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Theorem ruclem10 12564
Description: Lemma for ruc 12568. 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 12560 . . . 4  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
6 ruclem10.6 . . . 4  |-  ( ph  ->  M  e.  NN0 )
7 ffvelrn 5701 . . . 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 6191 . . 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 3635 . . . . 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 5701 . . . 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 6191 . . 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 5701 . . . 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 6192 . . 3  |-  ( ( G `  N )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  N
) )  e.  RR )
2119, 20syl 15 . 2  |-  ( ph  ->  ( 2nd `  ( G `  N )
)  e.  RR )
226nn0red 10066 . . . . . 6  |-  ( ph  ->  M  e.  RR )
2311nn0red 10066 . . . . . 6  |-  ( ph  ->  N  e.  RR )
24 max1 10561 . . . . . 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 10162 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
2713nn0zd 10162 . . . . . 6  |-  ( ph  ->  if ( M  <_  N ,  N ,  M )  e.  ZZ )
28 eluz 10288 . . . . . 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 12563 . . 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 6192 . . . 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 12562 . . . 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 10563 . . . . . . 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 10162 . . . . . . 7  |-  ( ph  ->  N  e.  ZZ )
40 eluz 10288 . . . . . . 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 12563 . . . 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 9021 . 2  |-  ( ph  ->  ( 1st `  ( G `  if ( M  <_  N ,  N ,  M ) ) )  <  ( 2nd `  ( G `  N )
) )
4610, 17, 21, 32, 45lelttrd 9019 1  |-  ( ph  ->  ( 1st `  ( G `  M )
)  <  ( 2nd `  ( G `  N
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1633    e. wcel 1701   [_csb 3115    u. cun 3184   ifcif 3599   {csn 3674   <.cop 3677   class class class wbr 4060    X. cxp 4724   -->wf 5288   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902   1stc1st 6162   2ndc2nd 6163   RRcr 8781   0cc0 8782   1c1 8783    + caddc 8785    < clt 8912    <_ cle 8913    / cdiv 9468   NNcn 9791   2c2 9840   NN0cn0 10012   ZZcz 10071   ZZ>=cuz 10277    seq cseq 11093
This theorem is referenced by:  ruclem11  12565  ruclem12  12566
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-seq 11094
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