . . . . 7  |-  ( ph  ->  A  <  B )
9 avglt1 9949 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A  <  ( ( A  +  B )  / 
2 ) ) )
101, 3, 9syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( A  <  B  <->  A  <  ( ( A  +  B )  / 
2 ) ) )
118, 10mpbid 201 . . . . . 6  |-  ( ph  ->  A  <  ( ( A  +  B )  /  2 ) )
12 avglt2 9950 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( ( A  +  B
)  /  2 )  <  B ) )
131, 3, 12syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( A  <  B  <->  ( ( A  +  B
)  /  2 )  <  B ) )
148, 13mpbid 201 . . . . . . 7  |-  ( ph  ->  ( ( A  +  B )  /  2
)  <  B )
15 avglt1 9949 . . . . . . . 8  |-  ( ( ( ( A  +  B )  /  2
)  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  / 
2 )  <  B  <->  ( ( A  +  B
)  /  2 )  <  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ) )
165, 3, 15syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  <  B  <->  ( ( A  +  B
)  /  2 )  <  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ) )
1714, 16mpbid 201 . . . . . 6  |-  ( ph  ->  ( ( A  +  B )  /  2
)  <  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) )
181, 5, 7, 11, 17lttrd 8977 . . . . 5  |-  ( ph  ->  A  <  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) )
191, 7, 18ltled 8967 . . . 4  |-  ( ph  ->  A  <_  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) )
20 breq2 4027 . . . . 5  |-  ( A  =  if ( ( ( A  +  B
)  /  2 )  <  M ,  A ,  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )  -> 
( A  <_  A  <->  A  <_  if ( ( ( A  +  B
)  /  2 )  <  M ,  A ,  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ) ) )
21 breq2 4027 . . . . 5  |-  ( ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 )  =  if ( ( ( A  +  B
)  /  2 )  <  M ,  A ,  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )  -> 
( A  <_  (
( ( ( A  +  B )  / 
2 )  +  B
)  /  2 )  <-> 
A  <_  if (
( ( A  +  B )  /  2
)  <  M ,  A ,  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) ) ) )
2220, 21ifboth 3596 . . . 4  |-  ( ( A  <_  A  /\  A  <_  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )  ->  A  <_  if ( ( ( A  +  B
)  /  2 )  <  M ,  A ,  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ) )
232, 19, 22syl2anc 642 . . 3  |-  ( ph  ->  A  <_  if (
( ( A  +  B )  /  2
)  <  M ,  A ,  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) ) )
24 ruc.1 . . . . 5  |-  ( ph  ->  F : NN --> RR )
25 ruc.2 . . . . 5  |-  ( 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
) >. ) ) )
26 ruclem1.5 . . . . 5  |-  ( ph  ->  M  e.  RR )
27 ruclem1.6 . . . . 5  |-  X  =  ( 1st `  ( <. A ,  B >. D M ) )
28 ruclem1.7 . . . . 5  |-  Y  =  ( 2nd `  ( <. A ,  B >. D M ) )
2924, 25, 1, 3, 26, 27, 28ruclem1 12509 . . . 4  |-  ( 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 ) ) )
3029simp2d 968 . . 3  |-  ( ph  ->  X  =  if ( ( ( A  +  B )  /  2
)  <  M ,  A ,  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) ) )
3123, 30breqtrrd 4049 . 2  |-  ( ph  ->  A  <_  X )
32 iftrue 3571 . . . . . 6  |-  ( ( ( A  +  B
)  /  2 )  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  A ,  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) )  =  A )
33 iftrue 3571 . . . . . 6  |-  ( ( ( A  +  B
)  /  2 )  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
)  =  ( ( A  +  B )  /  2 ) )
3432, 33breq12d 4036 . . . . 5  |-  ( ( ( A  +  B
)  /  2 )  <  M  ->  ( if ( ( ( A  +  B )  / 
2 )  <  M ,  A ,  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) )  <  if ( ( ( A  +  B
)  /  2 )  <  M ,  ( ( A  +  B
)  /  2 ) ,  B )  <->  A  <  ( ( A  +  B
)  /  2 ) ) )
3511, 34syl5ibrcom 213 . . . 4  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  <  M  ->  if ( ( ( A  +  B )  /  2 )  < 
M ,  A , 
( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) )  <  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
) ) )
36 avglt2 9950 . . . . . . 7  |-  ( ( ( ( A  +  B )  /  2
)  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  / 
2 )  <  B  <->  ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 )  <  B ) )
375, 3, 36syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  <  B  <->  ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 )  <  B ) )
3814, 37mpbid 201 . . . . 5  |-  ( ph  ->  ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  <  B )
39 iffalse 3572 . . . . . 6  |-  ( -.  ( ( A  +  B )  /  2
)  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  A ,  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) )  =  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )
40 iffalse 3572 . . . . . 6  |-  ( -.  ( ( A  +  B )  /  2
)  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
)  =  B )
4139, 40breq12d 4036 . . . . 5  |-  ( -.  ( ( A  +  B )  /  2
)  <  M  ->  ( if ( ( ( A  +  B )  /  2 )  < 
M ,  A , 
( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) )  <  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
)  <->  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 )  <  B
) )
4238, 41syl5ibrcom 213 . . . 4  |-  ( ph  ->  ( -.  ( ( A  +  B )  /  2 )  < 
M  ->  if (
( ( A  +  B )  /  2
)  <  M ,  A ,  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) )  <  if ( ( ( A  +  B
)  /  2 )  <  M ,  ( ( A  +  B
)  /  2 ) ,  B ) ) )
4335, 42pm2.61d 150 . . 3  |-  ( ph  ->  if ( ( ( A  +  B )  /  2 )  < 
M ,  A , 
( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) )  <  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
) )
4429simp3d 969 . . 3  |-  ( ph  ->  Y  =  if ( ( ( A  +  B )  /  2
)  <  M , 
( ( A  +  B )  /  2
) ,  B ) )
4543, 30, 443brtr4d 4053 . 2  |-  ( ph  ->  X  <  Y )
465, 3, 14ltled 8967 . . . 4  |-  ( ph  ->  ( ( A  +  B )  /  2
)  <_  B )
473leidd 9339 . . . 4  |-  ( ph  ->  B  <_  B )
48 breq1 4026 . . . . 5  |-  ( ( ( A  +  B
)  /  2 )  =  if ( ( ( A  +  B
)  /  2 )  <  M ,  ( ( A  +  B
)  /  2 ) ,  B )  -> 
( ( ( A  +  B )  / 
2 )  <_  B  <->  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
)  <_  B )
)
49 breq1 4026 . . . . 5  |-  ( B  =  if ( ( ( A  +  B
)  /  2 )  <  M ,  ( ( A  +  B
)  /  2 ) ,  B )  -> 
( B  <_  B  <->  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
)  <_  B )
)
5048, 49ifboth 3596 . . . 4  |-  ( ( ( ( A  +  B )  /  2
)  <_  B  /\  B  <_  B )  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
)  <_  B )
5146, 47, 50syl2anc 642 . . 3  |-  ( ph  ->  if ( ( ( A  +  B )  /  2 )  < 
M ,  ( ( A  +  B )  /  2 ) ,  B )  <_  B
)
5244, 51eqbrtrd 4043 . 2  |-  ( ph  ->  Y  <_  B )
5331, 45, 523jca 1132 1  |-  ( ph  ->  ( A  <_  X  /\  X  <  Y  /\  Y  <_  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   [_csb 3081   ifcif 3565   <.cop 3643   class class class wbr 4023    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   RRcr 8736    + caddc 8740    < clt 8867    <_ cle 8868    / cdiv 9423   NNcn 9746   2c2 9795
This theorem is referenced by:  ruclem8  12515  ruclem9  12516
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804
  Copyright terms: Public domain W3C validator