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Theorem ruclem8 13641
Description: Lemma for ruc 13647. 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 5802 . . . . . 6  |-  ( k  =  0  ->  ( G `  k )  =  ( G ` 
0 ) )
21fveq2d 5806 . . . . 5  |-  ( k  =  0  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  0 )
) )
31fveq2d 5806 . . . . 5  |-  ( k  =  0  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  0 )
) )
42, 3breq12d 4416 . . . 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 5802 . . . . . 6  |-  ( k  =  n  ->  ( G `  k )  =  ( G `  n ) )
76fveq2d 5806 . . . . 5  |-  ( k  =  n  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  n )
) )
86fveq2d 5806 . . . . 5  |-  ( k  =  n  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  n )
) )
97, 8breq12d 4416 . . . 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 5802 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( G `  k )  =  ( G `  ( n  +  1
) ) )
1211fveq2d 5806 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  ( n  +  1 ) ) ) )
1311fveq2d 5806 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  ( n  +  1 ) ) ) )
1412, 13breq12d 4416 . . . 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 5802 . . . . . 6  |-  ( k  =  N  ->  ( G `  k )  =  ( G `  N ) )
1716fveq2d 5806 . . . . 5  |-  ( k  =  N  ->  ( 1st `  ( G `  k ) )  =  ( 1st `  ( G `  N )
) )
1816fveq2d 5806 . . . . 5  |-  ( k  =  N  ->  ( 2nd `  ( G `  k ) )  =  ( 2nd `  ( G `  N )
) )
1917, 18breq12d 4416 . . . 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 9977 . . . . 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 13638 . . . . . 6  |-  ( ph  ->  ( G `  0
)  =  <. 0 ,  1 >. )
2827fveq2d 5806 . . . . 5  |-  ( ph  ->  ( 1st `  ( G `  0 )
)  =  ( 1st `  <. 0 ,  1
>. ) )
29 c0ex 9495 . . . . . 6  |-  0  e.  _V
30 1ex 9496 . . . . . 6  |-  1  e.  _V
3129, 30op1st 6698 . . . . 5  |-  ( 1st `  <. 0 ,  1
>. )  =  0
3228, 31syl6eq 2511 . . . 4  |-  ( ph  ->  ( 1st `  ( G `  0 )
)  =  0 )
3327fveq2d 5806 . . . . 5  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  ( 2nd `  <. 0 ,  1
>. ) )
3429, 30op2nd 6699 . . . . 5  |-  ( 2nd `  <. 0 ,  1
>. )  =  1
3533, 34syl6eq 2511 . . . 4  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  1 )
3622, 32, 353brtr4d 4433 . . 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 13639 . . . . . . . . . . . 12  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
4039ffvelrnda 5955 . . . . . . . . . . 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 6719 . . . . . . . . . 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 6720 . . . . . . . . . 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 10734 . . . . . . . . . . 11  |-  ( n  e.  NN0  ->  ( n  +  1 )  e.  NN )
47 ffvelrn 5953 . . . . . . . . . . 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 2454 . . . . . . . . 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 2454 . . . . . . . . 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 13636 . . . . . . . 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 1001 . . . . . . 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 13640 . . . . . . . . . 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 6726 . . . . . . . . . . 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 6218 . . . . . . . . 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 2495 . . . . . . . 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 5806 . . . . . . 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 5806 . . . . . . 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 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 10853 . 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 1370    e. wcel 1758   [_csb 3398    u. cun 3437   ifcif 3902   {csn 3988   <.cop 3994   class class class wbr 4403    X. cxp 4949   -->wf 5525   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   1stc1st 6688   2ndc2nd 6689   RRcr 9396   0cc0 9397   1c1 9398    + caddc 9400    < clt 9533    <_ cle 9534    / cdiv 10108   NNcn 10437   2c2 10486   NN0cn0 10694    seqcseq 11927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-seq 11928
This theorem is referenced by:  ruclem9  13642  ruclem10  13643  ruclem12  13645
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