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Theorem ruclem2 12831
Description: Lemma for ruc 12842. 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 9593 . . . 4  |-  ( ph  ->  A  <_  A )
3 ruclem1.4 . . . . . . . . 9  |-  ( ph  ->  B  e.  RR )
41, 3readdcld 9115 . . . . . . . 8  |-  ( ph  ->  ( A  +  B
)  e.  RR )
54rehalfcld 10214 . . . . . . 7  |-  ( ph  ->  ( ( A  +  B )  /  2
)  e.  RR )
65, 3readdcld 9115 . . . . . 6  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  +  B
)  e.  RR )
76rehalfcld 10214 . . . . 5  |-  ( ph  ->  ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  e.  RR )
8 ruclem2.8 . . . . . . 7  |-  ( ph  ->  A  <  B )
9 avglt1 10205 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A  <  ( ( A  +  B )  / 
2 ) ) )
101, 3, 9syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( A  <  B  <->  A  <  ( ( A  +  B )  / 
2 ) ) )
118, 10mpbid 202 . . . . . 6  |-  ( ph  ->  A  <  ( ( A  +  B )  /  2 ) )
12 avglt2 10206 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( ( A  +  B
)  /  2 )  <  B ) )
131, 3, 12syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( A  <  B  <->  ( ( A  +  B
)  /  2 )  <  B ) )
148, 13mpbid 202 . . . . . . 7  |-  ( ph  ->  ( ( A  +  B )  /  2
)  <  B )
15 avglt1 10205 . . . . . . . 8  |-  ( ( ( ( A  +  B )  /  2
)  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  / 
2 )  <  B  <->  ( ( A  +  B
)  /  2 )  <  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ) )
165, 3, 15syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  <  B  <->  ( ( A  +  B
)  /  2 )  <  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ) )
1714, 16mpbid 202 . . . . . 6  |-  ( ph  ->  ( ( A  +  B )  /  2
)  <  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) )
181, 5, 7, 11, 17lttrd 9231 . . . . 5  |-  ( ph  ->  A  <  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) )
191, 7, 18ltled 9221 . . . 4  |-  ( ph  ->  A  <_  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) )
20 breq2 4216 . . . . 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 4216 . . . . 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 3770 . . . 4  |-  ( ( A  <_  A  /\  A  <_  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )  ->  A  <_  if ( ( ( A  +  B
)  /  2 )  <  M ,  A ,  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) ) )
232, 19, 22syl2anc 643 . . 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 12830 . . . 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 970 . . 3  |-  ( ph  ->  X  =  if ( ( ( A  +  B )  /  2
)  <  M ,  A ,  ( (
( ( A  +  B )  /  2
)  +  B )  /  2 ) ) )
3123, 30breqtrrd 4238 . 2  |-  ( ph  ->  A  <_  X )
32 iftrue 3745 . . . . . 6  |-  ( ( ( A  +  B
)  /  2 )  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  A ,  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) )  =  A )
33 iftrue 3745 . . . . . 6  |-  ( ( ( A  +  B
)  /  2 )  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
)  =  ( ( A  +  B )  /  2 ) )
3432, 33breq12d 4225 . . . . 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 214 . . . 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 10206 . . . . . . 7  |-  ( ( ( ( A  +  B )  /  2
)  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  / 
2 )  <  B  <->  ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 )  <  B ) )
375, 3, 36syl2anc 643 . . . . . 6  |-  ( ph  ->  ( ( ( A  +  B )  / 
2 )  <  B  <->  ( ( ( ( A  +  B )  / 
2 )  +  B
)  /  2 )  <  B ) )
3814, 37mpbid 202 . . . . 5  |-  ( ph  ->  ( ( ( ( A  +  B )  /  2 )  +  B )  /  2
)  <  B )
39 iffalse 3746 . . . . . 6  |-  ( -.  ( ( A  +  B )  /  2
)  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  A ,  ( ( ( ( A  +  B )  /  2
)  +  B )  /  2 ) )  =  ( ( ( ( A  +  B
)  /  2 )  +  B )  / 
2 ) )
40 iffalse 3746 . . . . . 6  |-  ( -.  ( ( A  +  B )  /  2
)  <  M  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
)  =  B )
4139, 40breq12d 4225 . . . . 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 214 . . . 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 152 . . 3  |-  ( ph  ->  if ( ( ( A  +  B )  /  2 )  < 
M ,  A , 
( ( ( ( A  +  B )  /  2 )  +  B )  /  2
) )  <  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
) )
4429simp3d 971 . . 3  |-  ( ph  ->  Y  =  if ( ( ( A  +  B )  /  2
)  <  M , 
( ( A  +  B )  /  2
) ,  B ) )
4543, 30, 443brtr4d 4242 . 2  |-  ( ph  ->  X  <  Y )
465, 3, 14ltled 9221 . . . 4  |-  ( ph  ->  ( ( A  +  B )  /  2
)  <_  B )
473leidd 9593 . . . 4  |-  ( ph  ->  B  <_  B )
48 breq1 4215 . . . . 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 4215 . . . . 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 3770 . . . 4  |-  ( ( ( ( A  +  B )  /  2
)  <_  B  /\  B  <_  B )  ->  if ( ( ( A  +  B )  / 
2 )  <  M ,  ( ( A  +  B )  / 
2 ) ,  B
)  <_  B )
5146, 47, 50syl2anc 643 . . 3  |-  ( ph  ->  if ( ( ( A  +  B )  /  2 )  < 
M ,  ( ( A  +  B )  /  2 ) ,  B )  <_  B
)
5244, 51eqbrtrd 4232 . 2  |-  ( ph  ->  Y  <_  B )
5331, 45, 523jca 1134 1  |-  ( ph  ->  ( A  <_  X  /\  X  <  Y  /\  Y  <_  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725   [_csb 3251   ifcif 3739   <.cop 3817   class class class wbr 4212    X. cxp 4876   -->wf 5450   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1stc1st 6347   2ndc2nd 6348   RRcr 8989    + caddc 8993    < clt 9120    <_ cle 9121    / cdiv 9677   NNcn 10000   2c2 10049
This theorem is referenced by:  ruclem8  12836  ruclem9  12837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-2 10058
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