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Theorem stoweidlem61 27685
Description: This lemma proves that there exists a function  g as in the proof in [BrosowskiDeutsh] p. 92:  g is in the subalgebra, and for all  t in  T, abs( f(t) - g(t) ) < 2*ε. Here  F is used to represent f in the paper, and  E is used to represent ε. For this lemma there's the further assumption that the function  F to be approximated is nonnegative (this assumption is removed in a later theorem). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem61.1  |-  F/_ t F
stoweidlem61.2  |-  F/ t
ph
stoweidlem61.3  |-  K  =  ( topGen `  ran  (,) )
stoweidlem61.4  |-  ( ph  ->  J  e.  Comp )
stoweidlem61.5  |-  T  = 
U. J
stoweidlem61.6  |-  ( ph  ->  T  =/=  (/) )
stoweidlem61.7  |-  C  =  ( J  Cn  K
)
stoweidlem61.8  |-  ( ph  ->  A  C_  C )
stoweidlem61.9  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
stoweidlem61.10  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
stoweidlem61.11  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
stoweidlem61.12  |-  ( (
ph  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  E. q  e.  A  ( q `  r
)  =/=  ( q `
 t ) )
stoweidlem61.13  |-  ( ph  ->  F  e.  C )
stoweidlem61.14  |-  ( ph  ->  A. t  e.  T 
0  <_  ( F `  t ) )
stoweidlem61.15  |-  ( ph  ->  E  e.  RR+ )
stoweidlem61.16  |-  ( ph  ->  E  <  ( 1  /  3 ) )
Assertion
Ref Expression
stoweidlem61  |-  ( ph  ->  E. g  e.  A  A. t  e.  T  ( abs `  ( ( g `  t )  -  ( F `  t ) ) )  <  ( 2  x.  E ) )
Distinct variable groups:    f, g,
q, r, t, x, A    f, E, g, q, r, t, x   
f, F, g, q, r, x    f, J, g, r, t    T, f, g, q, r, t, x    ph, f, g, q, r, x    t, K
Allowed substitution hints:    ph( t)    C( x, t, f, g, r, q)    F( t)    J( x, q)    K( x, f, g, r, q)

Proof of Theorem stoweidlem61
Dummy variables  j  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 stoweidlem61.1 . . 3  |-  F/_ t F
2 stoweidlem61.2 . . 3  |-  F/ t
ph
3 stoweidlem61.3 . . 3  |-  K  =  ( topGen `  ran  (,) )
4 stoweidlem61.5 . . 3  |-  T  = 
U. J
5 stoweidlem61.7 . . 3  |-  C  =  ( J  Cn  K
)
6 eqid 2412 . . 3  |-  ( j  e.  ( 0 ... n )  |->  { t  e.  T  |  ( F `  t )  <_  ( ( j  -  ( 1  / 
3 ) )  x.  E ) } )  =  ( j  e.  ( 0 ... n
)  |->  { t  e.  T  |  ( F `
 t )  <_ 
( ( j  -  ( 1  /  3
) )  x.  E
) } )
7 eqid 2412 . . 3  |-  ( j  e.  ( 0 ... n )  |->  { t  e.  T  |  ( ( j  +  ( 1  /  3 ) )  x.  E )  <_  ( F `  t ) } )  =  ( j  e.  ( 0 ... n
)  |->  { t  e.  T  |  ( ( j  +  ( 1  /  3 ) )  x.  E )  <_ 
( F `  t
) } )
8 stoweidlem61.4 . . 3  |-  ( ph  ->  J  e.  Comp )
9 stoweidlem61.6 . . 3  |-  ( ph  ->  T  =/=  (/) )
10 stoweidlem61.8 . . 3  |-  ( ph  ->  A  C_  C )
11 stoweidlem61.9 . . 3  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
12 stoweidlem61.10 . . 3  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
13 stoweidlem61.11 . . 3  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
14 stoweidlem61.12 . . 3  |-  ( (
ph  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  E. q  e.  A  ( q `  r
)  =/=  ( q `
 t ) )
15 stoweidlem61.13 . . 3  |-  ( ph  ->  F  e.  C )
16 stoweidlem61.14 . . 3  |-  ( ph  ->  A. t  e.  T 
0  <_  ( F `  t ) )
17 stoweidlem61.15 . . 3  |-  ( ph  ->  E  e.  RR+ )
18 stoweidlem61.16 . . 3  |-  ( ph  ->  E  <  ( 1  /  3 ) )
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18stoweidlem60 27684 . 2  |-  ( ph  ->  E. g  e.  A  A. t  e.  T  E. j  e.  RR  ( ( ( ( j  -  ( 4  /  3 ) )  x.  E )  < 
( F `  t
)  /\  ( F `  t )  <_  (
( j  -  (
1  /  3 ) )  x.  E ) )  /\  ( ( g `  t )  <  ( ( j  +  ( 1  / 
3 ) )  x.  E )  /\  (
( j  -  (
4  /  3 ) )  x.  E )  <  ( g `  t ) ) ) )
20 nfv 1626 . . . . 5  |-  F/ t  g  e.  A
212, 20nfan 1842 . . . 4  |-  F/ t ( ph  /\  g  e.  A )
2217ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  g  e.  A )  /\  t  e.  T )  ->  E  e.  RR+ )
233, 4, 5, 15fcnre 27571 . . . . . . 7  |-  ( ph  ->  F : T --> RR )
2423fnvinran 27560 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
2524adantlr 696 . . . . 5  |-  ( ( ( ph  /\  g  e.  A )  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
2610sselda 3316 . . . . . . 7  |-  ( (
ph  /\  g  e.  A )  ->  g  e.  C )
273, 4, 5, 26fcnre 27571 . . . . . 6  |-  ( (
ph  /\  g  e.  A )  ->  g : T --> RR )
2827fnvinran 27560 . . . . 5  |-  ( ( ( ph  /\  g  e.  A )  /\  t  e.  T )  ->  (
g `  t )  e.  RR )
29 simpll1 996 . . . . . . . 8  |-  ( ( ( ( E  e.  RR+  /\  ( F `  t )  e.  RR  /\  ( g `  t
)  e.  RR )  /\  j  e.  RR )  /\  ( ( ( ( j  -  (
4  /  3 ) )  x.  E )  <  ( F `  t )  /\  ( F `  t )  <_  ( ( j  -  ( 1  /  3
) )  x.  E
) )  /\  (
( g `  t
)  <  ( (
j  +  ( 1  /  3 ) )  x.  E )  /\  ( ( j  -  ( 4  /  3
) )  x.  E
)  <  ( g `  t ) ) ) )  ->  E  e.  RR+ )
30 simpll2 997 . . . . . . . 8  |-  ( ( ( ( E  e.  RR+  /\  ( F `  t )  e.  RR  /\  ( g `  t
)  e.  RR )  /\  j  e.  RR )  /\  ( ( ( ( j  -  (
4  /  3 ) )  x.  E )  <  ( F `  t )  /\  ( F `  t )  <_  ( ( j  -  ( 1  /  3
) )  x.  E
) )  /\  (
( g `  t
)  <  ( (
j  +  ( 1  /  3 ) )  x.  E )  /\  ( ( j  -  ( 4  /  3
) )  x.  E
)  <  ( g `  t ) ) ) )  ->  ( F `  t )  e.  RR )
31 simpll3 998 . . . . . . . 8  |-  ( ( ( ( E  e.  RR+  /\  ( F `  t )  e.  RR  /\  ( g `  t
)  e.  RR )  /\  j  e.  RR )  /\  ( ( ( ( j  -  (
4  /  3 ) )  x.  E )  <  ( F `  t )  /\  ( F `  t )  <_  ( ( j  -  ( 1  /  3
) )  x.  E
) )  /\  (
( g `  t
)  <  ( (
j  +  ( 1  /  3 ) )  x.  E )  /\  ( ( j  -  ( 4  /  3
) )  x.  E
)  <  ( g `  t ) ) ) )  ->  ( g `  t )  e.  RR )
32 simplr 732 . . . . . . . 8  |-  ( ( ( ( E  e.  RR+  /\  ( F `  t )  e.  RR  /\  ( g `  t
)  e.  RR )  /\  j  e.  RR )  /\  ( ( ( ( j  -  (
4  /  3 ) )  x.  E )  <  ( F `  t )  /\  ( F `  t )  <_  ( ( j  -  ( 1  /  3
) )  x.  E
) )  /\  (
( g `  t
)  <  ( (
j  +  ( 1  /  3 ) )  x.  E )  /\  ( ( j  -  ( 4  /  3
) )  x.  E
)  <  ( g `  t ) ) ) )  ->  j  e.  RR )
33 simprll 739 . . . . . . . 8  |-  ( ( ( ( E  e.  RR+  /\  ( F `  t )  e.  RR  /\  ( g `  t
)  e.  RR )  /\  j  e.  RR )  /\  ( ( ( ( j  -  (
4  /  3 ) )  x.  E )  <  ( F `  t )  /\  ( F `  t )  <_  ( ( j  -  ( 1  /  3
) )  x.  E
) )  /\  (
( g `  t
)  <  ( (
j  +  ( 1  /  3 ) )  x.  E )  /\  ( ( j  -  ( 4  /  3
) )  x.  E
)  <  ( g `  t ) ) ) )  ->  ( (
j  -  ( 4  /  3 ) )  x.  E )  < 
( F `  t
) )
34 simprlr 740 . . . . . . . 8  |-  ( ( ( ( E  e.  RR+  /\  ( F `  t )  e.  RR  /\  ( g `  t
)  e.  RR )  /\  j  e.  RR )  /\  ( ( ( ( j  -  (
4  /  3 ) )  x.  E )  <  ( F `  t )  /\  ( F `  t )  <_  ( ( j  -  ( 1  /  3
) )  x.  E
) )  /\  (
( g `  t
)  <  ( (
j  +  ( 1  /  3 ) )  x.  E )  /\  ( ( j  -  ( 4  /  3
) )  x.  E
)  <  ( g `  t ) ) ) )  ->  ( F `  t )  <_  (
( j  -  (
1  /  3 ) )  x.  E ) )
35 simprrr 742 . . . . . . . 8  |-  ( ( ( ( E  e.  RR+  /\  ( F `  t )  e.  RR  /\  ( g `  t
)  e.  RR )  /\  j  e.  RR )  /\  ( ( ( ( j  -  (
4  /  3 ) )  x.  E )  <  ( F `  t )  /\  ( F `  t )  <_  ( ( j  -  ( 1  /  3
) )  x.  E
) )  /\  (
( g `  t
)  <  ( (
j  +  ( 1  /  3 ) )  x.  E )  /\  ( ( j  -  ( 4  /  3
) )  x.  E
)  <  ( g `  t ) ) ) )  ->  ( (
j  -  ( 4  /  3 ) )  x.  E )  < 
( g `  t
) )
36 simprrl 741 . . . . . . . 8  |-  ( ( ( ( E  e.  RR+  /\  ( F `  t )  e.  RR  /\  ( g `  t
)  e.  RR )  /\  j  e.  RR )  /\  ( ( ( ( j  -  (
4  /  3 ) )  x.  E )  <  ( F `  t )  /\  ( F `  t )  <_  ( ( j  -  ( 1  /  3
) )  x.  E
) )  /\  (
( g `  t
)  <  ( (
j  +  ( 1  /  3 ) )  x.  E )  /\  ( ( j  -  ( 4  /  3
) )  x.  E
)  <  ( g `  t ) ) ) )  ->  ( g `  t )  <  (
( j  +  ( 1  /  3 ) )  x.  E ) )
3729, 30, 31, 32, 33, 34, 35, 36stoweidlem13 27637 . . . . . . 7  |-  ( ( ( ( E  e.  RR+  /\  ( F `  t )  e.  RR  /\  ( g `  t
)  e.  RR )  /\  j  e.  RR )  /\  ( ( ( ( j  -  (
4  /  3 ) )  x.  E )  <  ( F `  t )  /\  ( F `  t )  <_  ( ( j  -  ( 1  /  3
) )  x.  E
) )  /\  (
( g `  t
)  <  ( (
j  +  ( 1  /  3 ) )  x.  E )  /\  ( ( j  -  ( 4  /  3
) )  x.  E
)  <  ( g `  t ) ) ) )  ->  ( abs `  ( ( g `  t )  -  ( F `  t )
) )  <  (
2  x.  E ) )
3837ex 424 . . . . . 6  |-  ( ( ( E  e.  RR+  /\  ( F `  t
)  e.  RR  /\  ( g `  t
)  e.  RR )  /\  j  e.  RR )  ->  ( ( ( ( ( j  -  ( 4  /  3
) )  x.  E
)  <  ( F `  t )  /\  ( F `  t )  <_  ( ( j  -  ( 1  /  3
) )  x.  E
) )  /\  (
( g `  t
)  <  ( (
j  +  ( 1  /  3 ) )  x.  E )  /\  ( ( j  -  ( 4  /  3
) )  x.  E
)  <  ( g `  t ) ) )  ->  ( abs `  (
( g `  t
)  -  ( F `
 t ) ) )  <  ( 2  x.  E ) ) )
3938rexlimdva 2798 . . . . 5  |-  ( ( E  e.  RR+  /\  ( F `  t )  e.  RR  /\  ( g `
 t )  e.  RR )  ->  ( E. j  e.  RR  ( ( ( ( j  -  ( 4  /  3 ) )  x.  E )  < 
( F `  t
)  /\  ( F `  t )  <_  (
( j  -  (
1  /  3 ) )  x.  E ) )  /\  ( ( g `  t )  <  ( ( j  +  ( 1  / 
3 ) )  x.  E )  /\  (
( j  -  (
4  /  3 ) )  x.  E )  <  ( g `  t ) ) )  ->  ( abs `  (
( g `  t
)  -  ( F `
 t ) ) )  <  ( 2  x.  E ) ) )
4022, 25, 28, 39syl3anc 1184 . . . 4  |-  ( ( ( ph  /\  g  e.  A )  /\  t  e.  T )  ->  ( E. j  e.  RR  ( ( ( ( j  -  ( 4  /  3 ) )  x.  E )  < 
( F `  t
)  /\  ( F `  t )  <_  (
( j  -  (
1  /  3 ) )  x.  E ) )  /\  ( ( g `  t )  <  ( ( j  +  ( 1  / 
3 ) )  x.  E )  /\  (
( j  -  (
4  /  3 ) )  x.  E )  <  ( g `  t ) ) )  ->  ( abs `  (
( g `  t
)  -  ( F `
 t ) ) )  <  ( 2  x.  E ) ) )
4121, 40ralimdaa 2751 . . 3  |-  ( (
ph  /\  g  e.  A )  ->  ( A. t  e.  T  E. j  e.  RR  ( ( ( ( j  -  ( 4  /  3 ) )  x.  E )  < 
( F `  t
)  /\  ( F `  t )  <_  (
( j  -  (
1  /  3 ) )  x.  E ) )  /\  ( ( g `  t )  <  ( ( j  +  ( 1  / 
3 ) )  x.  E )  /\  (
( j  -  (
4  /  3 ) )  x.  E )  <  ( g `  t ) ) )  ->  A. t  e.  T  ( abs `  ( ( g `  t )  -  ( F `  t ) ) )  <  ( 2  x.  E ) ) )
4241reximdva 2786 . 2  |-  ( ph  ->  ( E. g  e.  A  A. t  e.  T  E. j  e.  RR  ( ( ( ( j  -  (
4  /  3 ) )  x.  E )  <  ( F `  t )  /\  ( F `  t )  <_  ( ( j  -  ( 1  /  3
) )  x.  E
) )  /\  (
( g `  t
)  <  ( (
j  +  ( 1  /  3 ) )  x.  E )  /\  ( ( j  -  ( 4  /  3
) )  x.  E
)  <  ( g `  t ) ) )  ->  E. g  e.  A  A. t  e.  T  ( abs `  ( ( g `  t )  -  ( F `  t ) ) )  <  ( 2  x.  E ) ) )
4319, 42mpd 15 1  |-  ( ph  ->  E. g  e.  A  A. t  e.  T  ( abs `  ( ( g `  t )  -  ( F `  t ) ) )  <  ( 2  x.  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   F/wnf 1550    = wceq 1649    e. wcel 1721   F/_wnfc 2535    =/= wne 2575   A.wral 2674   E.wrex 2675   {crab 2678    C_ wss 3288   (/)c0 3596   U.cuni 3983   class class class wbr 4180    e. cmpt 4234   ran crn 4846   ` cfv 5421  (class class class)co 6048   RRcr 8953   0cc0 8954   1c1 8955    + caddc 8957    x. cmul 8959    < clt 9084    <_ cle 9085    - cmin 9255    / cdiv 9641   2c2 10013   3c3 10014   4c4 10015   RR+crp 10576   (,)cioo 10880   ...cfz 11007   abscabs 12002   topGenctg 13628    Cn ccn 17250   Compccmp 17411
This theorem is referenced by:  stoweidlem62  27686
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-mulf 9034
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-sup 7412  df-oi 7443  df-card 7790  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-ioo 10884  df-ioc 10885  df-ico 10886  df-icc 10887  df-fz 11008  df-fzo 11099  df-fl 11165  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-rlim 12246  df-sum 12443  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-hom 13516  df-cco 13517  df-rest 13613  df-topn 13614  df-topgen 13630  df-pt 13631  df-prds 13634  df-xrs 13689  df-0g 13690  df-gsum 13691  df-qtop 13696  df-imas 13697  df-xps 13699  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-submnd 14702  df-mulg 14778  df-cntz 15079  df-cmn 15377  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-cnfld 16667  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-cld 17046  df-cn 17253  df-cnp 17254  df-cmp 17412  df-tx 17555  df-hmeo 17748  df-xms 18311  df-ms 18312  df-tms 18313
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