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Theorem stowei 27915
Description: This theorem proves the Stone-Weierstrass theorem for real valued functions: let  J be a compact topology on  T, and  C be the set of real continuous functions on  T. Assume that  A is a subalgebra of  C (closed under addition and multiplication of functions) containing constant functions and discriminating points (if  r and  t are distinct points in  T, then there exists a function  h in  A such that h(r) is distinct from h(t) ). Then, for any continuous function 
F and for any positive real  E, there exists a function  f in the subalgebra  A, such that  f approximates  F up to  E ( E represents the usual ε value). As a classical example, given any a,b reals, the closed interval  T  =  [
a ,  b ] could be taken, along with the subalgebra  A of real polynomials on  T, and then use this theorem to easily prove that real polynomials are dense in the standard metric space of continuous functions on  [ a ,  b ]. The proof and lemmas are written following [BrosowskiDeutsh] p. 89 (through page 92). Some effort is put in avoiding the use of the axiom of choice. The deduction version of this theorem is stoweid 27914: often times it will be better to use stoweid 27914 in other proofs (but this version is probably easier to be read and understood). (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stowei.1  |-  K  =  ( topGen `  ran  (,) )
stowei.2  |-  J  e. 
Comp
stowei.3  |-  T  = 
U. J
stowei.4  |-  C  =  ( J  Cn  K
)
stowei.5  |-  A  C_  C
stowei.6  |-  ( ( f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A
)
stowei.7  |-  ( ( f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  x.  (
g `  t )
) )  e.  A
)
stowei.8  |-  ( x  e.  RR  ->  (
t  e.  T  |->  x )  e.  A )
stowei.9  |-  ( ( r  e.  T  /\  t  e.  T  /\  r  =/=  t )  ->  E. h  e.  A  ( h `  r
)  =/=  ( h `
 t ) )
stowei.10  |-  F  e.  C
stowei.11  |-  E  e.  RR+
Assertion
Ref Expression
stowei  |-  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t )
) )  <  E
Distinct variable groups:    f, g,
t, A    f, h, r, x, t, A    f, E, g, t    f, F, g, t    f, J, r, t    T, f, g, t    h, E, r, x    h, F, r, x    T, h, r, x    t, K
Allowed substitution hints:    C( x, t, f, g, h, r)    J( x, g, h)    K( x, f, g, h, r)

Proof of Theorem stowei
StepHypRef Expression
1 nfcv 2432 . . 3  |-  F/_ t F
2 nftru 1544 . . 3  |-  F/ t  T.
3 stowei.1 . . 3  |-  K  =  ( topGen `  ran  (,) )
4 stowei.2 . . . 4  |-  J  e. 
Comp
54a1i 10 . . 3  |-  (  T. 
->  J  e.  Comp )
6 stowei.3 . . 3  |-  T  = 
U. J
7 stowei.4 . . 3  |-  C  =  ( J  Cn  K
)
8 stowei.5 . . . 4  |-  A  C_  C
98a1i 10 . . 3  |-  (  T. 
->  A  C_  C )
10 3simpc 954 . . . 4  |-  ( (  T.  /\  f  e.  A  /\  g  e.  A )  ->  (
f  e.  A  /\  g  e.  A )
)
11 stowei.6 . . . 4  |-  ( ( f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A
)
1210, 11syl 15 . . 3  |-  ( (  T.  /\  f  e.  A  /\  g  e.  A )  ->  (
t  e.  T  |->  ( ( f `  t
)  +  ( g `
 t ) ) )  e.  A )
13 stowei.7 . . . 4  |-  ( ( f  e.  A  /\  g  e.  A )  ->  ( t  e.  T  |->  ( ( f `  t )  x.  (
g `  t )
) )  e.  A
)
1410, 13syl 15 . . 3  |-  ( (  T.  /\  f  e.  A  /\  g  e.  A )  ->  (
t  e.  T  |->  ( ( f `  t
)  x.  ( g `
 t ) ) )  e.  A )
15 stowei.8 . . . 4  |-  ( x  e.  RR  ->  (
t  e.  T  |->  x )  e.  A )
1615adantl 452 . . 3  |-  ( (  T.  /\  x  e.  RR )  ->  (
t  e.  T  |->  x )  e.  A )
17 simpr 447 . . . 4  |-  ( (  T.  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  -> 
( r  e.  T  /\  t  e.  T  /\  r  =/=  t
) )
18 stowei.9 . . . 4  |-  ( ( r  e.  T  /\  t  e.  T  /\  r  =/=  t )  ->  E. h  e.  A  ( h `  r
)  =/=  ( h `
 t ) )
1917, 18syl 15 . . 3  |-  ( (  T.  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  E. h  e.  A  ( h `  r
)  =/=  ( h `
 t ) )
20 stowei.10 . . . 4  |-  F  e.  C
2120a1i 10 . . 3  |-  (  T. 
->  F  e.  C
)
22 stowei.11 . . . 4  |-  E  e.  RR+
2322a1i 10 . . 3  |-  (  T. 
->  E  e.  RR+ )
241, 2, 3, 5, 6, 7, 9, 12, 14, 16, 19, 21, 23stoweid 27914 . 2  |-  (  T. 
->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E )
2524trud 1314 1  |-  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t )
) )  <  E
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    T. wtru 1307    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    C_ wss 3165   U.cuni 3843   class class class wbr 4039    e. cmpt 4093   ran crn 4706   ` cfv 5271  (class class class)co 5874   RRcr 8752    + caddc 8756    x. cmul 8758    < clt 8883    - cmin 9053   RR+crp 10370   (,)cioo 10672   abscabs 11735   topGenctg 13358    Cn ccn 16970   Compccmp 17129
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  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  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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-int 3879  df-iun 3923  df-iin 3924  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-se 4369  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-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  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-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-cn 16973  df-cnp 16974  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-xms 17901  df-ms 17902  df-tms 17903
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