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Theorem ruclem11 12831
Description: Lemma for ruc 12834. Closure lemmas for supremum. (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
ruclem11  |-  ( ph  ->  ( ran  ( 1st 
o.  G )  C_  RR  /\  ran  ( 1st 
o.  G )  =/=  (/)  /\  A. z  e. 
ran  ( 1st  o.  G ) z  <_ 
1 ) )
Distinct variable groups:    x, m, y    z, C    z, m, F, x, y    m, G, x, y, z    ph, z    z, D
Allowed substitution hints:    ph( x, y, m)    C( x, y, m)    D( x, y, m)

Proof of Theorem ruclem11
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 ruc.1 . . . . 5  |-  ( ph  ->  F : NN --> RR )
2 ruc.2 . . . . 5  |-  ( 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
) >. ) ) )
3 ruc.4 . . . . 5  |-  C  =  ( { <. 0 ,  <. 0 ,  1
>. >. }  u.  F
)
4 ruc.5 . . . . 5  |-  G  =  seq  0 ( D ,  C )
51, 2, 3, 4ruclem6 12826 . . . 4  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
6 1stcof 6366 . . . 4  |-  ( G : NN0 --> ( RR 
X.  RR )  -> 
( 1st  o.  G
) : NN0 --> RR )
75, 6syl 16 . . 3  |-  ( ph  ->  ( 1st  o.  G
) : NN0 --> RR )
8 frn 5589 . . 3  |-  ( ( 1st  o.  G ) : NN0 --> RR  ->  ran  ( 1st  o.  G
)  C_  RR )
97, 8syl 16 . 2  |-  ( ph  ->  ran  ( 1st  o.  G )  C_  RR )
10 fdm 5587 . . . . 5  |-  ( ( 1st  o.  G ) : NN0 --> RR  ->  dom  ( 1st  o.  G
)  =  NN0 )
117, 10syl 16 . . . 4  |-  ( ph  ->  dom  ( 1st  o.  G )  =  NN0 )
12 0nn0 10228 . . . . 5  |-  0  e.  NN0
13 ne0i 3626 . . . . 5  |-  ( 0  e.  NN0  ->  NN0  =/=  (/) )
1412, 13mp1i 12 . . . 4  |-  ( ph  ->  NN0  =/=  (/) )
1511, 14eqnetrd 2616 . . 3  |-  ( ph  ->  dom  ( 1st  o.  G )  =/=  (/) )
16 dm0rn0 5078 . . . 4  |-  ( dom  ( 1st  o.  G
)  =  (/)  <->  ran  ( 1st 
o.  G )  =  (/) )
1716necon3bii 2630 . . 3  |-  ( dom  ( 1st  o.  G
)  =/=  (/)  <->  ran  ( 1st 
o.  G )  =/=  (/) )
1815, 17sylib 189 . 2  |-  ( ph  ->  ran  ( 1st  o.  G )  =/=  (/) )
19 fvco3 5792 . . . . . 6  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  n  e.  NN0 )  -> 
( ( 1st  o.  G ) `  n
)  =  ( 1st `  ( G `  n
) ) )
205, 19sylan 458 . . . . 5  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( ( 1st  o.  G ) `  n )  =  ( 1st `  ( G `
 n ) ) )
211adantr 452 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  F : NN
--> RR )
222adantr 452 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  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
) >. ) ) )
23 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  n  e.  NN0 )
2412a1i 11 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  0  e.  NN0 )
2521, 22, 3, 4, 23, 24ruclem10 12830 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G ` 
0 ) ) )
261, 2, 3, 4ruclem4 12825 . . . . . . . . . 10  |-  ( ph  ->  ( G `  0
)  =  <. 0 ,  1 >. )
2726fveq2d 5724 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  ( 2nd `  <. 0 ,  1
>. ) )
2812elexi 2957 . . . . . . . . . 10  |-  0  e.  _V
29 1ex 9078 . . . . . . . . . 10  |-  1  e.  _V
3028, 29op2nd 6348 . . . . . . . . 9  |-  ( 2nd `  <. 0 ,  1
>. )  =  1
3127, 30syl6eq 2483 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  1 )
3231adantr 452 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 2nd `  ( G `  0
) )  =  1 )
3325, 32breqtrd 4228 . . . . . 6  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 1st `  ( G `  n
) )  <  1
)
345ffvelrnda 5862 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( G `  n )  e.  ( RR  X.  RR ) )
35 xp1st 6368 . . . . . . . 8  |-  ( ( G `  n )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  n
) )  e.  RR )
3634, 35syl 16 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 1st `  ( G `  n
) )  e.  RR )
37 1re 9082 . . . . . . 7  |-  1  e.  RR
38 ltle 9155 . . . . . . 7  |-  ( ( ( 1st `  ( G `  n )
)  e.  RR  /\  1  e.  RR )  ->  ( ( 1st `  ( G `  n )
)  <  1  ->  ( 1st `  ( G `
 n ) )  <_  1 ) )
3936, 37, 38sylancl 644 . . . . . 6  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( ( 1st `  ( G `  n ) )  <  1  ->  ( 1st `  ( G `  n
) )  <_  1
) )
4033, 39mpd 15 . . . . 5  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 1st `  ( G `  n
) )  <_  1
)
4120, 40eqbrtrd 4224 . . . 4  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( ( 1st  o.  G ) `  n )  <_  1
)
4241ralrimiva 2781 . . 3  |-  ( ph  ->  A. n  e.  NN0  ( ( 1st  o.  G ) `  n
)  <_  1 )
43 ffn 5583 . . . . 5  |-  ( ( 1st  o.  G ) : NN0 --> RR  ->  ( 1st  o.  G )  Fn  NN0 )
447, 43syl 16 . . . 4  |-  ( ph  ->  ( 1st  o.  G
)  Fn  NN0 )
45 breq1 4207 . . . . 5  |-  ( z  =  ( ( 1st 
o.  G ) `  n )  ->  (
z  <_  1  <->  ( ( 1st  o.  G ) `  n )  <_  1
) )
4645ralrn 5865 . . . 4  |-  ( ( 1st  o.  G )  Fn  NN0  ->  ( A. z  e.  ran  ( 1st 
o.  G ) z  <_  1  <->  A. n  e.  NN0  ( ( 1st 
o.  G ) `  n )  <_  1
) )
4744, 46syl 16 . . 3  |-  ( ph  ->  ( A. z  e. 
ran  ( 1st  o.  G ) z  <_ 
1  <->  A. n  e.  NN0  ( ( 1st  o.  G ) `  n
)  <_  1 ) )
4842, 47mpbird 224 . 2  |-  ( ph  ->  A. z  e.  ran  ( 1st  o.  G ) z  <_  1 )
499, 18, 483jca 1134 1  |-  ( ph  ->  ( ran  ( 1st 
o.  G )  C_  RR  /\  ran  ( 1st 
o.  G )  =/=  (/)  /\  A. z  e. 
ran  ( 1st  o.  G ) z  <_ 
1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   [_csb 3243    u. cun 3310    C_ wss 3312   (/)c0 3620   ifcif 3731   {csn 3806   <.cop 3809   class class class wbr 4204    X. cxp 4868   dom cdm 4870   ran crn 4871    o. ccom 4874    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    < clt 9112    <_ cle 9113    / cdiv 9669   NNcn 9992   2c2 10041   NN0cn0 10213    seq cseq 11315
This theorem is referenced by:  ruclem12  12832
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-seq 11316
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