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Theorem ruclem8 13843
Description: Lemma for ruc 13849. The intervals of the  G sequence are all nonempty. (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 )
Assertion
Ref Expression
ruclem8  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( 1st `  ( G `  N
) )  <  ( 2nd `  ( G `  N ) ) )
Distinct variable groups:    x, m, y, F    m, G, x, y    m, N, x, y
Allowed substitution hints:    ph( x, y, m)    C( x, y, m)    D( x, y, m)

Proof of Theorem ruclem8
Dummy variables  n  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5871 . . . . . 6  |-  ( k  =  0  ->  ( G `  k )  =  ( G ` 
0 ) )
21fveq2d 5875 . . . . 5  |-  ( k  =  0  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  0 )
) )
31fveq2d 5875 . . . . 5  |-  ( k  =  0  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  0 )
) )
42, 3breq12d 4465 . . . 4  |-  ( k  =  0  ->  (
( 1st `  ( G `  k )
)  <  ( 2nd `  ( G `  k
) )  <->  ( 1st `  ( G `  0
) )  <  ( 2nd `  ( G ` 
0 ) ) ) )
54imbi2d 316 . . 3  |-  ( k  =  0  ->  (
( ph  ->  ( 1st `  ( G `  k
) )  <  ( 2nd `  ( G `  k ) ) )  <-> 
( ph  ->  ( 1st `  ( G `  0
) )  <  ( 2nd `  ( G ` 
0 ) ) ) ) )
6 fveq2 5871 . . . . . 6  |-  ( k  =  n  ->  ( G `  k )  =  ( G `  n ) )
76fveq2d 5875 . . . . 5  |-  ( k  =  n  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  n )
) )
86fveq2d 5875 . . . . 5  |-  ( k  =  n  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  n )
) )
97, 8breq12d 4465 . . . 4  |-  ( k  =  n  ->  (
( 1st `  ( G `  k )
)  <  ( 2nd `  ( G `  k
) )  <->  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )
109imbi2d 316 . . 3  |-  ( k  =  n  ->  (
( ph  ->  ( 1st `  ( G `  k
) )  <  ( 2nd `  ( G `  k ) ) )  <-> 
( ph  ->  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) ) )
11 fveq2 5871 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( G `  k )  =  ( G `  ( n  +  1
) ) )
1211fveq2d 5875 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  ( n  +  1 ) ) ) )
1311fveq2d 5875 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  ( n  +  1 ) ) ) )
1412, 13breq12d 4465 . . . 4  |-  ( k  =  ( n  + 
1 )  ->  (
( 1st `  ( G `  k )
)  <  ( 2nd `  ( G `  k
) )  <->  ( 1st `  ( G `  (
n  +  1 ) ) )  <  ( 2nd `  ( G `  ( n  +  1
) ) ) ) )
1514imbi2d 316 . . 3  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( 1st `  ( G `  k
) )  <  ( 2nd `  ( G `  k ) ) )  <-> 
( ph  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  <  ( 2nd `  ( G `  ( n  +  1
) ) ) ) ) )
16 fveq2 5871 . . . . . 6  |-  ( k  =  N  ->  ( G `  k )  =  ( G `  N ) )
1716fveq2d 5875 . . . . 5  |-  ( k  =  N  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  N )
) )
1816fveq2d 5875 . . . . 5  |-  ( k  =  N  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  N )
) )
1917, 18breq12d 4465 . . . 4  |-  ( k  =  N  ->  (
( 1st `  ( G `  k )
)  <  ( 2nd `  ( G `  k
) )  <->  ( 1st `  ( G `  N
) )  <  ( 2nd `  ( G `  N ) ) ) )
2019imbi2d 316 . . 3  |-  ( k  =  N  ->  (
( ph  ->  ( 1st `  ( G `  k
) )  <  ( 2nd `  ( G `  k ) ) )  <-> 
( ph  ->  ( 1st `  ( G `  N
) )  <  ( 2nd `  ( G `  N ) ) ) ) )
21 0lt1 10085 . . . . 5  |-  0  <  1
2221a1i 11 . . . 4  |-  ( ph  ->  0  <  1 )
23 ruc.1 . . . . . . 7  |-  ( ph  ->  F : NN --> RR )
24 ruc.2 . . . . . . 7  |-  ( 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
) >. ) ) )
25 ruc.4 . . . . . . 7  |-  C  =  ( { <. 0 ,  <. 0 ,  1
>. >. }  u.  F
)
26 ruc.5 . . . . . . 7  |-  G  =  seq 0 ( D ,  C )
2723, 24, 25, 26ruclem4 13840 . . . . . 6  |-  ( ph  ->  ( G `  0
)  =  <. 0 ,  1 >. )
2827fveq2d 5875 . . . . 5  |-  ( ph  ->  ( 1st `  ( G `  0 )
)  =  ( 1st `  <. 0 ,  1
>. ) )
29 c0ex 9600 . . . . . 6  |-  0  e.  _V
30 1ex 9601 . . . . . 6  |-  1  e.  _V
3129, 30op1st 6802 . . . . 5  |-  ( 1st `  <. 0 ,  1
>. )  =  0
3228, 31syl6eq 2524 . . . 4  |-  ( ph  ->  ( 1st `  ( G `  0 )
)  =  0 )
3327fveq2d 5875 . . . . 5  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  ( 2nd `  <. 0 ,  1
>. ) )
3429, 30op2nd 6803 . . . . 5  |-  ( 2nd `  <. 0 ,  1
>. )  =  1
3533, 34syl6eq 2524 . . . 4  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  1 )
3622, 32, 353brtr4d 4482 . . 3  |-  ( ph  ->  ( 1st `  ( G `  0 )
)  <  ( 2nd `  ( G `  0
) ) )
3723adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  F : NN
--> RR )
3824adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  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
) >. ) ) )
3923, 24, 25, 26ruclem6 13841 . . . . . . . . . . . 12  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
4039ffvelrnda 6031 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( G `  n )  e.  ( RR  X.  RR ) )
4140adantrr 716 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( G `  n )  e.  ( RR  X.  RR ) )
42 xp1st 6824 . . . . . . . . . 10  |-  ( ( G `  n )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  n
) )  e.  RR )
4341, 42syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( 1st `  ( G `  n
) )  e.  RR )
44 xp2nd 6825 . . . . . . . . . 10  |-  ( ( G `  n )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  n
) )  e.  RR )
4541, 44syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( 2nd `  ( G `  n
) )  e.  RR )
46 nn0p1nn 10845 . . . . . . . . . . 11  |-  ( n  e.  NN0  ->  ( n  +  1 )  e.  NN )
47 ffvelrn 6029 . . . . . . . . . . 11  |-  ( ( F : NN --> RR  /\  ( n  +  1
)  e.  NN )  ->  ( F `  ( n  +  1
) )  e.  RR )
4823, 46, 47syl2an 477 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( F `  ( n  +  1 ) )  e.  RR )
4948adantrr 716 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( F `  ( n  +  1 ) )  e.  RR )
50 eqid 2467 . . . . . . . . 9  |-  ( 1st `  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )  =  ( 1st `  ( <.
( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )
51 eqid 2467 . . . . . . . . 9  |-  ( 2nd `  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )  =  ( 2nd `  ( <.
( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )
52 simprr 756 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) )
5337, 38, 43, 45, 49, 50, 51, 52ruclem2 13838 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( ( 1st `  ( G `  n ) )  <_ 
( 1st `  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )  /\  ( 1st `  ( <. ( 1st `  ( G `  n ) ) ,  ( 2nd `  ( G `  n )
) >. D ( F `
 ( n  + 
1 ) ) ) )  <  ( 2nd `  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )  /\  ( 2nd `  ( <. ( 1st `  ( G `  n ) ) ,  ( 2nd `  ( G `  n )
) >. D ( F `
 ( n  + 
1 ) ) ) )  <_  ( 2nd `  ( G `  n
) ) ) )
5453simp2d 1009 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( 1st `  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )  <  ( 2nd `  ( <. ( 1st `  ( G `  n ) ) ,  ( 2nd `  ( G `  n )
) >. D ( F `
 ( n  + 
1 ) ) ) ) )
5523, 24, 25, 26ruclem7 13842 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( G `  ( n  +  1 ) )  =  ( ( G `  n
) D ( F `
 ( n  + 
1 ) ) ) )
5655adantrr 716 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( G `  ( n  +  1 ) )  =  ( ( G `  n
) D ( F `
 ( n  + 
1 ) ) ) )
57 1st2nd2 6831 . . . . . . . . . . 11  |-  ( ( G `  n )  e.  ( RR  X.  RR )  ->  ( G `
 n )  = 
<. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. )
5841, 57syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( G `  n )  =  <. ( 1st `  ( G `
 n ) ) ,  ( 2nd `  ( G `  n )
) >. )
5958oveq1d 6309 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( ( G `  n ) D ( F `  ( n  +  1
) ) )  =  ( <. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )
6056, 59eqtrd 2508 . . . . . . . 8  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( G `  ( n  +  1 ) )  =  (
<. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) )
6160fveq2d 5875 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  =  ( 1st `  ( <.
( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) ) )
6260fveq2d 5875 . . . . . . 7  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( 2nd `  ( G `  (
n  +  1 ) ) )  =  ( 2nd `  ( <.
( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. D ( F `  ( n  +  1 ) ) ) ) )
6354, 61, 623brtr4d 4482 . . . . . 6  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  <  ( 2nd `  ( G `  ( n  +  1
) ) ) )
6463expr 615 . . . . 5  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( ( 1st `  ( G `  n ) )  < 
( 2nd `  ( G `  n )
)  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  <  ( 2nd `  ( G `  ( n  +  1
) ) ) ) )
6564expcom 435 . . . 4  |-  ( n  e.  NN0  ->  ( ph  ->  ( ( 1st `  ( G `  n )
)  <  ( 2nd `  ( G `  n
) )  ->  ( 1st `  ( G `  ( n  +  1
) ) )  < 
( 2nd `  ( G `  ( n  +  1 ) ) ) ) ) )
6665a2d 26 . . 3  |-  ( n  e.  NN0  ->  ( (
ph  ->  ( 1st `  ( G `  n )
)  <  ( 2nd `  ( G `  n
) ) )  -> 
( ph  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  <  ( 2nd `  ( G `  ( n  +  1
) ) ) ) ) )
675, 10, 15, 20, 36, 66nn0ind 10967 . 2  |-  ( N  e.  NN0  ->  ( ph  ->  ( 1st `  ( G `  N )
)  <  ( 2nd `  ( G `  N
) ) ) )
6867impcom 430 1  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( 1st `  ( G `  N
) )  <  ( 2nd `  ( G `  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   [_csb 3440    u. cun 3479   ifcif 3944   {csn 4032   <.cop 4038   class class class wbr 4452    X. cxp 5002   -->wf 5589   ` cfv 5593  (class class class)co 6294    |-> cmpt2 6296   1stc1st 6792   2ndc2nd 6793   RRcr 9501   0cc0 9502   1c1 9503    + caddc 9505    < clt 9638    <_ cle 9639    / cdiv 10216   NNcn 10546   2c2 10595   NN0cn0 10805    seqcseq 12085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-n0 10806  df-z 10875  df-uz 11093  df-fz 11683  df-seq 12086
This theorem is referenced by:  ruclem9  13844  ruclem10  13845  ruclem12  13847
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