Theorem stoweid 40280

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 477	. . . 4 |- ( ( ph /\ T = (/) ) -> T = (/) )
2		stoweid.10	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
3	2	ralrimiva 2966	. . . . . 6 |- ( ph -> A. x e. RR ( t e. T |-> x ) e. A )
4		1re 10039	. . . . . 6 |- 1 e. RR
5		id 22	. . . . . . . . 9 |- ( x = 1 -> x = 1 )
6	5	mpteq2dv 4745	. . . . . . . 8 |- ( x = 1 -> ( t e. T |-> x ) = ( t e. T |-> 1 ) )
7	6	eleq1d 2686	. . . . . . 7 |- ( x = 1 -> ( ( t e. T |-> x ) e. A <-> ( t e. T |-> 1 ) e. A ) )
8	7	rspccv 3306	. . . . . 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 21	. . . . 5 |- ( ph -> ( t e. T |-> 1 ) e. A )
10	9	adantr 481	. . . 4 |- ( ( ph /\ T = (/) ) -> ( t e. T |-> 1 ) e. A )
11	1, 10	stoweidlem9 40226	. . 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 1843	. . . . 5 |- F/ f ph
14		nfv 1843	. . . . 5 |- F/ f -. T = (/)
15	13, 14	nfan 1828	. . . 4 |- F/ f ( ph /\ -. T = (/) )
16		stoweid.2	. . . . 5 |- F/ t ph
17		nfv 1843	. . . . 5 |- F/ t -. T = (/)
18	16, 17	nfan 1828	. . . 4 |- F/ t ( ph /\ -. T = (/) )
19		eqid 2622	. . . 4 |- ( t e. T |-> ( ( F ` t ) - inf ( ran F , RR , < ) ) ) = ( t e. T |-> ( ( F ` t ) - inf ( 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 481	. . . 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 481	. . . 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 1319	. . . 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 1319	. . . 4 |- ( ( ( ph /\ -. T = (/) ) /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
31	2	adantlr 751	. . . 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 751	. . . 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 481	. . . 4 |- ( ( ph /\ -. T = (/) ) -> F e. C )
36		stoweid.13	. . . . . 6 |- ( ph -> E e. RR+ )
37		4re 11097	. . . . . . . . 9 |- 4 e. RR
38		4pos 11116	. . . . . . . . 9 |- 0 < 4
39	37, 38	elrpii 11835	. . . . . . . 8 |- 4 e. RR+
40	39	a1i 11	. . . . . . 7 |- ( ph -> 4 e. RR+ )
41	40	rpreccld 11882	. . . . . 6 |- ( ph -> ( 1 / 4 ) e. RR+ )
42	36, 41	ifcld 4131	. . . . 5 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR+ )
43	42	adantr 481	. . . 4 |- ( ( ph /\ -. T = (/) ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR+ )
44		neqne 2802	. . . . 5 |- ( -. T = (/) -> T =/= (/) )
45	44	adantl 482	. . . 4 |- ( ( ph /\ -. T = (/) ) -> T =/= (/) )
46	36	rpred 11872	. . . . . . 7 |- ( ph -> E e. RR )
47		4ne0 11117	. . . . . . . . 9 |- 4 =/= 0
48	37, 47	rereccli 10790	. . . . . . . 8 |- ( 1 / 4 ) e. RR
49	48	a1i 11	. . . . . . 7 |- ( ph -> ( 1 / 4 ) e. RR )
50	46, 49	ifcld 4131	. . . . . 6 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR )
51		3re 11094	. . . . . . . 8 |- 3 e. RR
52		3ne0 11115	. . . . . . . 8 |- 3 =/= 0
53	51, 52	rereccli 10790	. . . . . . 7 |- ( 1 / 3 ) e. RR
54	53	a1i 11	. . . . . 6 |- ( ph -> ( 1 / 3 ) e. RR )
55	36	rpxrd 11873	. . . . . . 7 |- ( ph -> E e. RR* )
56	41	rpxrd 11873	. . . . . . 7 |- ( ph -> ( 1 / 4 ) e. RR* )
57		xrmin2 12009	. . . . . . 7 |- ( ( E e. RR* /\ ( 1 / 4 ) e. RR* ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ ( 1 / 4 ) )
58	55, 56, 57	syl2anc 693	. . . . . 6 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ ( 1 / 4 ) )
59		3lt4 11197	. . . . . . . 8 |- 3 < 4
60		3pos 11114	. . . . . . . . 9 |- 0 < 3
61	51, 37, 60, 38	ltrecii 10940	. . . . . . . 8 |- ( 3 < 4 <-> ( 1 / 4 ) < ( 1 / 3 ) )
62	59, 61	mpbi 220	. . . . . . 7 |- ( 1 / 4 ) < ( 1 / 3 )
63	62	a1i 11	. . . . . 6 |- ( ph -> ( 1 / 4 ) < ( 1 / 3 ) )
64	50, 49, 54, 58, 63	lelttrd 10195	. . . . 5 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) < ( 1 / 3 ) )
65	64	adantr 481	. . . 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 40279	. . 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 832	. 2 |- ( ph -> E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) )
68		nfv 1843	. . . . 5 |- F/ t f e. A
69	16, 68	nfan 1828	. . . 4 |- F/ t ( ph /\ f e. A )
70		xrmin1 12008	. . . . . . 7 |- ( ( E e. RR* /\ ( 1 / 4 ) e. RR* ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ E )
71	55, 56, 70	syl2anc 693	. . . . . 6 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ E )
72	71	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ E )
73	25	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> A C_ C )
74		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> f e. A )
75	73, 74	sseldd 3604	. . . . . . . . . . 11 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> f e. C )
76	20, 21, 24, 75	fcnre 39184	. . . . . . . . . 10 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> f : T --> RR )
77		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> t e. T )
78	76, 77	jca 554	. . . . . . . . 9 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( f : T --> RR /\ t e. T ) )
79		ffvelrn 6357	. . . . . . . . 9 |- ( ( f : T --> RR /\ t e. T ) -> ( f ` t ) e. RR )
80		recn 10026	. . . . . . . . 9 |- ( ( f ` t ) e. RR -> ( f ` t ) e. CC )
81	78, 79, 80	3syl 18	. . . . . . . 8 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( f ` t ) e. CC )
82	34	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> F e. C )
83	20, 21, 24, 82	fcnre 39184	. . . . . . . . . 10 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> F : T --> RR )
84	83, 77	jca 554	. . . . . . . . 9 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( F : T --> RR /\ t e. T ) )
85		ffvelrn 6357	. . . . . . . . 9 |- ( ( F : T --> RR /\ t e. T ) -> ( F ` t ) e. RR )
86		recn 10026	. . . . . . . . 9 |- ( ( F ` t ) e. RR -> ( F ` t ) e. CC )
87	84, 85, 86	3syl 18	. . . . . . . 8 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( F ` t ) e. CC )
88	81, 87	subcld 10392	. . . . . . 7 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( ( f ` t ) - ( F ` t ) ) e. CC )
89	88	abscld 14175	. . . . . 6 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( abs ` ( ( f ` t ) - ( F ` t ) ) ) e. RR )
90	4, 37, 47	3pm3.2i 1239	. . . . . . . . 9 |- ( 1 e. RR /\ 4 e. RR /\ 4 =/= 0 )
91		redivcl 10744	. . . . . . . . 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 4131	. . . . . . 7 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR )
94	93	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR )
95	46	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> E e. RR )
96		ltletr 10129	. . . . . 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 1326	. . . . 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 710	. . . 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 2958	. . 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 3017	. 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 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   ifcif 4086   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650  infcinf 8347   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   3c3 11071   4c4 11072   RR+crp 11832   (,)cioo 12175   abscabs 13974   topGenctg 16098   Cn ccn 21028   Compccmp 21189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-cn 21031  df-cnp 21032  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127

This theorem is referenced by:  stowei  40281