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Theorem ruclem11 12534
Description: Lemma for ruc 12537. 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 12529 . . . 4  |-  ( ph  ->  G : NN0 --> ( RR 
X.  RR ) )
6 1stcof 6163 . . . 4  |-  ( G : NN0 --> ( RR 
X.  RR )  -> 
( 1st  o.  G
) : NN0 --> RR )
75, 6syl 15 . . 3  |-  ( ph  ->  ( 1st  o.  G
) : NN0 --> RR )
8 frn 5411 . . 3  |-  ( ( 1st  o.  G ) : NN0 --> RR  ->  ran  ( 1st  o.  G
)  C_  RR )
97, 8syl 15 . 2  |-  ( ph  ->  ran  ( 1st  o.  G )  C_  RR )
10 fdm 5409 . . . . 5  |-  ( ( 1st  o.  G ) : NN0 --> RR  ->  dom  ( 1st  o.  G
)  =  NN0 )
117, 10syl 15 . . . 4  |-  ( ph  ->  dom  ( 1st  o.  G )  =  NN0 )
12 0nn0 9996 . . . . 5  |-  0  e.  NN0
13 ne0i 3474 . . . . 5  |-  ( 0  e.  NN0  ->  NN0  =/=  (/) )
1412, 13mp1i 11 . . . 4  |-  ( ph  ->  NN0  =/=  (/) )
1511, 14eqnetrd 2477 . . 3  |-  ( ph  ->  dom  ( 1st  o.  G )  =/=  (/) )
16 dm0rn0 4911 . . . 4  |-  ( dom  ( 1st  o.  G
)  =  (/)  <->  ran  ( 1st 
o.  G )  =  (/) )
1716necon3bii 2491 . . 3  |-  ( dom  ( 1st  o.  G
)  =/=  (/)  <->  ran  ( 1st 
o.  G )  =/=  (/) )
1815, 17sylib 188 . 2  |-  ( ph  ->  ran  ( 1st  o.  G )  =/=  (/) )
19 fvco3 5612 . . . . . 6  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  n  e.  NN0 )  -> 
( ( 1st  o.  G ) `  n
)  =  ( 1st `  ( G `  n
) ) )
205, 19sylan 457 . . . . 5  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( ( 1st  o.  G ) `  n )  =  ( 1st `  ( G `
 n ) ) )
211adantr 451 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  F : NN
--> RR )
222adantr 451 . . . . . . . 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 447 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  n  e.  NN0 )
2412a1i 10 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  0  e.  NN0 )
2521, 22, 3, 4, 23, 24ruclem10 12533 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 1st `  ( G `  n
) )  <  ( 2nd `  ( G ` 
0 ) ) )
261, 2, 3, 4ruclem4 12528 . . . . . . . . . 10  |-  ( ph  ->  ( G `  0
)  =  <. 0 ,  1 >. )
2726fveq2d 5545 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  ( 2nd `  <. 0 ,  1
>. ) )
2812elexi 2810 . . . . . . . . . 10  |-  0  e.  _V
29 1re 8853 . . . . . . . . . . 11  |-  1  e.  RR
3029elexi 2810 . . . . . . . . . 10  |-  1  e.  _V
3128, 30op2nd 6145 . . . . . . . . 9  |-  ( 2nd `  <. 0 ,  1
>. )  =  1
3227, 31syl6eq 2344 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  ( G `  0 )
)  =  1 )
3332adantr 451 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 2nd `  ( G `  0
) )  =  1 )
3425, 33breqtrd 4063 . . . . . 6  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 1st `  ( G `  n
) )  <  1
)
35 ffvelrn 5679 . . . . . . . . 9  |-  ( ( G : NN0 --> ( RR 
X.  RR )  /\  n  e.  NN0 )  -> 
( G `  n
)  e.  ( RR 
X.  RR ) )
365, 35sylan 457 . . . . . . . 8  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( G `  n )  e.  ( RR  X.  RR ) )
37 xp1st 6165 . . . . . . . 8  |-  ( ( G `  n )  e.  ( RR  X.  RR )  ->  ( 1st `  ( G `  n
) )  e.  RR )
3836, 37syl 15 . . . . . . 7  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 1st `  ( G `  n
) )  e.  RR )
39 ltle 8926 . . . . . . 7  |-  ( ( ( 1st `  ( G `  n )
)  e.  RR  /\  1  e.  RR )  ->  ( ( 1st `  ( G `  n )
)  <  1  ->  ( 1st `  ( G `
 n ) )  <_  1 ) )
4038, 29, 39sylancl 643 . . . . . 6  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( ( 1st `  ( G `  n ) )  <  1  ->  ( 1st `  ( G `  n
) )  <_  1
) )
4134, 40mpd 14 . . . . 5  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( 1st `  ( G `  n
) )  <_  1
)
4220, 41eqbrtrd 4059 . . . 4  |-  ( (
ph  /\  n  e.  NN0 )  ->  ( ( 1st  o.  G ) `  n )  <_  1
)
4342ralrimiva 2639 . . 3  |-  ( ph  ->  A. n  e.  NN0  ( ( 1st  o.  G ) `  n
)  <_  1 )
44 ffn 5405 . . . . 5  |-  ( ( 1st  o.  G ) : NN0 --> RR  ->  ( 1st  o.  G )  Fn  NN0 )
457, 44syl 15 . . . 4  |-  ( ph  ->  ( 1st  o.  G
)  Fn  NN0 )
46 breq1 4042 . . . . 5  |-  ( z  =  ( ( 1st 
o.  G ) `  n )  ->  (
z  <_  1  <->  ( ( 1st  o.  G ) `  n )  <_  1
) )
4746ralrn 5684 . . . 4  |-  ( ( 1st  o.  G )  Fn  NN0  ->  ( A. z  e.  ran  ( 1st 
o.  G ) z  <_  1  <->  A. n  e.  NN0  ( ( 1st 
o.  G ) `  n )  <_  1
) )
4845, 47syl 15 . . 3  |-  ( ph  ->  ( A. z  e. 
ran  ( 1st  o.  G ) z  <_ 
1  <->  A. n  e.  NN0  ( ( 1st  o.  G ) `  n
)  <_  1 ) )
4943, 48mpbird 223 . 2  |-  ( ph  ->  A. z  e.  ran  ( 1st  o.  G ) z  <_  1 )
509, 18, 493jca 1132 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   [_csb 3094    u. cun 3163    C_ wss 3165   (/)c0 3468   ifcif 3578   {csn 3653   <.cop 3656   class class class wbr 4039    X. cxp 4703   dom cdm 4705   ran crn 4706    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    <_ cle 8884    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981    seq cseq 11062
This theorem is referenced by:  ruclem12  12535
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063
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