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Theorem stowei 27113
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 27112: often times it will be better to use stoweid 27112 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 2392 . . 3  |-  F/_ t F
2 nftru 1559 . . 3  |-  F/ t  T.
3 stowei.1 . . 3  |-  K  =  ( topGen `  ran  (,) )
4 stowei.2 . . . 4  |-  J  e. 
Comp
54a1i 12 . . 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 12 . . 3  |-  (  T. 
->  A  C_  C )
10 3simpc 959 . . . 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 17 . . 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 17 . . 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 454 . . 3  |-  ( (  T.  /\  x  e.  RR )  ->  (
t  e.  T  |->  x )  e.  A )
17 simpr 449 . . . 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 17 . . 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 12 . . 3  |-  (  T. 
->  F  e.  C
)
22 stowei.11 . . . 4  |-  E  e.  RR+
2322a1i 12 . . 3  |-  (  T. 
->  E  e.  RR+ )
241, 2, 3, 5, 6, 7, 9, 12, 14, 16, 19, 21, 23stoweid 27112 . 2  |-  (  T. 
->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E )
2524trud 1320 1  |-  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t )
) )  <  E
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    T. wtru 1312    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   E.wrex 2517    C_ wss 3094   U.cuni 3768   class class class wbr 3963    e. cmpt 4017   ran crn 4627   ` cfv 4638  (class class class)co 5757   RRcr 8669    + caddc 8673    x. cmul 8675    < clt 8800    - cmin 8970   RR+crp 10286   (,)cioo 10587   abscabs 11649   topGenctg 13269    Cn ccn 16881   Compccmp 17040
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748  ax-addf 8749  ax-mulf 8750
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-of 5977  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-omul 6417  df-er 6593  df-ec 6595  df-qs 6599  df-map 6707  df-pm 6708  df-ixp 6751  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-fi 7098  df-sup 7127  df-oi 7158  df-card 7505  df-cda 7727  df-ni 8429  df-pli 8430  df-mi 8431  df-lti 8432  df-plpq 8465  df-mpq 8466  df-ltpq 8467  df-enq 8468  df-nq 8469  df-erq 8470  df-plq 8471  df-mq 8472  df-1nq 8473  df-rq 8474  df-ltnq 8475  df-np 8538  df-1p 8539  df-plp 8540  df-mp 8541  df-ltp 8542  df-plpr 8612  df-mpr 8613  df-enr 8614  df-nr 8615  df-plr 8616  df-mr 8617  df-ltr 8618  df-0r 8619  df-1r 8620  df-m1r 8621  df-c 8676  df-0 8677  df-1 8678  df-r 8680  df-plus 8681  df-mul 8682  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-7 9742  df-8 9743  df-9 9744  df-10 9745  df-n0 9898  df-z 9957  df-dec 10057  df-uz 10163  df-q 10249  df-rp 10287  df-xneg 10384  df-xadd 10385  df-xmul 10386  df-ioo 10591  df-ioc 10592  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-fl 10856  df-seq 10978  df-exp 11036  df-hash 11269  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-clim 11892  df-rlim 11893  df-sum 12089  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-starv 13150  df-sca 13151  df-vsca 13152  df-tset 13154  df-ple 13155  df-ds 13157  df-hom 13159  df-cco 13160  df-rest 13254  df-topn 13255  df-topgen 13271  df-pt 13272  df-prds 13275  df-xrs 13330  df-0g 13331  df-gsum 13332  df-qtop 13337  df-imas 13338  df-xps 13340  df-mre 13415  df-mrc 13416  df-acs 13418  df-mnd 14294  df-submnd 14343  df-mulg 14419  df-cntz 14720  df-cmn 15018  df-xmet 16300  df-met 16301  df-bl 16302  df-mopn 16303  df-cnfld 16305  df-top 16563  df-bases 16565  df-topon 16566  df-topsp 16567  df-cld 16683  df-cn 16884  df-cnp 16885  df-cmp 17041  df-tx 17184  df-hmeo 17373  df-xms 17812  df-ms 17813  df-tms 17814
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