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Theorem ruclem2 12526
Description: Lemma for ruc 12537. 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 9355 . . . 4  |-  ( ph  ->  A  <_  A )
3 ruclem1.4 . . . . . . . . 9  |-  ( ph  ->  B  e.  RR )
41, 3readdcld 8878 . . . . . . . 8  |-  ( ph  ->  ( A  +  B
)  e.  RR )
54rehalfcld 9974 . . . . . . 7  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  RR )
65, 3readdcld 8878 . . . . . 6  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  +  B
)  e.  RR )
76rehalfcld 9974 . . . . 5  |-  ( ph  ->  ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  RR )
8 ruclem2.8 . . . . . . 7  |-  ( ph  ->  A  <  B )
9 avglt1 9965 . . . . . . . 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 9966 . . . . . . . . 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 9965 . . . . . . . 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 8993 . . . . 5  |-  ( ph  ->  A  <  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) )
191, 7, 18ltled 8983 . . . 4  |-  ( ph  ->  A  <_  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) )
20 breq2 4043 . . . . 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 4043 . . . . 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 3609 . . . 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 12525 . . . 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 4065 . 2  |-  ( ph  ->  A  <_  X )
32 iftrue 3584 . . . . . 6  |-  ( ( ( A  +  B
)  /  2 )  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  A ,  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) )  =  A )
33 iftrue 3584 . . . . . 6  |-  ( ( ( A  +  B
)  /  2 )  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
)  =  ( ( A  +  B )  /  2 ) )
3432, 33breq12d 4052 . . . . 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 9966 . . . . . . 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 3585 . . . . . 6  |-  ( -.  ( ( A  +  B )  /  2
)  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  A ,  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) )  =  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )
40 iffalse 3585 . . . . . 6  |-  ( -.  ( ( A  +  B )  /  2
)  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
)  =  B )
4139, 40breq12d 4052 . . . . 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 4069 . 2  |-  ( ph  ->  X  <  Y )
465, 3, 14ltled 8983 . . . 4  |-  ( ph  ->  ( ( A  +  B )  /  2
)  <_  B )
473leidd 9355 . . . 4  |-  ( ph  ->  B  <_  B )
48 breq1 4042 . . . . 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 4042 . . . . 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 3609 . . . 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 4059 . 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 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:  ruclem8  12531  ruclem9  12532
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
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