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Mirrors > Home > MPE Home > Th. List > Mathboxes > stowei | Structured version Visualization version GIF version |
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 45594: often times it will be better to use stoweid 45594 in other proofs (but this version is probably easier to be read and understood). (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
stowei.1 | ⊢ 𝐾 = (topGen‘ran (,)) |
stowei.2 | ⊢ 𝐽 ∈ Comp |
stowei.3 | ⊢ 𝑇 = ∪ 𝐽 |
stowei.4 | ⊢ 𝐶 = (𝐽 Cn 𝐾) |
stowei.5 | ⊢ 𝐴 ⊆ 𝐶 |
stowei.6 | ⊢ ((𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) |
stowei.7 | ⊢ ((𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
stowei.8 | ⊢ (𝑥 ∈ ℝ → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) |
stowei.9 | ⊢ ((𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡) → ∃ℎ ∈ 𝐴 (ℎ‘𝑟) ≠ (ℎ‘𝑡)) |
stowei.10 | ⊢ 𝐹 ∈ 𝐶 |
stowei.11 | ⊢ 𝐸 ∈ ℝ+ |
Ref | Expression |
---|---|
stowei | ⊢ ∃𝑓 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2891 | . . 3 ⊢ Ⅎ𝑡𝐹 | |
2 | nftru 1798 | . . 3 ⊢ Ⅎ𝑡⊤ | |
3 | stowei.1 | . . 3 ⊢ 𝐾 = (topGen‘ran (,)) | |
4 | stowei.2 | . . . 4 ⊢ 𝐽 ∈ Comp | |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → 𝐽 ∈ Comp) |
6 | stowei.3 | . . 3 ⊢ 𝑇 = ∪ 𝐽 | |
7 | stowei.4 | . . 3 ⊢ 𝐶 = (𝐽 Cn 𝐾) | |
8 | stowei.5 | . . . 4 ⊢ 𝐴 ⊆ 𝐶 | |
9 | 8 | a1i 11 | . . 3 ⊢ (⊤ → 𝐴 ⊆ 𝐶) |
10 | stowei.6 | . . . 4 ⊢ ((𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) | |
11 | 10 | 3adant1 1127 | . . 3 ⊢ ((⊤ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) |
12 | stowei.7 | . . . 4 ⊢ ((𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) | |
13 | 12 | 3adant1 1127 | . . 3 ⊢ ((⊤ ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) |
14 | stowei.8 | . . . 4 ⊢ (𝑥 ∈ ℝ → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) | |
15 | 14 | adantl 480 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) |
16 | stowei.9 | . . . 4 ⊢ ((𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡) → ∃ℎ ∈ 𝐴 (ℎ‘𝑟) ≠ (ℎ‘𝑡)) | |
17 | 16 | adantl 480 | . . 3 ⊢ ((⊤ ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃ℎ ∈ 𝐴 (ℎ‘𝑟) ≠ (ℎ‘𝑡)) |
18 | stowei.10 | . . . 4 ⊢ 𝐹 ∈ 𝐶 | |
19 | 18 | a1i 11 | . . 3 ⊢ (⊤ → 𝐹 ∈ 𝐶) |
20 | stowei.11 | . . . 4 ⊢ 𝐸 ∈ ℝ+ | |
21 | 20 | a1i 11 | . . 3 ⊢ (⊤ → 𝐸 ∈ ℝ+) |
22 | 1, 2, 3, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21 | stoweid 45594 | . 2 ⊢ (⊤ → ∃𝑓 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸) |
23 | 22 | mptru 1540 | 1 ⊢ ∃𝑓 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 ≠ wne 2929 ∀wral 3050 ∃wrex 3059 ⊆ wss 3944 ∪ cuni 4909 class class class wbr 5149 ↦ cmpt 5232 ran crn 5679 ‘cfv 6549 (class class class)co 7419 ℝcr 11144 + caddc 11148 · cmul 11150 < clt 11285 − cmin 11481 ℝ+crp 13014 (,)cioo 13364 abscabs 15225 topGenctg 17438 Cn ccn 23189 Compccmp 23351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9671 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 ax-addf 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9393 df-fi 9441 df-sup 9472 df-inf 9473 df-oi 9540 df-card 9969 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13798 df-seq 14008 df-exp 14068 df-hash 14334 df-cj 15090 df-re 15091 df-im 15092 df-sqrt 15226 df-abs 15227 df-clim 15476 df-rlim 15477 df-sum 15677 df-struct 17135 df-sets 17152 df-slot 17170 df-ndx 17182 df-base 17200 df-ress 17229 df-plusg 17265 df-mulr 17266 df-starv 17267 df-sca 17268 df-vsca 17269 df-ip 17270 df-tset 17271 df-ple 17272 df-ds 17274 df-unif 17275 df-hom 17276 df-cco 17277 df-rest 17423 df-topn 17424 df-0g 17442 df-gsum 17443 df-topgen 17444 df-pt 17445 df-prds 17448 df-xrs 17503 df-qtop 17508 df-imas 17509 df-xps 17511 df-mre 17585 df-mrc 17586 df-acs 17588 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-submnd 18760 df-mulg 19048 df-cntz 19297 df-cmn 19766 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-cnfld 21314 df-top 22857 df-topon 22874 df-topsp 22896 df-bases 22910 df-cld 22984 df-cn 23192 df-cnp 23193 df-cmp 23352 df-tx 23527 df-hmeo 23720 df-xms 24287 df-ms 24288 df-tms 24289 |
This theorem is referenced by: (None) |
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