Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stowei Structured version   Visualization version   GIF version

Theorem stowei 38957
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 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 38956: often times it will be better to use stoweid 38956 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 ∈ Comp
stowei.3 𝑇 = 𝐽
stowei.4 𝐶 = (𝐽 Cn 𝐾)
stowei.5 𝐴𝐶
stowei.6 ((𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
stowei.7 ((𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
stowei.8 (𝑥 ∈ ℝ → (𝑡𝑇𝑥) ∈ 𝐴)
stowei.9 ((𝑟𝑇𝑡𝑇𝑟𝑡) → ∃𝐴 (𝑟) ≠ (𝑡))
stowei.10 𝐹𝐶
stowei.11 𝐸 ∈ ℝ+
Assertion
Ref Expression
stowei 𝑓𝐴𝑡𝑇 (abs‘((𝑓𝑡) − (𝐹𝑡))) < 𝐸
Distinct variable groups:   𝑓,𝑔,𝑡,𝐴   𝑓,,𝑟,𝑥,𝑡,𝐴   𝑓,𝐸,𝑔,𝑡   𝑓,𝐹,𝑔,𝑡   𝑓,𝐽,𝑟,𝑡   𝑇,𝑓,𝑔,𝑡   ,𝐸,𝑟,𝑥   ,𝐹,𝑟,𝑥   𝑇,,𝑟,𝑥   𝑡,𝐾
Allowed substitution hints:   𝐶(𝑥,𝑡,𝑓,𝑔,,𝑟)   𝐽(𝑥,𝑔,)   𝐾(𝑥,𝑓,𝑔,,𝑟)

Proof of Theorem stowei
StepHypRef Expression
1 nfcv 2751 . . 3 𝑡𝐹
2 nftru 1721 . . 3 𝑡
3 stowei.1 . . 3 𝐾 = (topGen‘ran (,))
4 stowei.2 . . . 4 𝐽 ∈ Comp
54a1i 11 . . 3 (⊤ → 𝐽 ∈ Comp)
6 stowei.3 . . 3 𝑇 = 𝐽
7 stowei.4 . . 3 𝐶 = (𝐽 Cn 𝐾)
8 stowei.5 . . . 4 𝐴𝐶
98a1i 11 . . 3 (⊤ → 𝐴𝐶)
10 stowei.6 . . . 4 ((𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
11103adant1 1072 . . 3 ((⊤ ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
12 stowei.7 . . . 4 ((𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
13123adant1 1072 . . 3 ((⊤ ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
14 stowei.8 . . . 4 (𝑥 ∈ ℝ → (𝑡𝑇𝑥) ∈ 𝐴)
1514adantl 481 . . 3 ((⊤ ∧ 𝑥 ∈ ℝ) → (𝑡𝑇𝑥) ∈ 𝐴)
16 stowei.9 . . . 4 ((𝑟𝑇𝑡𝑇𝑟𝑡) → ∃𝐴 (𝑟) ≠ (𝑡))
1716adantl 481 . . 3 ((⊤ ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝐴 (𝑟) ≠ (𝑡))
18 stowei.10 . . . 4 𝐹𝐶
1918a1i 11 . . 3 (⊤ → 𝐹𝐶)
20 stowei.11 . . . 4 𝐸 ∈ ℝ+
2120a1i 11 . . 3 (⊤ → 𝐸 ∈ ℝ+)
221, 2, 3, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21stoweid 38956 . 2 (⊤ → ∃𝑓𝐴𝑡𝑇 (abs‘((𝑓𝑡) − (𝐹𝑡))) < 𝐸)
2322trud 1484 1 𝑓𝐴𝑡𝑇 (abs‘((𝑓𝑡) − (𝐹𝑡))) < 𝐸
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wtru 1476  wcel 1977  wne 2780  wral 2896  wrex 2897  wss 3540   cuni 4372   class class class wbr 4583  cmpt 4643  ran crn 5039  cfv 5804  (class class class)co 6549  cr 9814   + caddc 9818   · cmul 9820   < clt 9953  cmin 10145  +crp 11708  (,)cioo 12046  abscabs 13822  topGenctg 15921   Cn ccn 20838  Compccmp 20999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-cn 20841  df-cnp 20842  df-cmp 21000  df-tx 21175  df-hmeo 21368  df-xms 21935  df-ms 21936  df-tms 21937
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator