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Theorem ruclem10 12838
Description: Lemma for ruc 12842. Every first component of the  G sequence is less than every second component. That is, the sequences form a chain a1  < a2 
<...  < b2  < b1, where ai are the first components and bi are the second components. (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
) >. ) ) )
ruc.4  |-  C  =  ( { <. 0 ,  <. 0 ,  1
>. >. }  u.  F
)
ruc.5  |-  G  =  seq  0 ( D ,  C )
ruclem10.6  |-  ( ph  ->  M  e.  NN0 )
ruclem10.7  |-  ( ph  ->  N  e.  NN0 )
Assertion
Ref Expression
ruclem10  |-  ( ph  ->  ( 1st `  ( G `  M )
)  <  ( 2nd `  ( G `  N
) ) )
Distinct variable groups:    x, m, y, F    m, G, x, y    m, M, x, y    m, N, x, y
Allowed substitution hints:    ph( x, y, m)    C( x, y, m)    D( x, y, m)

Proof of Theorem ruclem10
StepHypRef Expression
1 ruc.1 . . . . 5  |-  ( ph  ->  F : NN --> RR )
2 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
) >. ) ) )
3 ruc.4 . . . . 5  |-  C  =  ( { <. 0 ,  <. 0 ,  1
>. >. }  u.  F
)
4 ruc.5 . . . . 5  |-  G  =  seq  0 ( D ,  C )
51, 2, 3, 4ruclem6 12834 . . . 4  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
6 ruclem10.6 . . . 4  |-  ( ph  ->  M  e.  NN0 )
75, 6ffvelrnd 5871 . . 3  |-  ( ph  ->  ( G `  M
)  e.  ( RR 
X.  RR ) )
8 xp1st 6376 . . 3  |-  ( ( G `  M )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  M
) )  e.  RR )
97, 8syl 16 . 2  |-  ( ph  ->  ( 1st `  ( G `  M )
)  e.  RR )
10 ruclem10.7 . . . . 5  |-  ( ph  ->  N  e.  NN0 )
11 ifcl 3775 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
1210, 6, 11syl2anc 643 . . . 4  |-  ( ph  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
135, 12ffvelrnd 5871 . . 3  |-  ( ph  ->  ( G `  if ( M  <_  N ,  N ,  M )
)  e.  ( RR 
X.  RR ) )
14 xp1st 6376 . . 3  |-  ( ( G `  if ( M  <_  N ,  N ,  M )
)  e.  ( RR 
X.  RR )  -> 
( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  e.  RR )
1513, 14syl 16 . 2  |-  ( ph  ->  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  e.  RR )
165, 10ffvelrnd 5871 . . 3  |-  ( ph  ->  ( G `  N
)  e.  ( RR 
X.  RR ) )
17 xp2nd 6377 . . 3  |-  ( ( G `  N )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  N
) )  e.  RR )
1816, 17syl 16 . 2  |-  ( ph  ->  ( 2nd `  ( G `  N )
)  e.  RR )
196nn0red 10275 . . . . . 6  |-  ( ph  ->  M  e.  RR )
2010nn0red 10275 . . . . . 6  |-  ( ph  ->  N  e.  RR )
21 max1 10773 . . . . . 6  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  M  <_  if ( M  <_  N ,  N ,  M ) )
2219, 20, 21syl2anc 643 . . . . 5  |-  ( ph  ->  M  <_  if ( M  <_  N ,  N ,  M ) )
236nn0zd 10373 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
2412nn0zd 10373 . . . . . 6  |-  ( ph  ->  if ( M  <_  N ,  N ,  M )  e.  ZZ )
25 eluz 10499 . . . . . 6  |-  ( ( M  e.  ZZ  /\  if ( M  <_  N ,  N ,  M )  e.  ZZ )  -> 
( if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  M )  <->  M  <_  if ( M  <_  N ,  N ,  M ) ) )
2623, 24, 25syl2anc 643 . . . . 5  |-  ( ph  ->  ( if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  M )  <->  M  <_  if ( M  <_  N ,  N ,  M ) ) )
2722, 26mpbird 224 . . . 4  |-  ( ph  ->  if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  M ) )
281, 2, 3, 4, 6, 27ruclem9 12837 . . 3  |-  ( ph  ->  ( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M )
) )  /\  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <_  ( 2nd `  ( G `  M ) ) ) )
2928simpld 446 . 2  |-  ( ph  ->  ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M )
) ) )
30 xp2nd 6377 . . . 4  |-  ( ( G `  if ( M  <_  N ,  N ,  M )
)  e.  ( RR 
X.  RR )  -> 
( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  e.  RR )
3113, 30syl 16 . . 3  |-  ( ph  ->  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  e.  RR )
321, 2, 3, 4ruclem8 12836 . . . 4  |-  ( (
ph  /\  if ( M  <_  N ,  N ,  M )  e.  NN0 )  ->  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) ) )
3312, 32mpdan 650 . . 3  |-  ( ph  ->  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) ) )
34 max2 10775 . . . . . . 7  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  N  <_  if ( M  <_  N ,  N ,  M ) )
3519, 20, 34syl2anc 643 . . . . . 6  |-  ( ph  ->  N  <_  if ( M  <_  N ,  N ,  M ) )
3610nn0zd 10373 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
37 eluz 10499 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  if ( M  <_  N ,  N ,  M )  e.  ZZ )  -> 
( if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  N )  <->  N  <_  if ( M  <_  N ,  N ,  M ) ) )
3836, 24, 37syl2anc 643 . . . . . 6  |-  ( ph  ->  ( if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  N )  <->  N  <_  if ( M  <_  N ,  N ,  M ) ) )
3935, 38mpbird 224 . . . . 5  |-  ( ph  ->  if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  N ) )
401, 2, 3, 4, 10, 39ruclem9 12837 . . . 4  |-  ( ph  ->  ( ( 1st `  ( G `  N )
)  <_  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M )
) )  /\  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <_  ( 2nd `  ( G `  N ) ) ) )
4140simprd 450 . . 3  |-  ( ph  ->  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <_  ( 2nd `  ( G `  N )
) )
4215, 31, 18, 33, 41ltletrd 9230 . 2  |-  ( ph  ->  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <  ( 2nd `  ( G `  N )
) )
439, 15, 18, 29, 42lelttrd 9228 1  |-  ( ph  ->  ( 1st `  ( G `  M )
)  <  ( 2nd `  ( G `  N
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   [_csb 3251    u. cun 3318   ifcif 3739   {csn 3814   <.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   0cc0 8990   1c1 8991    + caddc 8993    < clt 9120    <_ cle 9121    / cdiv 9677   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488    seq cseq 11323
This theorem is referenced by:  ruclem11  12839  ruclem12  12840
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-cnex 9046  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-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  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-recs 6633  df-rdg 6668  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-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-seq 11324
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