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Theorem ruclem8 14289
Description: Lemma for ruc 14295. 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 5865 . . . . . 6  |-  ( k  =  0  ->  ( G `  k )  =  ( G ` 
0 ) )
21fveq2d 5869 . . . . 5  |-  ( k  =  0  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  0 )
) )
31fveq2d 5869 . . . . 5  |-  ( k  =  0  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  0 )
) )
42, 3breq12d 4415 . . . 4  |-  ( k  =  0  ->  (
( 1st `  ( G `  k )
)  <  ( 2nd `  ( G `  k
) )  <->  ( 1st `  ( G `  0
) )  <  ( 2nd `  ( G ` 
0 ) ) ) )
54imbi2d 318 . . 3  |-  ( k  =  0  ->  (
( ph  ->  ( 1st `  ( G `  k
) )  <  ( 2nd `  ( G `  k ) ) )  <-> 
( ph  ->  ( 1st `  ( G `  0
) )  <  ( 2nd `  ( G ` 
0 ) ) ) ) )
6 fveq2 5865 . . . . . 6  |-  ( k  =  n  ->  ( G `  k )  =  ( G `  n ) )
76fveq2d 5869 . . . . 5  |-  ( k  =  n  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  n )
) )
86fveq2d 5869 . . . . 5  |-  ( k  =  n  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  n )
) )
97, 8breq12d 4415 . . . 4  |-  ( k  =  n  ->  (
( 1st `  ( G `  k )
)  <  ( 2nd `  ( G `  k
) )  <->  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )
109imbi2d 318 . . 3  |-  ( k  =  n  ->  (
( ph  ->  ( 1st `  ( G `  k
) )  <  ( 2nd `  ( G `  k ) ) )  <-> 
( ph  ->  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) ) )
11 fveq2 5865 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( G `  k )  =  ( G `  ( n  +  1
) ) )
1211fveq2d 5869 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  ( n  +  1 ) ) ) )
1311fveq2d 5869 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  ( n  +  1 ) ) ) )
1412, 13breq12d 4415 . . . 4  |-  ( k  =  ( n  + 
1 )  ->  (
( 1st `  ( G `  k )
)  <  ( 2nd `  ( G `  k
) )  <->  ( 1st `  ( G `  (
n  +  1 ) ) )  <  ( 2nd `  ( G `  ( n  +  1
) ) ) ) )
1514imbi2d 318 . . 3  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( 1st `  ( G `  k
) )  <  ( 2nd `  ( G `  k ) ) )  <-> 
( ph  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  <  ( 2nd `  ( G `  ( n  +  1
) ) ) ) ) )
16 fveq2 5865 . . . . . 6  |-  ( k  =  N  ->  ( G `  k )  =  ( G `  N ) )
1716fveq2d 5869 . . . . 5  |-  ( k  =  N  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  N )
) )
1816fveq2d 5869 . . . . 5  |-  ( k  =  N  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  N )
) )
1917, 18breq12d 4415 . . . 4  |-  ( k  =  N  ->  (
( 1st `  ( G `  k )
)  <  ( 2nd `  ( G `  k
) )  <->  ( 1st `  ( G `  N
) )  <  ( 2nd `  ( G `  N ) ) ) )
2019imbi2d 318 . . 3  |-  ( k  =  N  ->  (
( ph  ->  ( 1st `  ( G `  k
) )  <  ( 2nd `  ( G `  k ) ) )  <-> 
( ph  ->  ( 1st `  ( G `  N
) )  <  ( 2nd `  ( G `  N ) ) ) ) )
21 0lt1 10136 . . . . 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 14286 . . . . . 6  |-  ( ph  ->  ( G `  0
)  =  <. 0 ,  1 >. )
2827fveq2d 5869 . . . . 5  |-  ( ph  ->  ( 1st `  ( G `  0 )
)  =  ( 1st `  <. 0 ,  1
>. ) )
29 c0ex 9637 . . . . . 6  |-  0  e.  _V
30 1ex 9638 . . . . . 6  |-  1  e.  _V
3129, 30op1st 6801 . . . . 5  |-  ( 1st `  <. 0 ,  1
>. )  =  0
3228, 31syl6eq 2501 . . . 4  |-  ( ph  ->  ( 1st `  ( G `  0 )
)  =  0 )
3327fveq2d 5869 . . . . 5  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  ( 2nd `  <. 0 ,  1
>. ) )
3429, 30op2nd 6802 . . . . 5  |-  ( 2nd `  <. 0 ,  1
>. )  =  1
3533, 34syl6eq 2501 . . . 4  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  1 )
3622, 32, 353brtr4d 4433 . . 3  |-  ( ph  ->  ( 1st `  ( G `  0 )
)  <  ( 2nd `  ( G `  0
) ) )
3723adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  F : NN
--> RR )
3824adantr 467 . . . . . . . . 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 14287 . . . . . . . . . . . 12  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
4039ffvelrnda 6022 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( G `  n )  e.  ( RR  X.  RR ) )
4140adantrr 723 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( G `  n )  e.  ( RR  X.  RR ) )
42 xp1st 6823 . . . . . . . . . 10  |-  ( ( G `  n )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  n
) )  e.  RR )
4341, 42syl 17 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( 1st `  ( G `  n
) )  e.  RR )
44 xp2nd 6824 . . . . . . . . . 10  |-  ( ( G `  n )  e.  ( RR  X.  RR )  ->  ( 2nd `  ( G `  n
) )  e.  RR )
4541, 44syl 17 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( 2nd `  ( G `  n
) )  e.  RR )
46 nn0p1nn 10909 . . . . . . . . . . 11  |-  ( n  e.  NN0  ->  ( n  +  1 )  e.  NN )
47 ffvelrn 6020 . . . . . . . . . . 11  |-  ( ( F : NN --> RR  /\  ( n  +  1
)  e.  NN )  ->  ( F `  ( n  +  1
) )  e.  RR )
4823, 46, 47syl2an 480 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( F `  ( n  +  1 ) )  e.  RR )
4948adantrr 723 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( F `  ( n  +  1 ) )  e.  RR )
50 eqid 2451 . . . . . . . . 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 2451 . . . . . . . . 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 766 . . . . . . . . 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 14284 . . . . . . . 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 1021 . . . . . . 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 14288 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( G `  ( n  +  1 ) )  =  ( ( G `  n
) D ( F `
 ( n  + 
1 ) ) ) )
5655adantrr 723 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( G `  ( n  +  1 ) )  =  ( ( G `  n
) D ( F `
 ( n  + 
1 ) ) ) )
57 1st2nd2 6830 . . . . . . . . . . 11  |-  ( ( G `  n )  e.  ( RR  X.  RR )  ->  ( G `
 n )  = 
<. ( 1st `  ( G `  n )
) ,  ( 2nd `  ( G `  n
) ) >. )
5841, 57syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( G `  n )  =  <. ( 1st `  ( G `
 n ) ) ,  ( 2nd `  ( G `  n )
) >. )
5958oveq1d 6305 . . . . . . . . 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 2485 . . . . . . . 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 5869 . . . . . . 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 5869 . . . . . . 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 4433 . . . . . 6  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  <  ( 2nd `  ( G `  ( n  +  1
) ) ) )
6463expr 620 . . . . 5  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( ( 1st `  ( G `  n ) )  < 
( 2nd `  ( G `  n )
)  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  <  ( 2nd `  ( G `  ( n  +  1
) ) ) ) )
6564expcom 437 . . . 4  |-  ( n  e.  NN0  ->  ( ph  ->  ( ( 1st `  ( G `  n )
)  <  ( 2nd `  ( G `  n
) )  ->  ( 1st `  ( G `  ( n  +  1
) ) )  < 
( 2nd `  ( G `  ( n  +  1 ) ) ) ) ) )
6665a2d 29 . . 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 11030 . 2  |-  ( N  e.  NN0  ->  ( ph  ->  ( 1st `  ( G `  N )
)  <  ( 2nd `  ( G `  N
) ) ) )
6867impcom 432 1  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( 1st `  ( G `  N
) )  <  ( 2nd `  ( G `  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   [_csb 3363    u. cun 3402   ifcif 3881   {csn 3968   <.cop 3974   class class class wbr 4402    X. cxp 4832   -->wf 5578   ` cfv 5582  (class class class)co 6290    |-> cmpt2 6292   1stc1st 6791   2ndc2nd 6792   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    < clt 9675    <_ cle 9676    / cdiv 10269   NNcn 10609   2c2 10659   NN0cn0 10869    seqcseq 12213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-seq 12214
This theorem is referenced by:  ruclem9  14290  ruclem10  14291  ruclem12  14293
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