Theorem stowei 40281

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 40280: often times it will be better to use stoweid 40280 in other proofs (but this version is probably easier to be read and understood). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

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

Assertion

Ref	Expression
stowei	|- 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, t   f, J, r, t   T, f, g, t   h, E, r, x   h, F, r, x   T, h, r, x   t, K

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

Proof of Theorem stowei

Step	Hyp	Ref	Expression
1		nfcv 2764	. . 3 |- F/_ t F
2		nftru 1730	. . 3 |- F/ t T.
3		stowei.1	. . 3 |- K = ( topGen ` ran (,) )
4		stowei.2	. . . 4 |- J e. Comp
5	4	a1i 11	. . 3 |- ( T. -> J e. Comp )
6		stowei.3	. . 3 |- T = U. J
7		stowei.4	. . 3 |- C = ( J Cn K )
8		stowei.5	. . . 4 |- A C_ C
9	8	a1i 11	. . 3 |- ( T. -> A C_ C )
10		stowei.6	. . . 4 |- ( ( f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A )
11	10	3adant1 1079	. . 3 |- ( ( T. /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A )
12		stowei.7	. . . 4 |- ( ( f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
13	12	3adant1 1079	. . 3 |- ( ( T. /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
14		stowei.8	. . . 4 |- ( x e. RR -> ( t e. T |-> x ) e. A )
15	14	adantl 482	. . 3 |- ( ( T. /\ x e. RR ) -> ( t e. T |-> x ) e. A )
16		stowei.9	. . . 4 |- ( ( r e. T /\ t e. T /\ r =/= t ) -> E. h e. A ( h ` r ) =/= ( h ` t ) )
17	16	adantl 482	. . 3 |- ( ( T. /\ ( r e. T /\ t e. T /\ r =/= t ) ) -> E. h e. A ( h ` r ) =/= ( h ` t ) )
18		stowei.10	. . . 4 |- F e. C
19	18	a1i 11	. . 3 |- ( T. -> F e. C )
20		stowei.11	. . . 4 |- E e. RR+
21	20	a1i 11	. . 3 |- ( T. -> E e. RR+ )
22	1, 2, 3, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21	stoweid 40280	. 2 |- ( T. -> E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < E )
23	22	trud 1493	1 |- E. f e. A A. t e. T ( abs ` ( ( f ` t ) - ( F ` t ) ) ) < E

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   T. wtru 1484   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   RRcr 9935   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   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: (None)