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Theorem stoweidlem61 27479
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 2388 . . 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 2388 . . 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 27478 . 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 1836 . . . 4  |-  F/ t ( ph  /\  g  e.  A )
2217ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  g  e.  A )  /\  t  e.  T )  ->  E  e.  RR+ )
233, 4, 5, 15fcnre 27365 . . . . . . 7  |-  ( ph  ->  F : T --> RR )
2423fnvinran 27354 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
2524adantlr 696 . . . . 5  |-  ( ( ( ph  /\  g  e.  A )  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
2610sselda 3292 . . . . . . 7  |-  ( (
ph  /\  g  e.  A )  ->  g  e.  C )
273, 4, 5, 26fcnre 27365 . . . . . 6  |-  ( (
ph  /\  g  e.  A )  ->  g : T --> RR )
2827fnvinran 27354 . . . . 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 27431 . . . . . . 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 2774 . . . . 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 2727 . . 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 2762 . 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 1717   F/_wnfc 2511    =/= wne 2551   A.wral 2650   E.wrex 2651   {crab 2654    C_ wss 3264   (/)c0 3572   U.cuni 3958   class class class wbr 4154    e. cmpt 4208   ran crn 4820   ` cfv 5395  (class class class)co 6021   RRcr 8923   0cc0 8924   1c1 8925    + caddc 8927    x. cmul 8929    < clt 9054    <_ cle 9055    - cmin 9224    / cdiv 9610   2c2 9982   3c3 9983   4c4 9984   RR+crp 10545   (,)cioo 10849   ...cfz 10976   abscabs 11967   topGenctg 13593    Cn ccn 17211   Compccmp 17372
This theorem is referenced by:  stoweidlem62  27480
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-mulf 9004
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-oi 7413  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-ioc 10854  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-fl 11130  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-clim 12210  df-rlim 12211  df-sum 12408  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-starv 13472  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-hom 13481  df-cco 13482  df-rest 13578  df-topn 13579  df-topgen 13595  df-pt 13596  df-prds 13599  df-xrs 13654  df-0g 13655  df-gsum 13656  df-qtop 13661  df-imas 13662  df-xps 13664  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-submnd 14667  df-mulg 14743  df-cntz 15044  df-cmn 15342  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cld 17007  df-cn 17214  df-cnp 17215  df-cmp 17373  df-tx 17516  df-hmeo 17709  df-xms 18260  df-ms 18261  df-tms 18262
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