Theorem stoweid 29829

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 461	. . . 4 |- ( ( ph /\ T = (/) ) -> T = (/) )
2		stoweid.10	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
3	2	ralrimiva 2794	. . . . . 6 |- ( ph -> A. x e. RR ( t e. T |-> x ) e. A )
4		1re 9377	. . . . . 6 |- 1 e. RR
5		id 22	. . . . . . . . 9 |- ( x = 1 -> x = 1 )
6	5	mpteq2dv 4374	. . . . . . . 8 |- ( x = 1 -> ( t e. T |-> x ) = ( t e. T |-> 1 ) )
7	6	eleq1d 2504	. . . . . . 7 |- ( x = 1 -> ( ( t e. T |-> x ) e. A <-> ( t e. T |-> 1 ) e. A ) )
8	7	rspccv 3065	. . . . . 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 18	. . . . 5 |- ( ph -> ( t e. T |-> 1 ) e. A )
10	9	adantr 465	. . . 4 |- ( ( ph /\ T = (/) ) -> ( t e. T |-> 1 ) e. A )
11	1, 10	stoweidlem9 29775	. . 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 1673	. . . . 5 |- F/ f ph
14		nfv 1673	. . . . 5 |- F/ f -. T = (/)
15	13, 14	nfan 1861	. . . 4 |- F/ f ( ph /\ -. T = (/) )
16		stoweid.2	. . . . 5 |- F/ t ph
17		nfv 1673	. . . . 5 |- F/ t -. T = (/)
18	16, 17	nfan 1861	. . . 4 |- F/ t ( ph /\ -. T = (/) )
19		eqid 2438	. . . 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 465	. . . 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 465	. . . 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 1211	. . . 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 1211	. . . 4 |- ( ( ( ph /\ -. T = (/) ) /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
31	2	adantlr 714	. . . 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 714	. . . 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 465	. . . 4 |- ( ( ph /\ -. T = (/) ) -> F e. C )
36		stoweid.13	. . . . . 6 |- ( ph -> E e. RR+ )
37		4re 10390	. . . . . . . . 9 |- 4 e. RR
38		4pos 10409	. . . . . . . . 9 |- 0 < 4
39	37, 38	elrpii 10986	. . . . . . . 8 |- 4 e. RR+
40	39	a1i 11	. . . . . . 7 |- ( ph -> 4 e. RR+ )
41	40	rpreccld 11029	. . . . . 6 |- ( ph -> ( 1 / 4 ) e. RR+ )
42	36, 41	ifcld 3827	. . . . 5 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR+ )
43	42	adantr 465	. . . 4 |- ( ( ph /\ -. T = (/) ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR+ )
44		df-ne 2603	. . . . . 6 |- ( T =/= (/) <-> -. T = (/) )
45	44	biimpri 206	. . . . 5 |- ( -. T = (/) -> T =/= (/) )
46	45	adantl 466	. . . 4 |- ( ( ph /\ -. T = (/) ) -> T =/= (/) )
47	36	rpred 11019	. . . . . . 7 |- ( ph -> E e. RR )
48		4ne0 10410	. . . . . . . . 9 |- 4 =/= 0
49	37, 48	rereccli 10088	. . . . . . . 8 |- ( 1 / 4 ) e. RR
50	49	a1i 11	. . . . . . 7 |- ( ph -> ( 1 / 4 ) e. RR )
51	47, 50	ifcld 3827	. . . . . 6 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR )
52		3re 10387	. . . . . . . 8 |- 3 e. RR
53		3ne0 10408	. . . . . . . 8 |- 3 =/= 0
54	52, 53	rereccli 10088	. . . . . . 7 |- ( 1 / 3 ) e. RR
55	54	a1i 11	. . . . . 6 |- ( ph -> ( 1 / 3 ) e. RR )
56	36	rpxrd 11020	. . . . . . 7 |- ( ph -> E e. RR* )
57	41	rpxrd 11020	. . . . . . 7 |- ( ph -> ( 1 / 4 ) e. RR* )
58		xrmin2 11142	. . . . . . 7 |- ( ( E e. RR* /\ ( 1 / 4 ) e. RR* ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ ( 1 / 4 ) )
59	56, 57, 58	syl2anc 661	. . . . . 6 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ ( 1 / 4 ) )
60		3lt4 10483	. . . . . . . 8 |- 3 < 4
61		3pos 10407	. . . . . . . . 9 |- 0 < 3
62	52, 37, 61, 38	ltrecii 10241	. . . . . . . 8 |- ( 3 < 4 <-> ( 1 / 4 ) < ( 1 / 3 ) )
63	60, 62	mpbi 208	. . . . . . 7 |- ( 1 / 4 ) < ( 1 / 3 )
64	63	a1i 11	. . . . . 6 |- ( ph -> ( 1 / 4 ) < ( 1 / 3 ) )
65	51, 50, 55, 59, 64	lelttrd 9521	. . . . 5 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) < ( 1 / 3 ) )
66	65	adantr 465	. . . 4 |- ( ( ph /\ -. T = (/) ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) < ( 1 / 3 ) )
67	12, 15, 18, 19, 20, 21, 23, 24, 26, 28, 30, 31, 33, 35, 43, 46, 66	stoweidlem62 29828	. . 3 |- ( ( ph /\ -. T = (/) ) -> E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) )
68	11, 67	pm2.61dan 789	. 2 |- ( ph -> E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) )
69		nfv 1673	. . . . 5 |- F/ t f e. A
70	16, 69	nfan 1861	. . . 4 |- F/ t ( ph /\ f e. A )
71		xrmin1 11141	. . . . . . 7 |- ( ( E e. RR* /\ ( 1 / 4 ) e. RR* ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ E )
72	56, 57, 71	syl2anc 661	. . . . . 6 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ E )
73	72	ad2antrr 725	. . . . 5 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ E )
74	25	ad2antrr 725	. . . . . . . . . . . 12 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> A C_ C )
75		simplr 754	. . . . . . . . . . . 12 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> f e. A )
76	74, 75	sseldd 3352	. . . . . . . . . . 11 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> f e. C )
77	20, 21, 24, 76	fcnre 29718	. . . . . . . . . 10 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> f : T --> RR )
78		simpr 461	. . . . . . . . . 10 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> t e. T )
79	77, 78	jca 532	. . . . . . . . 9 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( f : T --> RR /\ t e. T ) )
80		ffvelrn 5836	. . . . . . . . 9 |- ( ( f : T --> RR /\ t e. T ) -> ( f ` t ) e. RR )
81		recn 9364	. . . . . . . . 9 |- ( ( f ` t ) e. RR -> ( f ` t ) e. CC )
82	79, 80, 81	3syl 20	. . . . . . . 8 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( f ` t ) e. CC )
83	34	ad2antrr 725	. . . . . . . . . . 11 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> F e. C )
84	20, 21, 24, 83	fcnre 29718	. . . . . . . . . 10 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> F : T --> RR )
85	84, 78	jca 532	. . . . . . . . 9 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( F : T --> RR /\ t e. T ) )
86		ffvelrn 5836	. . . . . . . . 9 |- ( ( F : T --> RR /\ t e. T ) -> ( F ` t ) e. RR )
87		recn 9364	. . . . . . . . 9 |- ( ( F ` t ) e. RR -> ( F ` t ) e. CC )
88	85, 86, 87	3syl 20	. . . . . . . 8 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( F ` t ) e. CC )
89	82, 88	subcld 9711	. . . . . . 7 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( ( f ` t ) - ( F ` t ) ) e. CC )
90	89	abscld 12914	. . . . . 6 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( abs ` ( ( f ` t ) - ( F ` t ) ) ) e. RR )
91	4, 37, 48	3pm3.2i 1166	. . . . . . . . 9 |- ( 1 e. RR /\ 4 e. RR /\ 4 =/= 0 )
92		redivcl 10042	. . . . . . . . 9 |- ( ( 1 e. RR /\ 4 e. RR /\ 4 =/= 0 ) -> ( 1 / 4 ) e. RR )
93	91, 92	mp1i 12	. . . . . . . 8 |- ( ph -> ( 1 / 4 ) e. RR )
94	47, 93	ifcld 3827	. . . . . . 7 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR )
95	94	ad2antrr 725	. . . . . 6 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR )
96	47	ad2antrr 725	. . . . . 6 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> E e. RR )
97		ltletr 9458	. . . . . 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 ) )
98	90, 95, 96, 97	syl3anc 1218	. . . . 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 ) )
99	73, 98	mpan2d 674	. . . 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 ) )
100	70, 99	ralimdaa 2788	. . 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 ) )
101	100	reximdva 2823	. 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 ) )
102	68, 101	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 369   /\ w3a 965   = wceq 1369   F/wnf 1589   e. wcel 1756   F/_wnfc 2561   =/= wne 2601   A.wral 2710   E.wrex 2711   C_ wss 3323   (/)c0 3632   ifcif 3786   U.cuni 4086   class class class wbr 4287   e. cmpt 4345   `'ccnv 4834   ran crn 4836   -->wf 5409   ` cfv 5413  (class class class)co 6086   supcsup 7682   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275   + caddc 9277   x. cmul 9279   RR*cxr 9409   < clt 9410   <_ cle 9411   - cmin 9587   / cdiv 9985   3c3 10364   4c4 10365   RR+crp 10983   (,)cioo 11292   abscabs 12715   topGenctg 14368   Cn ccn 18808   Compccmp 18969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-cnfld 17799  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cld 18603  df-cn 18811  df-cnp 18812  df-cmp 18970  df-tx 19115  df-hmeo 19308  df-xms 19875  df-ms 19876  df-tms 19877

This theorem is referenced by:  stowei  29830