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Theorem ruclem8 12836
Description: Lemma for ruc 12842. 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 5728 . . . . . 6  |-  ( k  =  0  ->  ( G `  k )  =  ( G ` 
0 ) )
21fveq2d 5732 . . . . 5  |-  ( k  =  0  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  0 )
) )
31fveq2d 5732 . . . . 5  |-  ( k  =  0  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  0 )
) )
42, 3breq12d 4225 . . . 4  |-  ( k  =  0  ->  (
( 1st `  ( G `  k )
)  <  ( 2nd `  ( G `  k
) )  <->  ( 1st `  ( G `  0
) )  <  ( 2nd `  ( G ` 
0 ) ) ) )
54imbi2d 308 . . 3  |-  ( k  =  0  ->  (
( ph  ->  ( 1st `  ( G `  k
) )  <  ( 2nd `  ( G `  k ) ) )  <-> 
( ph  ->  ( 1st `  ( G `  0
) )  <  ( 2nd `  ( G ` 
0 ) ) ) ) )
6 fveq2 5728 . . . . . 6  |-  ( k  =  n  ->  ( G `  k )  =  ( G `  n ) )
76fveq2d 5732 . . . . 5  |-  ( k  =  n  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  n )
) )
86fveq2d 5732 . . . . 5  |-  ( k  =  n  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  n )
) )
97, 8breq12d 4225 . . . 4  |-  ( k  =  n  ->  (
( 1st `  ( G `  k )
)  <  ( 2nd `  ( G `  k
) )  <->  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )
109imbi2d 308 . . 3  |-  ( k  =  n  ->  (
( ph  ->  ( 1st `  ( G `  k
) )  <  ( 2nd `  ( G `  k ) ) )  <-> 
( ph  ->  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) ) )
11 fveq2 5728 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( G `  k )  =  ( G `  ( n  +  1
) ) )
1211fveq2d 5732 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  ( n  +  1 ) ) ) )
1311fveq2d 5732 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  ( n  +  1 ) ) ) )
1412, 13breq12d 4225 . . . 4  |-  ( k  =  ( n  + 
1 )  ->  (
( 1st `  ( G `  k )
)  <  ( 2nd `  ( G `  k
) )  <->  ( 1st `  ( G `  (
n  +  1 ) ) )  <  ( 2nd `  ( G `  ( n  +  1
) ) ) ) )
1514imbi2d 308 . . 3  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( 1st `  ( G `  k
) )  <  ( 2nd `  ( G `  k ) ) )  <-> 
( ph  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  <  ( 2nd `  ( G `  ( n  +  1
) ) ) ) ) )
16 fveq2 5728 . . . . . 6  |-  ( k  =  N  ->  ( G `  k )  =  ( G `  N ) )
1716fveq2d 5732 . . . . 5  |-  ( k  =  N  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  N )
) )
1816fveq2d 5732 . . . . 5  |-  ( k  =  N  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  N )
) )
1917, 18breq12d 4225 . . . 4  |-  ( k  =  N  ->  (
( 1st `  ( G `  k )
)  <  ( 2nd `  ( G `  k
) )  <->  ( 1st `  ( G `  N
) )  <  ( 2nd `  ( G `  N ) ) ) )
2019imbi2d 308 . . 3  |-  ( k  =  N  ->  (
( ph  ->  ( 1st `  ( G `  k
) )  <  ( 2nd `  ( G `  k ) ) )  <-> 
( ph  ->  ( 1st `  ( G `  N
) )  <  ( 2nd `  ( G `  N ) ) ) ) )
21 0lt1 9550 . . . . 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 12833 . . . . . 6  |-  ( ph  ->  ( G `  0
)  =  <. 0 ,  1 >. )
2827fveq2d 5732 . . . . 5  |-  ( ph  ->  ( 1st `  ( G `  0 )
)  =  ( 1st `  <. 0 ,  1
>. ) )
29 c0ex 9085 . . . . . 6  |-  0  e.  _V
30 1ex 9086 . . . . . 6  |-  1  e.  _V
3129, 30op1st 6355 . . . . 5  |-  ( 1st `  <. 0 ,  1
>. )  =  0
3228, 31syl6eq 2484 . . . 4  |-  ( ph  ->  ( 1st `  ( G `  0 )
)  =  0 )
3327fveq2d 5732 . . . . 5  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  ( 2nd `  <. 0 ,  1
>. ) )
3429, 30op2nd 6356 . . . . 5  |-  ( 2nd `  <. 0 ,  1
>. )  =  1
3533, 34syl6eq 2484 . . . 4  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  1 )
3622, 32, 353brtr4d 4242 . . 3  |-  ( ph  ->  ( 1st `  ( G `  0 )
)  <  ( 2nd `  ( G `  0
) ) )
3723adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  F : NN
--> RR )
3824adantr 452 . . . . . . . . 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 12834 . . . . . . . . . . . 12  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
4039ffvelrnda 5870 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( G `  n )  e.  ( RR  X.  RR ) )
4140adantrr 698 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( G `  n )  e.  ( RR  X.  RR ) )
42 xp1st 6376 . . . . . . . . . 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 6377 . . . . . . . . . 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 10259 . . . . . . . . . . 11  |-  ( n  e.  NN0  ->  ( n  +  1 )  e.  NN )
47 ffvelrn 5868 . . . . . . . . . . 11  |-  ( ( F : NN --> RR  /\  ( n  +  1
)  e.  NN )  ->  ( F `  ( n  +  1
) )  e.  RR )
4823, 46, 47syl2an 464 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( F `  ( n  +  1 ) )  e.  RR )
4948adantrr 698 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( F `  ( n  +  1 ) )  e.  RR )
50 eqid 2436 . . . . . . . . 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 2436 . . . . . . . . 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 734 . . . . . . . . 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 12831 . . . . . . . 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 970 . . . . . . 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 12835 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( G `  ( n  +  1 ) )  =  ( ( G `  n
) D ( F `
 ( n  + 
1 ) ) ) )
5655adantrr 698 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( G `  ( n  +  1 ) )  =  ( ( G `  n
) D ( F `
 ( n  + 
1 ) ) ) )
57 1st2nd2 6386 . . . . . . . . . . 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 6096 . . . . . . . . 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 2468 . . . . . . . 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 5732 . . . . . . 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 5732 . . . . . . 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 4242 . . . . . 6  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  <  ( 2nd `  ( G `  ( n  +  1
) ) ) )
6463expr 599 . . . . 5  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( ( 1st `  ( G `  n ) )  < 
( 2nd `  ( G `  n )
)  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  <  ( 2nd `  ( G `  ( n  +  1
) ) ) ) )
6564expcom 425 . . . 4  |-  ( n  e.  NN0  ->  ( ph  ->  ( ( 1st `  ( G `  n )
)  <  ( 2nd `  ( G `  n
) )  ->  ( 1st `  ( G `  ( n  +  1
) ) )  < 
( 2nd `  ( G `  ( n  +  1 ) ) ) ) ) )
6665a2d 24 . . 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 10366 . 2  |-  ( N  e.  NN0  ->  ( ph  ->  ( 1st `  ( G `  N )
)  <  ( 2nd `  ( G `  N
) ) ) )
6867impcom 420 1  |-  ( (
ph  /\  N  e.  NN0 )  ->  ( 1st `  ( G `  N
) )  <  ( 2nd `  ( G `  N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   [_csb 3251    u. cun 3318   ifcif 3739   {csn 3814   <.cop 3817   class class class wbr 4212    X. cxp 4876   -->wf 5450   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1stc1st 6347   2ndc2nd 6348   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    < clt 9120    <_ cle 9121    / cdiv 9677   NNcn 10000   2c2 10049   NN0cn0 10221    seq cseq 11323
This theorem is referenced by:  ruclem9  12837  ruclem10  12838  ruclem12  12840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-seq 11324
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