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Theorem ruclem11 13534
Description: Lemma for ruc 13537. 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 13529 . . . 4  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
6 1stcof 6616 . . . 4  |-  ( G : NN0 --> ( RR 
X.  RR )  -> 
( 1st  o.  G
) : NN0 --> RR )
75, 6syl 16 . . 3  |-  ( ph  ->  ( 1st  o.  G
) : NN0 --> RR )
8 frn 5577 . . 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 5575 . . . . 5  |-  ( ( 1st  o.  G ) : NN0 --> RR  ->  dom  ( 1st  o.  G
)  =  NN0 )
117, 10syl 16 . . . 4  |-  ( ph  ->  dom  ( 1st  o.  G )  =  NN0 )
12 0nn0 10606 . . . . 5  |-  0  e.  NN0
13 ne0i 3655 . . . . 5  |-  ( 0  e.  NN0  ->  NN0  =/=  (/) )
1412, 13mp1i 12 . . . 4  |-  ( ph  ->  NN0  =/=  (/) )
1511, 14eqnetrd 2638 . . 3  |-  ( ph  ->  dom  ( 1st  o.  G )  =/=  (/) )
16 dm0rn0 5068 . . . 4  |-  ( dom  ( 1st  o.  G
)  =  (/)  <->  ran  ( 1st 
o.  G )  =  (/) )
1716necon3bii 2652 . . 3  |-  ( dom  ( 1st  o.  G
)  =/=  (/)  <->  ran  ( 1st 
o.  G )  =/=  (/) )
1815, 17sylib 196 . 2  |-  ( ph  ->  ran  ( 1st  o.  G )  =/=  (/) )
19 fvco3 5780 . . . . . 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 13533 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G ` 
0 ) ) )
261, 2, 3, 4ruclem4 13528 . . . . . . . . . 10  |-  ( ph  ->  ( G `  0
)  =  <. 0 ,  1 >. )
2726fveq2d 5707 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  ( 2nd `  <. 0 ,  1
>. ) )
2812elexi 2994 . . . . . . . . . 10  |-  0  e.  _V
29 1ex 9393 . . . . . . . . . 10  |-  1  e.  _V
3028, 29op2nd 6598 . . . . . . . . 9  |-  ( 2nd `  <. 0 ,  1
>. )  =  1
3127, 30syl6eq 2491 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  1 )
3231adantr 465 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 2nd `  ( G `  0
) )  =  1 )
3325, 32breqtrd 4328 . . . . . 6  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 1st `  ( G `  n
) )  <  1
)
345ffvelrnda 5855 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( G `  n )  e.  ( RR  X.  RR ) )
35 xp1st 6618 . . . . . . . 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 9397 . . . . . . 7  |-  1  e.  RR
38 ltle 9475 . . . . . . 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 4324 . . . 4  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( ( 1st  o.  G ) `  n )  <_  1
)
4241ralrimiva 2811 . . 3  |-  ( ph  ->  A. n  e.  NN0  ( ( 1st  o.  G ) `  n
)  <_  1 )
43 ffn 5571 . . . . 5  |-  ( ( 1st  o.  G ) : NN0 --> RR  ->  ( 1st  o.  G )  Fn  NN0 )
447, 43syl 16 . . . 4  |-  ( ph  ->  ( 1st  o.  G
)  Fn  NN0 )
45 breq1 4307 . . . . 5  |-  ( z  =  ( ( 1st 
o.  G ) `  n )  ->  (
z  <_  1  <->  ( ( 1st  o.  G ) `  n )  <_  1
) )
4645ralrn 5858 . . . 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 1168 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 965    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727   [_csb 3300    u. cun 3338    C_ wss 3340   (/)c0 3649   ifcif 3803   {csn 3889   <.cop 3895   class class class wbr 4304    X. cxp 4850   dom cdm 4852   ran crn 4853    o. ccom 4856    Fn wfn 5425   -->wf 5426   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105   1stc1st 6587   2ndc2nd 6588   RRcr 9293   0cc0 9294   1c1 9295    + caddc 9297    < clt 9430    <_ cle 9431    / cdiv 10005   NNcn 10334   2c2 10383   NN0cn0 10591    seqcseq 11818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-n0 10592  df-z 10659  df-uz 10874  df-fz 11450  df-seq 11819
This theorem is referenced by:  ruclem12  13535
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