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Theorem stoweidlem61 37862
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 2422 . . 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 2422 . . 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 37861 . 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 1755 . . . . 5  |-  F/ t  g  e.  A
212, 20nfan 1988 . . . 4  |-  F/ t ( ph  /\  g  e.  A )
2217ad2antrr 730 . . . . 5  |-  ( ( ( ph  /\  g  e.  A )  /\  t  e.  T )  ->  E  e.  RR+ )
233, 4, 5, 15fcnre 37319 . . . . . . 7  |-  ( ph  ->  F : T --> RR )
2423fnvinran 37308 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
2524adantlr 719 . . . . 5  |-  ( ( ( ph  /\  g  e.  A )  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
2610sselda 3464 . . . . . . 7  |-  ( (
ph  /\  g  e.  A )  ->  g  e.  C )
273, 4, 5, 26fcnre 37319 . . . . . 6  |-  ( (
ph  /\  g  e.  A )  ->  g : T --> RR )
2827fnvinran 37308 . . . . 5  |-  ( ( ( ph  /\  g  e.  A )  /\  t  e.  T )  ->  (
g `  t )  e.  RR )
29 simpll1 1044 . . . . . . . 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 1045 . . . . . . . 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 1046 . . . . . . . 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 760 . . . . . . . 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 770 . . . . . . . 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 771 . . . . . . . 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 773 . . . . . . . 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 772 . . . . . . . 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 37813 . . . . . . 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 435 . . . . . 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 2914 . . . . 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 1264 . . . 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 2824 . . 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 2897 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437   F/wnf 1661    e. wcel 1872   F/_wnfc 2566    =/= wne 2614   A.wral 2771   E.wrex 2772   {crab 2775    C_ wss 3436   (/)c0 3761   U.cuni 4219   class class class wbr 4423    |-> cmpt 4482   ran crn 4854   ` cfv 5601  (class class class)co 6305   RRcr 9545   0cc0 9546   1c1 9547    + caddc 9549    x. cmul 9551    < clt 9682    <_ cle 9683    - cmin 9867    / cdiv 10276   2c2 10666   3c3 10667   4c4 10668   RR+crp 11309   (,)cioo 11642   ...cfz 11791   abscabs 13297   topGenctg 15335    Cn ccn 20238   Compccmp 20399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-inf2 8155  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624  ax-mulf 9626
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-2o 7194  df-oadd 7197  df-er 7374  df-map 7485  df-pm 7486  df-ixp 7534  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-fsupp 7893  df-fi 7934  df-sup 7965  df-inf 7966  df-oi 8034  df-card 8381  df-cda 8605  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-ioo 11646  df-ioc 11647  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-fl 12034  df-seq 12220  df-exp 12279  df-hash 12522  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13551  df-rlim 13552  df-sum 13752  df-struct 15122  df-ndx 15123  df-slot 15124  df-base 15125  df-sets 15126  df-ress 15127  df-plusg 15202  df-mulr 15203  df-starv 15204  df-sca 15205  df-vsca 15206  df-ip 15207  df-tset 15208  df-ple 15209  df-ds 15211  df-unif 15212  df-hom 15213  df-cco 15214  df-rest 15320  df-topn 15321  df-0g 15339  df-gsum 15340  df-topgen 15341  df-pt 15342  df-prds 15345  df-xrs 15399  df-qtop 15405  df-imas 15406  df-xps 15409  df-mre 15491  df-mrc 15492  df-acs 15494  df-mgm 16487  df-sgrp 16526  df-mnd 16536  df-submnd 16582  df-mulg 16675  df-cntz 16970  df-cmn 17431  df-psmet 18961  df-xmet 18962  df-met 18963  df-bl 18964  df-mopn 18965  df-cnfld 18970  df-top 19919  df-bases 19920  df-topon 19921  df-topsp 19922  df-cld 20032  df-cn 20241  df-cnp 20242  df-cmp 20400  df-tx 20575  df-hmeo 20768  df-xms 21333  df-ms 21334  df-tms 21335
This theorem is referenced by:  stoweidlem62  37863  stoweidlem62OLD  37864
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