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Theorem ruclem10 14056
Description: Lemma for ruc 14060. 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 14052 . . . 4  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
6 ruclem10.6 . . . 4  |-  ( ph  ->  M  e.  NN0 )
75, 6ffvelrnd 6008 . . 3  |-  ( ph  ->  ( G `  M
)  e.  ( RR 
X.  RR ) )
8 xp1st 6803 . . 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 )
1110, 6ifcld 3972 . . . 4  |-  ( ph  ->  if ( M  <_  N ,  N ,  M )  e.  NN0 )
125, 11ffvelrnd 6008 . . 3  |-  ( ph  ->  ( G `  if ( M  <_  N ,  N ,  M )
)  e.  ( RR 
X.  RR ) )
13 xp1st 6803 . . 3  |-  ( ( G `  if ( M  <_  N ,  N ,  M )
)  e.  ( RR 
X.  RR )  -> 
( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  e.  RR )
1412, 13syl 16 . 2  |-  ( ph  ->  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  e.  RR )
155, 10ffvelrnd 6008 . . 3  |-  ( ph  ->  ( G `  N
)  e.  ( RR 
X.  RR ) )
16 xp2nd 6804 . . 3  |-  ( ( G `  N )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  N
) )  e.  RR )
1715, 16syl 16 . 2  |-  ( ph  ->  ( 2nd `  ( G `  N )
)  e.  RR )
186nn0red 10849 . . . . . 6  |-  ( ph  ->  M  e.  RR )
1910nn0red 10849 . . . . . 6  |-  ( ph  ->  N  e.  RR )
20 max1 11389 . . . . . 6  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  M  <_  if ( M  <_  N ,  N ,  M ) )
2118, 19, 20syl2anc 659 . . . . 5  |-  ( ph  ->  M  <_  if ( M  <_  N ,  N ,  M ) )
226nn0zd 10963 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
2311nn0zd 10963 . . . . . 6  |-  ( ph  ->  if ( M  <_  N ,  N ,  M )  e.  ZZ )
24 eluz 11095 . . . . . 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 ) ) )
2522, 23, 24syl2anc 659 . . . . 5  |-  ( ph  ->  ( if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  M )  <->  M  <_  if ( M  <_  N ,  N ,  M ) ) )
2621, 25mpbird 232 . . . 4  |-  ( ph  ->  if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  M ) )
271, 2, 3, 4, 6, 26ruclem9 14055 . . 3  |-  ( ph  ->  ( ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M )
) )  /\  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <_  ( 2nd `  ( G `  M ) ) ) )
2827simpld 457 . 2  |-  ( ph  ->  ( 1st `  ( G `  M )
)  <_  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M )
) ) )
29 xp2nd 6804 . . . 4  |-  ( ( G `  if ( M  <_  N ,  N ,  M )
)  e.  ( RR 
X.  RR )  -> 
( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  e.  RR )
3012, 29syl 16 . . 3  |-  ( ph  ->  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  e.  RR )
311, 2, 3, 4ruclem8 14054 . . . 4  |-  ( (
ph  /\  if ( M  <_  N ,  N ,  M )  e.  NN0 )  ->  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) ) )
3211, 31mpdan 666 . . 3  |-  ( ph  ->  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) ) )
33 max2 11391 . . . . . . 7  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  N  <_  if ( M  <_  N ,  N ,  M ) )
3418, 19, 33syl2anc 659 . . . . . 6  |-  ( ph  ->  N  <_  if ( M  <_  N ,  N ,  M ) )
3510nn0zd 10963 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
36 eluz 11095 . . . . . . 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 ) ) )
3735, 23, 36syl2anc 659 . . . . . 6  |-  ( ph  ->  ( if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  N )  <->  N  <_  if ( M  <_  N ,  N ,  M ) ) )
3834, 37mpbird 232 . . . . 5  |-  ( ph  ->  if ( M  <_  N ,  N ,  M )  e.  (
ZZ>= `  N ) )
391, 2, 3, 4, 10, 38ruclem9 14055 . . . 4  |-  ( ph  ->  ( ( 1st `  ( G `  N )
)  <_  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M )
) )  /\  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <_  ( 2nd `  ( G `  N ) ) ) )
4039simprd 461 . . 3  |-  ( ph  ->  ( 2nd `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <_  ( 2nd `  ( G `  N )
) )
4114, 30, 17, 32, 40ltletrd 9731 . 2  |-  ( ph  ->  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <  ( 2nd `  ( G `  N )
) )
429, 14, 17, 28, 41lelttrd 9729 1  |-  ( ph  ->  ( 1st `  ( G `  M )
)  <  ( 2nd `  ( G `  N
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1398    e. wcel 1823   [_csb 3420    u. cun 3459   ifcif 3929   {csn 4016   <.cop 4022   class class class wbr 4439    X. cxp 4986   -->wf 5566   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   1stc1st 6771   2ndc2nd 6772   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    < clt 9617    <_ cle 9618    / cdiv 10202   NNcn 10531   2c2 10581   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082    seqcseq 12089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-seq 12090
This theorem is referenced by:  ruclem11  14057  ruclem12  14058
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