MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ruclem11 Unicode version

Theorem ruclem11 12768
Description: Lemma for ruc 12771. 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 12763 . . . 4  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
6 1stcof 6315 . . . 4  |-  ( G : NN0 --> ( RR 
X.  RR )  -> 
( 1st  o.  G
) : NN0 --> RR )
75, 6syl 16 . . 3  |-  ( ph  ->  ( 1st  o.  G
) : NN0 --> RR )
8 frn 5539 . . 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 5537 . . . . 5  |-  ( ( 1st  o.  G ) : NN0 --> RR  ->  dom  ( 1st  o.  G
)  =  NN0 )
117, 10syl 16 . . . 4  |-  ( ph  ->  dom  ( 1st  o.  G )  =  NN0 )
12 0nn0 10170 . . . . 5  |-  0  e.  NN0
13 ne0i 3579 . . . . 5  |-  ( 0  e.  NN0  ->  NN0  =/=  (/) )
1412, 13mp1i 12 . . . 4  |-  ( ph  ->  NN0  =/=  (/) )
1511, 14eqnetrd 2570 . . 3  |-  ( ph  ->  dom  ( 1st  o.  G )  =/=  (/) )
16 dm0rn0 5028 . . . 4  |-  ( dom  ( 1st  o.  G
)  =  (/)  <->  ran  ( 1st 
o.  G )  =  (/) )
1716necon3bii 2584 . . 3  |-  ( dom  ( 1st  o.  G
)  =/=  (/)  <->  ran  ( 1st 
o.  G )  =/=  (/) )
1815, 17sylib 189 . 2  |-  ( ph  ->  ran  ( 1st  o.  G )  =/=  (/) )
19 fvco3 5741 . . . . . 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 12767 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G ` 
0 ) ) )
261, 2, 3, 4ruclem4 12762 . . . . . . . . . 10  |-  ( ph  ->  ( G `  0
)  =  <. 0 ,  1 >. )
2726fveq2d 5674 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  ( 2nd `  <. 0 ,  1
>. ) )
2812elexi 2910 . . . . . . . . . 10  |-  0  e.  _V
29 1ex 9021 . . . . . . . . . 10  |-  1  e.  _V
3028, 29op2nd 6297 . . . . . . . . 9  |-  ( 2nd `  <. 0 ,  1
>. )  =  1
3127, 30syl6eq 2437 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  1 )
3231adantr 452 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 2nd `  ( G `  0
) )  =  1 )
3325, 32breqtrd 4179 . . . . . 6  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 1st `  ( G `  n
) )  <  1
)
345ffvelrnda 5811 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( G `  n )  e.  ( RR  X.  RR ) )
35 xp1st 6317 . . . . . . . 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 9025 . . . . . . 7  |-  1  e.  RR
38 ltle 9098 . . . . . . 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 4175 . . . 4  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( ( 1st  o.  G ) `  n )  <_  1
)
4241ralrimiva 2734 . . 3  |-  ( ph  ->  A. n  e.  NN0  ( ( 1st  o.  G ) `  n
)  <_  1 )
43 ffn 5533 . . . . 5  |-  ( ( 1st  o.  G ) : NN0 --> RR  ->  ( 1st  o.  G )  Fn  NN0 )
447, 43syl 16 . . . 4  |-  ( ph  ->  ( 1st  o.  G
)  Fn  NN0 )
45 breq1 4158 . . . . 5  |-  ( z  =  ( ( 1st 
o.  G ) `  n )  ->  (
z  <_  1  <->  ( ( 1st  o.  G ) `  n )  <_  1
) )
4645ralrn 5814 . . . 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 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   [_csb 3196    u. cun 3263    C_ wss 3265   (/)c0 3573   ifcif 3684   {csn 3759   <.cop 3762   class class class wbr 4155    X. cxp 4818   dom cdm 4820   ran crn 4821    o. ccom 4824    Fn wfn 5391   -->wf 5392   ` cfv 5396  (class class class)co 6022    e. cmpt2 6024   1stc1st 6288   2ndc2nd 6289   RRcr 8924   0cc0 8925   1c1 8926    + caddc 8928    < clt 9055    <_ cle 9056    / cdiv 9611   NNcn 9934   2c2 9983   NN0cn0 10155    seq cseq 11252
This theorem is referenced by:  ruclem12  12769
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-seq 11253
  Copyright terms: Public domain W3C validator