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Theorem ruclem11 13850
Description: Lemma for ruc 13853. 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 13845 . . . 4  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
6 1stcof 6823 . . . 4  |-  ( G : NN0 --> ( RR 
X.  RR )  -> 
( 1st  o.  G
) : NN0 --> RR )
75, 6syl 16 . . 3  |-  ( ph  ->  ( 1st  o.  G
) : NN0 --> RR )
8 frn 5743 . . 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 5741 . . . . 5  |-  ( ( 1st  o.  G ) : NN0 --> RR  ->  dom  ( 1st  o.  G
)  =  NN0 )
117, 10syl 16 . . . 4  |-  ( ph  ->  dom  ( 1st  o.  G )  =  NN0 )
12 0nn0 10822 . . . . 5  |-  0  e.  NN0
13 ne0i 3796 . . . . 5  |-  ( 0  e.  NN0  ->  NN0  =/=  (/) )
1412, 13mp1i 12 . . . 4  |-  ( ph  ->  NN0  =/=  (/) )
1511, 14eqnetrd 2760 . . 3  |-  ( ph  ->  dom  ( 1st  o.  G )  =/=  (/) )
16 dm0rn0 5225 . . . 4  |-  ( dom  ( 1st  o.  G
)  =  (/)  <->  ran  ( 1st 
o.  G )  =  (/) )
1716necon3bii 2735 . . 3  |-  ( dom  ( 1st  o.  G
)  =/=  (/)  <->  ran  ( 1st 
o.  G )  =/=  (/) )
1815, 17sylib 196 . 2  |-  ( ph  ->  ran  ( 1st  o.  G )  =/=  (/) )
19 fvco3 5951 . . . . . 6  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  n  e.  NN0 )  -> 
( ( 1st  o.  G ) `  n
)  =  ( 1st `  ( G `  n
) ) )
205, 19sylan 471 . . . . 5  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( ( 1st  o.  G ) `  n )  =  ( 1st `  ( G `
 n ) ) )
211adantr 465 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  F : NN
--> RR )
222adantr 465 . . . . . . . 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 461 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  n  e.  NN0 )
2412a1i 11 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  0  e.  NN0 )
2521, 22, 3, 4, 23, 24ruclem10 13849 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G ` 
0 ) ) )
261, 2, 3, 4ruclem4 13844 . . . . . . . . . 10  |-  ( ph  ->  ( G `  0
)  =  <. 0 ,  1 >. )
2726fveq2d 5876 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  ( 2nd `  <. 0 ,  1
>. ) )
28 c0ex 9602 . . . . . . . . . 10  |-  0  e.  _V
29 1ex 9603 . . . . . . . . . 10  |-  1  e.  _V
3028, 29op2nd 6804 . . . . . . . . 9  |-  ( 2nd `  <. 0 ,  1
>. )  =  1
3127, 30syl6eq 2524 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  1 )
3231adantr 465 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 2nd `  ( G `  0
) )  =  1 )
3325, 32breqtrd 4477 . . . . . 6  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 1st `  ( G `  n
) )  <  1
)
345ffvelrnda 6032 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( G `  n )  e.  ( RR  X.  RR ) )
35 xp1st 6825 . . . . . . . 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 9607 . . . . . . 7  |-  1  e.  RR
38 ltle 9685 . . . . . . 7  |-  ( ( ( 1st `  ( G `  n )
)  e.  RR  /\  1  e.  RR )  ->  ( ( 1st `  ( G `  n )
)  <  1  ->  ( 1st `  ( G `
 n ) )  <_  1 ) )
3936, 37, 38sylancl 662 . . . . . 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 4473 . . . 4  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( ( 1st  o.  G ) `  n )  <_  1
)
4241ralrimiva 2881 . . 3  |-  ( ph  ->  A. n  e.  NN0  ( ( 1st  o.  G ) `  n
)  <_  1 )
43 ffn 5737 . . . . 5  |-  ( ( 1st  o.  G ) : NN0 --> RR  ->  ( 1st  o.  G )  Fn  NN0 )
447, 43syl 16 . . . 4  |-  ( ph  ->  ( 1st  o.  G
)  Fn  NN0 )
45 breq1 4456 . . . . 5  |-  ( z  =  ( ( 1st 
o.  G ) `  n )  ->  (
z  <_  1  <->  ( ( 1st  o.  G ) `  n )  <_  1
) )
4645ralrn 6035 . . . 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 232 . 2  |-  ( ph  ->  A. z  e.  ran  ( 1st  o.  G ) z  <_  1 )
499, 18, 483jca 1176 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   [_csb 3440    u. cun 3479    C_ wss 3481   (/)c0 3790   ifcif 3945   {csn 4033   <.cop 4039   class class class wbr 4453    X. cxp 5003   dom cdm 5005   ran crn 5006    o. ccom 5009    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   1stc1st 6793   2ndc2nd 6794   RRcr 9503   0cc0 9504   1c1 9505    + caddc 9507    < clt 9640    <_ cle 9641    / cdiv 10218   NNcn 10548   2c2 10597   NN0cn0 10807    seqcseq 12087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-seq 12088
This theorem is referenced by:  ruclem12  13851
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