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Theorem ruclem10 12517
Description: Lemma for ruc 12521. 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 12513 . . . 4  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
6 ruclem10.6 . . . 4  |-  ( ph  ->  M  e.  NN0 )
7 ffvelrn 5663 . . . 4  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  M  e.  NN0 )  -> 
( G `  M
)  e.  ( RR 
X.  RR ) )
85, 6, 7syl2anc 642 . . 3  |-  ( ph  ->  ( G `  M
)  e.  ( RR 
X.  RR ) )
9 xp1st 6149 . . 3  |-  ( ( G `  M )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  M
) )  e.  RR )
108, 9syl 15 . 2  |-  ( ph  ->  ( 1st `  ( G `  M )
)  e.  RR )
11 ruclem10.7 . . . . 5  |-  ( ph  ->  N  e.  NN0 )
12 ifcl 3601 . . . . 5  |-  ( ( N  e.  NN0  /\  M  e.  NN0 )  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
1311, 6, 12syl2anc 642 . . . 4  |-  ( ph  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
14 ffvelrn 5663 . . . 4  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  if ( M  <_  N ,  N ,  M )  e.  NN0 )  -> 
( G `  if ( M  <_  N ,  N ,  M )
)  e.  ( RR 
X.  RR ) )
155, 13, 14syl2anc 642 . . 3  |-  ( ph  ->  ( G `  if ( M  <_  N ,  N ,  M )
)  e.  ( RR 
X.  RR ) )
16 xp1st 6149 . . 3  |-  ( ( G `  if ( M  <_  N ,  N ,  M )
)  e.  ( RR 
X.  RR )  -> 
( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  e.  RR )
1715, 16syl 15 . 2  |-  ( ph  ->  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  e.  RR )
18 ffvelrn 5663 . . . 4  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  N  e.  NN0 )  -> 
( G `  N
)  e.  ( RR 
X.  RR ) )
195, 11, 18syl2anc 642 . . 3  |-  ( ph  ->  ( G `  N
)  e.  ( RR 
X.  RR ) )
20 xp2nd 6150 . . 3  |-  ( ( G `  N )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  N
) )  e.  RR )
2119, 20syl 15 . 2  |-  ( ph  ->  ( 2nd `  ( G `  N )
)  e.  RR )
226nn0red 10019 . . . . . 6  |-  ( ph  ->  M  e.  RR )
2311nn0red 10019 . . . . . 6  |-  ( ph  ->  N  e.  RR )
24 max1 10514 . . . . . 6  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  M  <_  if ( M  <_  N ,  N ,  M ) )
2522, 23, 24syl2anc 642 . . . . 5  |-  ( ph  ->  M  <_  if ( M  <_  N ,  N ,  M ) )
266nn0zd 10115 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
2713nn0zd 10115 . . . . . 6  |-  ( ph  ->  if ( M  <_  N ,  N ,  M )  e.  ZZ )
28 eluz 10241 . . . . . 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 ) ) )
2926, 27, 28syl2anc 642 . . . . 5  |-  ( ph  ->  ( if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  M )  <->  M  <_  if ( M  <_  N ,  N ,  M ) ) )
3025, 29mpbird 223 . . . 4  |-  ( ph  ->  if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  M ) )
311, 2, 3, 4, 6, 30ruclem9 12516 . . 3  |-  ( ph  ->  ( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M )
) )  /\  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <_  ( 2nd `  ( G `  M ) ) ) )
3231simpld 445 . 2  |-  ( ph  ->  ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M )
) ) )
33 xp2nd 6150 . . . 4  |-  ( ( G `  if ( M  <_  N ,  N ,  M )
)  e.  ( RR 
X.  RR )  -> 
( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  e.  RR )
3415, 33syl 15 . . 3  |-  ( ph  ->  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  e.  RR )
351, 2, 3, 4ruclem8 12515 . . . 4  |-  ( (
ph  /\  if ( M  <_  N ,  N ,  M )  e.  NN0 )  ->  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) ) )
3613, 35mpdan 649 . . 3  |-  ( ph  ->  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) ) )
37 max2 10516 . . . . . . 7  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  N  <_  if ( M  <_  N ,  N ,  M ) )
3822, 23, 37syl2anc 642 . . . . . 6  |-  ( ph  ->  N  <_  if ( M  <_  N ,  N ,  M ) )
3911nn0zd 10115 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
40 eluz 10241 . . . . . . 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 ) ) )
4139, 27, 40syl2anc 642 . . . . . 6  |-  ( ph  ->  ( if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  N )  <->  N  <_  if ( M  <_  N ,  N ,  M ) ) )
4238, 41mpbird 223 . . . . 5  |-  ( ph  ->  if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  N ) )
431, 2, 3, 4, 11, 42ruclem9 12516 . . . 4  |-  ( ph  ->  ( ( 1st `  ( G `  N )
)  <_  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M )
) )  /\  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <_  ( 2nd `  ( G `  N ) ) ) )
4443simprd 449 . . 3  |-  ( ph  ->  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <_  ( 2nd `  ( G `  N )
) )
4517, 34, 21, 36, 44ltletrd 8976 . 2  |-  ( ph  ->  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <  ( 2nd `  ( G `  N )
) )
4610, 17, 21, 32, 45lelttrd 8974 1  |-  ( ph  ->  ( 1st `  ( G `  M )
)  <  ( 2nd `  ( G `  N
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   [_csb 3081    u. cun 3150   ifcif 3565   {csn 3640   <.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   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230    seq cseq 11046
This theorem is referenced by:  ruclem11  12518  ruclem12  12519
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-cnex 8793  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-recs 6388  df-rdg 6423  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-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047
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