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Theorem ruclem8 13502
Description: Lemma for ruc 13508. 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 5679 . . . . . 6  |-  ( k  =  0  ->  ( G `  k )  =  ( G ` 
0 ) )
21fveq2d 5683 . . . . 5  |-  ( k  =  0  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  0 )
) )
31fveq2d 5683 . . . . 5  |-  ( k  =  0  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  0 )
) )
42, 3breq12d 4293 . . . 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 5679 . . . . . 6  |-  ( k  =  n  ->  ( G `  k )  =  ( G `  n ) )
76fveq2d 5683 . . . . 5  |-  ( k  =  n  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  n )
) )
86fveq2d 5683 . . . . 5  |-  ( k  =  n  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  n )
) )
97, 8breq12d 4293 . . . 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 5679 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( G `  k )  =  ( G `  ( n  +  1
) ) )
1211fveq2d 5683 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  ( n  +  1 ) ) ) )
1311fveq2d 5683 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  ( n  +  1 ) ) ) )
1412, 13breq12d 4293 . . . 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 5679 . . . . . 6  |-  ( k  =  N  ->  ( G `  k )  =  ( G `  N ) )
1716fveq2d 5683 . . . . 5  |-  ( k  =  N  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  N )
) )
1816fveq2d 5683 . . . . 5  |-  ( k  =  N  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  N )
) )
1917, 18breq12d 4293 . . . 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 9850 . . . . 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 13499 . . . . . 6  |-  ( ph  ->  ( G `  0
)  =  <. 0 ,  1 >. )
2827fveq2d 5683 . . . . 5  |-  ( ph  ->  ( 1st `  ( G `  0 )
)  =  ( 1st `  <. 0 ,  1
>. ) )
29 c0ex 9368 . . . . . 6  |-  0  e.  _V
30 1ex 9369 . . . . . 6  |-  1  e.  _V
3129, 30op1st 6574 . . . . 5  |-  ( 1st `  <. 0 ,  1
>. )  =  0
3228, 31syl6eq 2481 . . . 4  |-  ( ph  ->  ( 1st `  ( G `  0 )
)  =  0 )
3327fveq2d 5683 . . . . 5  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  ( 2nd `  <. 0 ,  1
>. ) )
3429, 30op2nd 6575 . . . . 5  |-  ( 2nd `  <. 0 ,  1
>. )  =  1
3533, 34syl6eq 2481 . . . 4  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  1 )
3622, 32, 353brtr4d 4310 . . 3  |-  ( ph  ->  ( 1st `  ( G `  0 )
)  <  ( 2nd `  ( G `  0
) ) )
3723adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  F : NN
--> RR )
3824adantr 462 . . . . . . . . 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 13500 . . . . . . . . . . . 12  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
4039ffvelrnda 5831 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( G `  n )  e.  ( RR  X.  RR ) )
4140adantrr 709 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( G `  n )  e.  ( RR  X.  RR ) )
42 xp1st 6595 . . . . . . . . . 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 6596 . . . . . . . . . 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 10607 . . . . . . . . . . 11  |-  ( n  e.  NN0  ->  ( n  +  1 )  e.  NN )
47 ffvelrn 5829 . . . . . . . . . . 11  |-  ( ( F : NN --> RR  /\  ( n  +  1
)  e.  NN )  ->  ( F `  ( n  +  1
) )  e.  RR )
4823, 46, 47syl2an 474 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( F `  ( n  +  1 ) )  e.  RR )
4948adantrr 709 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( F `  ( n  +  1 ) )  e.  RR )
50 eqid 2433 . . . . . . . . 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 2433 . . . . . . . . 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 749 . . . . . . . . 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 13497 . . . . . . . 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 994 . . . . . . 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 13501 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( G `  ( n  +  1 ) )  =  ( ( G `  n
) D ( F `
 ( n  + 
1 ) ) ) )
5655adantrr 709 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( G `  ( n  +  1 ) )  =  ( ( G `  n
) D ( F `
 ( n  + 
1 ) ) ) )
57 1st2nd2 6602 . . . . . . . . . . 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 6095 . . . . . . . . 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 2465 . . . . . . . 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 5683 . . . . . . 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 5683 . . . . . . 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 4310 . . . . . 6  |-  ( (
ph  /\  ( n  e.  NN0  /\  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G `  n ) ) ) )  ->  ( 1st `  ( G `  (
n  +  1 ) ) )  <  ( 2nd `  ( G `  ( n  +  1
) ) ) )
6463expr 610 . . . . 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 10726 . 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 1362    e. wcel 1755   [_csb 3276    u. cun 3314   ifcif 3779   {csn 3865   <.cop 3871   class class class wbr 4280    X. cxp 4825   -->wf 5402   ` cfv 5406  (class class class)co 6080    e. cmpt2 6082   1stc1st 6564   2ndc2nd 6565   RRcr 9269   0cc0 9270   1c1 9271    + caddc 9273    < clt 9406    <_ cle 9407    / cdiv 9981   NNcn 10310   2c2 10359   NN0cn0 10567    seqcseq 11790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-n0 10568  df-z 10635  df-uz 10850  df-fz 11425  df-seq 11791
This theorem is referenced by:  ruclem9  13503  ruclem10  13504  ruclem12  13506
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