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Theorem stowei 27212
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 27211: often times it will be better to use stoweid 27211 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 2420 . . 3  |-  F/_ t F
2 nftru 1542 . . 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 956 . . . 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 27211 . 2  |-  (  T. 
->  E. f  e.  A  A. t  e.  T  ( abs `  ( ( f `  t )  -  ( F `  t ) ) )  <  E )
2524trud 1316 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 936    T. wtru 1309    = wceq 1624    e. wcel 1685    =/= wne 2447   A.wral 2544   E.wrex 2545    C_ wss 3153   U.cuni 3828   class class class wbr 4024    e. cmpt 4078   ran crn 4689   ` cfv 5221  (class class class)co 5819   RRcr 8731    + caddc 8735    x. cmul 8737    < clt 8862    - cmin 9032   RR+crp 10349   (,)cioo 10650   abscabs 11713   topGenctg 13336    Cn ccn 16948   Compccmp 17107
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810  ax-addf 8811  ax-mulf 8812
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6655  df-map 6769  df-pm 6770  df-ixp 6813  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-fi 7160  df-sup 7189  df-oi 7220  df-card 7567  df-cda 7789  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-7 9804  df-8 9805  df-9 9806  df-10 9807  df-n0 9961  df-z 10020  df-dec 10120  df-uz 10226  df-q 10312  df-rp 10350  df-xneg 10447  df-xadd 10448  df-xmul 10449  df-ioo 10654  df-ioc 10655  df-ico 10656  df-icc 10657  df-fz 10777  df-fzo 10865  df-fl 10919  df-seq 11041  df-exp 11099  df-hash 11332  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-clim 11956  df-rlim 11957  df-sum 12153  df-struct 13144  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-mulr 13216  df-starv 13217  df-sca 13218  df-vsca 13219  df-tset 13221  df-ple 13222  df-ds 13224  df-hom 13226  df-cco 13227  df-rest 13321  df-topn 13322  df-topgen 13338  df-pt 13339  df-prds 13342  df-xrs 13397  df-0g 13398  df-gsum 13399  df-qtop 13404  df-imas 13405  df-xps 13407  df-mre 13482  df-mrc 13483  df-acs 13485  df-mnd 14361  df-submnd 14410  df-mulg 14486  df-cntz 14787  df-cmn 15085  df-xmet 16367  df-met 16368  df-bl 16369  df-mopn 16370  df-cnfld 16372  df-top 16630  df-bases 16632  df-topon 16633  df-topsp 16634  df-cld 16750  df-cn 16951  df-cnp 16952  df-cmp 17108  df-tx 17251  df-hmeo 17440  df-xms 17879  df-ms 17880  df-tms 17881
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