Theorem stoweid 37194

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. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweid.1	|- F/_ t F
stoweid.2	|- F/ t ph
stoweid.3	|- K = ( topGen ` ran (,) )
stoweid.4	|- ( ph -> J e. Comp )
stoweid.5	|- T = U. J
stoweid.6	|- C = ( J Cn K )
stoweid.7	|- ( ph -> A C_ C )
stoweid.8	|- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A )
stoweid.9	|- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
stoweid.10	|- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
stoweid.11	|- ( ( ph /\ ( r e. T /\ t e. T /\ r =/= t ) ) -> E. h e. A ( h ` r ) =/= ( h ` t ) )
stoweid.12	|- ( ph -> F e. C )
stoweid.13	|- ( ph -> E e. RR+ )

Assertion

Ref	Expression
stoweid	|- ( ph -> 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   f, J, r, t   T, f, g, t   ph, f, g   h, E, r, x   h, F, r, x   T, h, r, x   ph, h, r, x   t, K

Allowed substitution hints:   ph( t)   C( x, t, f, g, h, r)   F( t)   J( x, g, h)   K( x, f, g, h, r)

Proof of Theorem stoweid

Step	Hyp	Ref	Expression
1		simpr 459	. . . 4 |- ( ( ph /\ T = (/) ) -> T = (/) )
2		stoweid.10	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
3	2	ralrimiva 2817	. . . . . 6 |- ( ph -> A. x e. RR ( t e. T |-> x ) e. A )
4		1re 9624	. . . . . 6 |- 1 e. RR
5		id 22	. . . . . . . . 9 |- ( x = 1 -> x = 1 )
6	5	mpteq2dv 4481	. . . . . . . 8 |- ( x = 1 -> ( t e. T |-> x ) = ( t e. T |-> 1 ) )
7	6	eleq1d 2471	. . . . . . 7 |- ( x = 1 -> ( ( t e. T |-> x ) e. A <-> ( t e. T |-> 1 ) e. A ) )
8	7	rspccv 3156	. . . . . 6 |- ( A. x e. RR ( t e. T |-> x ) e. A -> ( 1 e. RR -> ( t e. T |-> 1 ) e. A ) )
9	3, 4, 8	mpisyl 19	. . . . 5 |- ( ph -> ( t e. T |-> 1 ) e. A )
10	9	adantr 463	. . . 4 |- ( ( ph /\ T = (/) ) -> ( t e. T |-> 1 ) e. A )
11	1, 10	stoweidlem9 37140	. . 3 |- ( ( ph /\ T = (/) ) -> E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) )
12		stoweid.1	. . . 4 |- F/_ t F
13		nfv 1728	. . . . 5 |- F/ f ph
14		nfv 1728	. . . . 5 |- F/ f -. T = (/)
15	13, 14	nfan 1956	. . . 4 |- F/ f ( ph /\ -. T = (/) )
16		stoweid.2	. . . . 5 |- F/ t ph
17		nfv 1728	. . . . 5 |- F/ t -. T = (/)
18	16, 17	nfan 1956	. . . 4 |- F/ t ( ph /\ -. T = (/) )
19		eqid 2402	. . . 4 |- ( t e. T |-> ( ( F ` t ) - sup ( ran F , RR , `' < ) ) ) = ( t e. T |-> ( ( F ` t ) - sup ( ran F , RR , `' < ) ) )
20		stoweid.3	. . . 4 |- K = ( topGen ` ran (,) )
21		stoweid.5	. . . 4 |- T = U. J
22		stoweid.4	. . . . 5 |- ( ph -> J e. Comp )
23	22	adantr 463	. . . 4 |- ( ( ph /\ -. T = (/) ) -> J e. Comp )
24		stoweid.6	. . . 4 |- C = ( J Cn K )
25		stoweid.7	. . . . 5 |- ( ph -> A C_ C )
26	25	adantr 463	. . . 4 |- ( ( ph /\ -. T = (/) ) -> A C_ C )
27		stoweid.8	. . . . 5 |- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A )
28	27	3adant1r 1223	. . . 4 |- ( ( ( ph /\ -. T = (/) ) /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A )
29		stoweid.9	. . . . 5 |- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
30	29	3adant1r 1223	. . . 4 |- ( ( ( ph /\ -. T = (/) ) /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
31	2	adantlr 713	. . . 4 |- ( ( ( ph /\ -. T = (/) ) /\ x e. RR ) -> ( t e. T |-> x ) e. A )
32		stoweid.11	. . . . 5 |- ( ( ph /\ ( r e. T /\ t e. T /\ r =/= t ) ) -> E. h e. A ( h ` r ) =/= ( h ` t ) )
33	32	adantlr 713	. . . 4 |- ( ( ( ph /\ -. T = (/) ) /\ ( r e. T /\ t e. T /\ r =/= t ) ) -> E. h e. A ( h ` r ) =/= ( h ` t ) )
34		stoweid.12	. . . . 5 |- ( ph -> F e. C )
35	34	adantr 463	. . . 4 |- ( ( ph /\ -. T = (/) ) -> F e. C )
36		stoweid.13	. . . . . 6 |- ( ph -> E e. RR+ )
37		4re 10652	. . . . . . . . 9 |- 4 e. RR
38		4pos 10671	. . . . . . . . 9 |- 0 < 4
39	37, 38	elrpii 11267	. . . . . . . 8 |- 4 e. RR+
40	39	a1i 11	. . . . . . 7 |- ( ph -> 4 e. RR+ )
41	40	rpreccld 11313	. . . . . 6 |- ( ph -> ( 1 / 4 ) e. RR+ )
42	36, 41	ifcld 3927	. . . . 5 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR+ )
43	42	adantr 463	. . . 4 |- ( ( ph /\ -. T = (/) ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR+ )
44		neqne 36790	. . . . 5 |- ( -. T = (/) -> T =/= (/) )
45	44	adantl 464	. . . 4 |- ( ( ph /\ -. T = (/) ) -> T =/= (/) )
46	36	rpred 11303	. . . . . . 7 |- ( ph -> E e. RR )
47		4ne0 10672	. . . . . . . . 9 |- 4 =/= 0
48	37, 47	rereccli 10349	. . . . . . . 8 |- ( 1 / 4 ) e. RR
49	48	a1i 11	. . . . . . 7 |- ( ph -> ( 1 / 4 ) e. RR )
50	46, 49	ifcld 3927	. . . . . 6 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR )
51		3re 10649	. . . . . . . 8 |- 3 e. RR
52		3ne0 10670	. . . . . . . 8 |- 3 =/= 0
53	51, 52	rereccli 10349	. . . . . . 7 |- ( 1 / 3 ) e. RR
54	53	a1i 11	. . . . . 6 |- ( ph -> ( 1 / 3 ) e. RR )
55	36	rpxrd 11304	. . . . . . 7 |- ( ph -> E e. RR* )
56	41	rpxrd 11304	. . . . . . 7 |- ( ph -> ( 1 / 4 ) e. RR* )
57		xrmin2 11431	. . . . . . 7 |- ( ( E e. RR* /\ ( 1 / 4 ) e. RR* ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ ( 1 / 4 ) )
58	55, 56, 57	syl2anc 659	. . . . . 6 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ ( 1 / 4 ) )
59		3lt4 10745	. . . . . . . 8 |- 3 < 4
60		3pos 10669	. . . . . . . . 9 |- 0 < 3
61	51, 37, 60, 38	ltrecii 10501	. . . . . . . 8 |- ( 3 < 4 <-> ( 1 / 4 ) < ( 1 / 3 ) )
62	59, 61	mpbi 208	. . . . . . 7 |- ( 1 / 4 ) < ( 1 / 3 )
63	62	a1i 11	. . . . . 6 |- ( ph -> ( 1 / 4 ) < ( 1 / 3 ) )
64	50, 49, 54, 58, 63	lelttrd 9773	. . . . 5 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) < ( 1 / 3 ) )
65	64	adantr 463	. . . 4 |- ( ( ph /\ -. T = (/) ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) < ( 1 / 3 ) )
66	12, 15, 18, 19, 20, 21, 23, 24, 26, 28, 30, 31, 33, 35, 43, 45, 65	stoweidlem62 37193	. . 3 |- ( ( ph /\ -. T = (/) ) -> E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) )
67	11, 66	pm2.61dan 792	. 2 |- ( ph -> E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) )
68		nfv 1728	. . . . 5 |- F/ t f e. A
69	16, 68	nfan 1956	. . . 4 |- F/ t ( ph /\ f e. A )
70		xrmin1 11430	. . . . . . 7 |- ( ( E e. RR* /\ ( 1 / 4 ) e. RR* ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ E )
71	55, 56, 70	syl2anc 659	. . . . . 6 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ E )
72	71	ad2antrr 724	. . . . 5 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ E )
73	25	ad2antrr 724	. . . . . . . . . . . 12 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> A C_ C )
74		simplr 754	. . . . . . . . . . . 12 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> f e. A )
75	73, 74	sseldd 3442	. . . . . . . . . . 11 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> f e. C )
76	20, 21, 24, 75	fcnre 36760	. . . . . . . . . 10 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> f : T --> RR )
77		simpr 459	. . . . . . . . . 10 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> t e. T )
78	76, 77	jca 530	. . . . . . . . 9 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( f : T --> RR /\ t e. T ) )
79		ffvelrn 6006	. . . . . . . . 9 |- ( ( f : T --> RR /\ t e. T ) -> ( f ` t ) e. RR )
80		recn 9611	. . . . . . . . 9 |- ( ( f ` t ) e. RR -> ( f ` t ) e. CC )
81	78, 79, 80	3syl 20	. . . . . . . 8 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( f ` t ) e. CC )
82	34	ad2antrr 724	. . . . . . . . . . 11 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> F e. C )
83	20, 21, 24, 82	fcnre 36760	. . . . . . . . . 10 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> F : T --> RR )
84	83, 77	jca 530	. . . . . . . . 9 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( F : T --> RR /\ t e. T ) )
85		ffvelrn 6006	. . . . . . . . 9 |- ( ( F : T --> RR /\ t e. T ) -> ( F ` t ) e. RR )
86		recn 9611	. . . . . . . . 9 |- ( ( F ` t ) e. RR -> ( F ` t ) e. CC )
87	84, 85, 86	3syl 20	. . . . . . . 8 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( F ` t ) e. CC )
88	81, 87	subcld 9966	. . . . . . 7 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( ( f ` t ) - ( F ` t ) ) e. CC )
89	88	abscld 13414	. . . . . 6 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( abs ` ( ( f ` t ) - ( F ` t ) ) ) e. RR )
90	4, 37, 47	3pm3.2i 1175	. . . . . . . . 9 |- ( 1 e. RR /\ 4 e. RR /\ 4 =/= 0 )
91		redivcl 10303	. . . . . . . . 9 |- ( ( 1 e. RR /\ 4 e. RR /\ 4 =/= 0 ) -> ( 1 / 4 ) e. RR )
92	90, 91	mp1i 13	. . . . . . . 8 |- ( ph -> ( 1 / 4 ) e. RR )
93	46, 92	ifcld 3927	. . . . . . 7 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR )
94	93	ad2antrr 724	. . . . . 6 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR )
95	46	ad2antrr 724	. . . . . 6 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> E e. RR )
96		ltletr 9706	. . . . . 6 |- ( ( ( abs ` ( ( f ` t ) - ( F ` t ) ) ) e. RR /\ if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR /\ E e. RR ) -> ( ( ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) /\ if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ E ) -> ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < E ) )
97	89, 94, 95, 96	syl3anc 1230	. . . . 5 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( ( ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) /\ if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ E ) -> ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < E ) )
98	72, 97	mpan2d 672	. . . 4 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) -> ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < E ) )
99	69, 98	ralimdaa 2805	. . 3 |- ( ( ph /\ f e. A ) -> ( A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) -> A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < E ) )
100	99	reximdva 2878	. 2 |- ( ph -> ( E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) -> E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < E ) )
101	67, 100	mpd 15	1 |- ( ph -> E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < E )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 367   /\ w3a 974   = wceq 1405   F/wnf 1637   e. wcel 1842   F/_wnfc 2550   =/= wne 2598   A.wral 2753   E.wrex 2754   C_ wss 3413   (/)c0 3737   ifcif 3884   U.cuni 4190   class class class wbr 4394   |-> cmpt 4452   `'ccnv 4821   ran crn 4823   -->wf 5564   ` cfv 5568  (class class class)co 6277   supcsup 7933   CCcc 9519   RRcr 9520   0cc0 9521   1c1 9522   + caddc 9524   x. cmul 9526   RR*cxr 9656   < clt 9657   <_ cle 9658   - cmin 9840   / cdiv 10246   3c3 10626   4c4 10627   RR+crp 11264   (,)cioo 11581   abscabs 13214   topGenctg 15050   Cn ccn 20016   Compccmp 20177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599  ax-addf 9600  ax-mulf 9601

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6520  df-om 6683  df-1st 6783  df-2nd 6784  df-supp 6902  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-ixp 7507  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fsupp 7863  df-fi 7904  df-sup 7934  df-oi 7968  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-q 11227  df-rp 11265  df-xneg 11370  df-xadd 11371  df-xmul 11372  df-ioo 11585  df-ioc 11586  df-ico 11587  df-icc 11588  df-fz 11725  df-fzo 11853  df-fl 11964  df-seq 12150  df-exp 12209  df-hash 12451  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-clim 13458  df-rlim 13459  df-sum 13656  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-starv 14922  df-sca 14923  df-vsca 14924  df-ip 14925  df-tset 14926  df-ple 14927  df-ds 14929  df-unif 14930  df-hom 14931  df-cco 14932  df-rest 15035  df-topn 15036  df-0g 15054  df-gsum 15055  df-topgen 15056  df-pt 15057  df-prds 15060  df-xrs 15114  df-qtop 15119  df-imas 15120  df-xps 15122  df-mre 15198  df-mrc 15199  df-acs 15201  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-submnd 16289  df-mulg 16382  df-cntz 16677  df-cmn 17122  df-psmet 18729  df-xmet 18730  df-met 18731  df-bl 18732  df-mopn 18733  df-cnfld 18739  df-top 19689  df-bases 19691  df-topon 19692  df-topsp 19693  df-cld 19810  df-cn 20019  df-cnp 20020  df-cmp 20178  df-tx 20353  df-hmeo 20546  df-xms 21113  df-ms 21114  df-tms 21115

This theorem is referenced by:  stowei  37195