Theorem stoweid 37919

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 463	. . . 4 |- ( ( ph /\ T = (/) ) -> T = (/) )
2		stoweid.10	. . . . . . 7 |- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
3	2	ralrimiva 2801	. . . . . 6 |- ( ph -> A. x e. RR ( t e. T |-> x ) e. A )
4		1re 9639	. . . . . 6 |- 1 e. RR
5		id 22	. . . . . . . . 9 |- ( x = 1 -> x = 1 )
6	5	mpteq2dv 4489	. . . . . . . 8 |- ( x = 1 -> ( t e. T |-> x ) = ( t e. T |-> 1 ) )
7	6	eleq1d 2512	. . . . . . 7 |- ( x = 1 -> ( ( t e. T |-> x ) e. A <-> ( t e. T |-> 1 ) e. A ) )
8	7	rspccv 3146	. . . . . 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 467	. . . 4 |- ( ( ph /\ T = (/) ) -> ( t e. T |-> 1 ) e. A )
11	1, 10	stoweidlem9 37863	. . 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 1760	. . . . 5 |- F/ f ph
14		nfv 1760	. . . . 5 |- F/ f -. T = (/)
15	13, 14	nfan 2010	. . . 4 |- F/ f ( ph /\ -. T = (/) )
16		stoweid.2	. . . . 5 |- F/ t ph
17		nfv 1760	. . . . 5 |- F/ t -. T = (/)
18	16, 17	nfan 2010	. . . 4 |- F/ t ( ph /\ -. T = (/) )
19		eqid 2450	. . . 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 467	. . . 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 467	. . . 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 1260	. . . 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 1260	. . . 4 |- ( ( ( ph /\ -. T = (/) ) /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
31	2	adantlr 720	. . . 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 720	. . . 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 467	. . . 4 |- ( ( ph /\ -. T = (/) ) -> F e. C )
36		stoweid.13	. . . . . 6 |- ( ph -> E e. RR+ )
37		4re 10683	. . . . . . . . 9 |- 4 e. RR
38		4pos 10702	. . . . . . . . 9 |- 0 < 4
39	37, 38	elrpii 11302	. . . . . . . 8 |- 4 e. RR+
40	39	a1i 11	. . . . . . 7 |- ( ph -> 4 e. RR+ )
41	40	rpreccld 11348	. . . . . 6 |- ( ph -> ( 1 / 4 ) e. RR+ )
42	36, 41	ifcld 3923	. . . . 5 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR+ )
43	42	adantr 467	. . . 4 |- ( ( ph /\ -. T = (/) ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR+ )
44		neqne 37368	. . . . 5 |- ( -. T = (/) -> T =/= (/) )
45	44	adantl 468	. . . 4 |- ( ( ph /\ -. T = (/) ) -> T =/= (/) )
46	36	rpred 11338	. . . . . . 7 |- ( ph -> E e. RR )
47		4ne0 10703	. . . . . . . . 9 |- 4 =/= 0
48	37, 47	rereccli 10369	. . . . . . . 8 |- ( 1 / 4 ) e. RR
49	48	a1i 11	. . . . . . 7 |- ( ph -> ( 1 / 4 ) e. RR )
50	46, 49	ifcld 3923	. . . . . 6 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR )
51		3re 10680	. . . . . . . 8 |- 3 e. RR
52		3ne0 10701	. . . . . . . 8 |- 3 =/= 0
53	51, 52	rereccli 10369	. . . . . . 7 |- ( 1 / 3 ) e. RR
54	53	a1i 11	. . . . . 6 |- ( ph -> ( 1 / 3 ) e. RR )
55	36	rpxrd 11339	. . . . . . 7 |- ( ph -> E e. RR* )
56	41	rpxrd 11339	. . . . . . 7 |- ( ph -> ( 1 / 4 ) e. RR* )
57		xrmin2 11470	. . . . . . 7 |- ( ( E e. RR* /\ ( 1 / 4 ) e. RR* ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ ( 1 / 4 ) )
58	55, 56, 57	syl2anc 666	. . . . . 6 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ ( 1 / 4 ) )
59		3lt4 10776	. . . . . . . 8 |- 3 < 4
60		3pos 10700	. . . . . . . . 9 |- 0 < 3
61	51, 37, 60, 38	ltrecii 10520	. . . . . . . 8 |- ( 3 < 4 <-> ( 1 / 4 ) < ( 1 / 3 ) )
62	59, 61	mpbi 212	. . . . . . 7 |- ( 1 / 4 ) < ( 1 / 3 )
63	62	a1i 11	. . . . . 6 |- ( ph -> ( 1 / 4 ) < ( 1 / 3 ) )
64	50, 49, 54, 58, 63	lelttrd 9790	. . . . 5 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) < ( 1 / 3 ) )
65	64	adantr 467	. . . 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 37917	. . 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 799	. 2 |- ( ph -> E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) )
68		nfv 1760	. . . . 5 |- F/ t f e. A
69	16, 68	nfan 2010	. . . 4 |- F/ t ( ph /\ f e. A )
70		xrmin1 11469	. . . . . . 7 |- ( ( E e. RR* /\ ( 1 / 4 ) e. RR* ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ E )
71	55, 56, 70	syl2anc 666	. . . . . 6 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ E )
72	71	ad2antrr 731	. . . . 5 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) <_ E )
73	25	ad2antrr 731	. . . . . . . . . . . 12 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> A C_ C )
74		simplr 761	. . . . . . . . . . . 12 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> f e. A )
75	73, 74	sseldd 3432	. . . . . . . . . . 11 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> f e. C )
76	20, 21, 24, 75	fcnre 37340	. . . . . . . . . 10 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> f : T --> RR )
77		simpr 463	. . . . . . . . . 10 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> t e. T )
78	76, 77	jca 535	. . . . . . . . 9 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( f : T --> RR /\ t e. T ) )
79		ffvelrn 6018	. . . . . . . . 9 |- ( ( f : T --> RR /\ t e. T ) -> ( f ` t ) e. RR )
80		recn 9626	. . . . . . . . 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 731	. . . . . . . . . . 11 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> F e. C )
83	20, 21, 24, 82	fcnre 37340	. . . . . . . . . 10 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> F : T --> RR )
84	83, 77	jca 535	. . . . . . . . 9 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( F : T --> RR /\ t e. T ) )
85		ffvelrn 6018	. . . . . . . . 9 |- ( ( F : T --> RR /\ t e. T ) -> ( F ` t ) e. RR )
86		recn 9626	. . . . . . . . 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 9983	. . . . . . 7 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( ( f ` t ) - ( F ` t ) ) e. CC )
89	88	abscld 13491	. . . . . 6 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> ( abs ` ( ( f ` t ) - ( F ` t ) ) ) e. RR )
90	4, 37, 47	3pm3.2i 1185	. . . . . . . . 9 |- ( 1 e. RR /\ 4 e. RR /\ 4 =/= 0 )
91		redivcl 10323	. . . . . . . . 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 3923	. . . . . . 7 |- ( ph -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR )
94	93	ad2antrr 731	. . . . . 6 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> if ( E <_ ( 1 / 4 ) , E , ( 1 / 4 ) ) e. RR )
95	46	ad2antrr 731	. . . . . 6 |- ( ( ( ph /\ f e. A ) /\ t e. T ) -> E e. RR )
96		ltletr 9722	. . . . . 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 1267	. . . . 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 679	. . . 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 2789	. . 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 2861	. 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 371   /\ w3a 984   = wceq 1443   F/wnf 1666   e. wcel 1886   F/_wnfc 2578   =/= wne 2621   A.wral 2736   E.wrex 2737   C_ wss 3403   (/)c0 3730   ifcif 3880   U.cuni 4197   class class class wbr 4401   |-> cmpt 4460   ran crn 4834   -->wf 5577   ` cfv 5581  (class class class)co 6288  infcinf 7952   CCcc 9534   RRcr 9535   0cc0 9536   1c1 9537   + caddc 9539   x. cmul 9541   RR*cxr 9671   < clt 9672   <_ cle 9673   - cmin 9857   / cdiv 10266   3c3 10657   4c4 10658   RR+crp 11299   (,)cioo 11632   abscabs 13290   topGenctg 15329   Cn ccn 20233   Compccmp 20394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ioc 11637  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-rlim 13546  df-sum 13746  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-cn 20236  df-cnp 20237  df-cmp 20395  df-tx 20570  df-hmeo 20763  df-xms 21328  df-ms 21329  df-tms 21330

This theorem is referenced by:  stowei  37920