Theorem stoweidlem61 40278

Description: This lemma proves that there exists a function g as in the proof in [BrosowskiDeutsh] p. 92: g is in the subalgebra, and for all t in T, abs( f(t) - g(t) ) < 2*ε. Here F is used to represent f in the paper, and E is used to represent ε. For this lemma there's the further assumption that the function F to be approximated is nonnegative (this assumption is removed in a later theorem). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem61.1	|- F/_ t F
stoweidlem61.2	|- F/ t ph
stoweidlem61.3	|- K = ( topGen ` ran (,) )
stoweidlem61.4	|- ( ph -> J e. Comp )
stoweidlem61.5	|- T = U. J
stoweidlem61.6	|- ( ph -> T =/= (/) )
stoweidlem61.7	|- C = ( J Cn K )
stoweidlem61.8	|- ( ph -> A C_ C )
stoweidlem61.9	|- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A )
stoweidlem61.10	|- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
stoweidlem61.11	|- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
stoweidlem61.12	|- ( ( ph /\ ( r e. T /\ t e. T /\ r =/= t ) ) -> E. q e. A ( q ` r ) =/= ( q ` t ) )
stoweidlem61.13	|- ( ph -> F e. C )
stoweidlem61.14	|- ( ph -> A. t e. T 0 <_ ( F ` t ) )
stoweidlem61.15	|- ( ph -> E e. RR+ )
stoweidlem61.16	|- ( ph -> E < ( 1 / 3 ) )

Assertion

Ref	Expression
stoweidlem61	|- ( ph -> E. g e. A A. t e. T ( abs ` ( ( g ` t ) - ( F ` t ) ) ) < ( 2 x. E ) )

Distinct variable groups:   f, g, q, r, t, x, A   f, E, g, q, r, t, x   f, F, g, q, r, x   f, J, g, r, t   T, f, g, q, r, t, x   ph, f, g, q, r, x   t, K

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

Proof of Theorem stoweidlem61

Dummy variables j n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem61.1	. . 3 |- F/_ t F
2		stoweidlem61.2	. . 3 |- F/ t ph
3		stoweidlem61.3	. . 3 |- K = ( topGen ` ran (,) )
4		stoweidlem61.5	. . 3 |- T = U. J
5		stoweidlem61.7	. . 3 |- C = ( J Cn K )
6		eqid 2622	. . 3 |- ( j e. ( 0 ... n ) |-> { t e. T | ( F ` t ) <_ ( ( j - ( 1 / 3 ) ) x. E ) } ) = ( j e. ( 0 ... n ) |-> { t e. T | ( F ` t ) <_ ( ( j - ( 1 / 3 ) ) x. E ) } )
7		eqid 2622	. . 3 |- ( j e. ( 0 ... n ) |-> { t e. T | ( ( j + ( 1 / 3 ) ) x. E ) <_ ( F ` t ) } ) = ( j e. ( 0 ... n ) |-> { t e. T | ( ( j + ( 1 / 3 ) ) x. E ) <_ ( F ` t ) } )
8		stoweidlem61.4	. . 3 |- ( ph -> J e. Comp )
9		stoweidlem61.6	. . 3 |- ( ph -> T =/= (/) )
10		stoweidlem61.8	. . 3 |- ( ph -> A C_ C )
11		stoweidlem61.9	. . 3 |- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) + ( g ` t ) ) ) e. A )
12		stoweidlem61.10	. . 3 |- ( ( ph /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
13		stoweidlem61.11	. . 3 |- ( ( ph /\ x e. RR ) -> ( t e. T |-> x ) e. A )
14		stoweidlem61.12	. . 3 |- ( ( ph /\ ( r e. T /\ t e. T /\ r =/= t ) ) -> E. q e. A ( q ` r ) =/= ( q ` t ) )
15		stoweidlem61.13	. . 3 |- ( ph -> F e. C )
16		stoweidlem61.14	. . 3 |- ( ph -> A. t e. T 0 <_ ( F ` t ) )
17		stoweidlem61.15	. . 3 |- ( ph -> E e. RR+ )
18		stoweidlem61.16	. . 3 |- ( ph -> E < ( 1 / 3 ) )
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	stoweidlem60 40277	. 2 |- ( ph -> E. g e. A A. t e. T E. j e. RR ( ( ( ( j - ( 4 / 3 ) ) x. E ) < ( F ` t ) /\ ( F ` t ) <_ ( ( j - ( 1 / 3 ) ) x. E ) ) /\ ( ( g ` t ) < ( ( j + ( 1 / 3 ) ) x. E ) /\ ( ( j - ( 4 / 3 ) ) x. E ) < ( g ` t ) ) ) )
20		nfv 1843	. . . . 5 |- F/ t g e. A
21	2, 20	nfan 1828	. . . 4 |- F/ t ( ph /\ g e. A )
22	17	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ g e. A ) /\ t e. T ) -> E e. RR+ )
23	3, 4, 5, 15	fcnre 39184	. . . . . . 7 |- ( ph -> F : T --> RR )
24	23	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ t e. T ) -> ( F ` t ) e. RR )
25	24	adantlr 751	. . . . 5 |- ( ( ( ph /\ g e. A ) /\ t e. T ) -> ( F ` t ) e. RR )
26	10	sselda 3603	. . . . . . 7 |- ( ( ph /\ g e. A ) -> g e. C )
27	3, 4, 5, 26	fcnre 39184	. . . . . 6 |- ( ( ph /\ g e. A ) -> g : T --> RR )
28	27	ffvelrnda 6359	. . . . 5 |- ( ( ( ph /\ g e. A ) /\ t e. T ) -> ( g ` t ) e. RR )
29		simpll1 1100	. . . . . . . 8 |- ( ( ( ( E e. RR+ /\ ( F ` t ) e. RR /\ ( g ` t ) e. RR ) /\ j e. RR ) /\ ( ( ( ( j - ( 4 / 3 ) ) x. E ) < ( F ` t ) /\ ( F ` t ) <_ ( ( j - ( 1 / 3 ) ) x. E ) ) /\ ( ( g ` t ) < ( ( j + ( 1 / 3 ) ) x. E ) /\ ( ( j - ( 4 / 3 ) ) x. E ) < ( g ` t ) ) ) ) -> E e. RR+ )
30		simpll2 1101	. . . . . . . 8 |- ( ( ( ( E e. RR+ /\ ( F ` t ) e. RR /\ ( g ` t ) e. RR ) /\ j e. RR ) /\ ( ( ( ( j - ( 4 / 3 ) ) x. E ) < ( F ` t ) /\ ( F ` t ) <_ ( ( j - ( 1 / 3 ) ) x. E ) ) /\ ( ( g ` t ) < ( ( j + ( 1 / 3 ) ) x. E ) /\ ( ( j - ( 4 / 3 ) ) x. E ) < ( g ` t ) ) ) ) -> ( F ` t ) e. RR )
31		simpll3 1102	. . . . . . . 8 |- ( ( ( ( E e. RR+ /\ ( F ` t ) e. RR /\ ( g ` t ) e. RR ) /\ j e. RR ) /\ ( ( ( ( j - ( 4 / 3 ) ) x. E ) < ( F ` t ) /\ ( F ` t ) <_ ( ( j - ( 1 / 3 ) ) x. E ) ) /\ ( ( g ` t ) < ( ( j + ( 1 / 3 ) ) x. E ) /\ ( ( j - ( 4 / 3 ) ) x. E ) < ( g ` t ) ) ) ) -> ( g ` t ) e. RR )
32		simplr 792	. . . . . . . 8 |- ( ( ( ( E e. RR+ /\ ( F ` t ) e. RR /\ ( g ` t ) e. RR ) /\ j e. RR ) /\ ( ( ( ( j - ( 4 / 3 ) ) x. E ) < ( F ` t ) /\ ( F ` t ) <_ ( ( j - ( 1 / 3 ) ) x. E ) ) /\ ( ( g ` t ) < ( ( j + ( 1 / 3 ) ) x. E ) /\ ( ( j - ( 4 / 3 ) ) x. E ) < ( g ` t ) ) ) ) -> j e. RR )
33		simprll 802	. . . . . . . 8 |- ( ( ( ( E e. RR+ /\ ( F ` t ) e. RR /\ ( g ` t ) e. RR ) /\ j e. RR ) /\ ( ( ( ( j - ( 4 / 3 ) ) x. E ) < ( F ` t ) /\ ( F ` t ) <_ ( ( j - ( 1 / 3 ) ) x. E ) ) /\ ( ( g ` t ) < ( ( j + ( 1 / 3 ) ) x. E ) /\ ( ( j - ( 4 / 3 ) ) x. E ) < ( g ` t ) ) ) ) -> ( ( j - ( 4 / 3 ) ) x. E ) < ( F ` t ) )
34		simprlr 803	. . . . . . . 8 |- ( ( ( ( E e. RR+ /\ ( F ` t ) e. RR /\ ( g ` t ) e. RR ) /\ j e. RR ) /\ ( ( ( ( j - ( 4 / 3 ) ) x. E ) < ( F ` t ) /\ ( F ` t ) <_ ( ( j - ( 1 / 3 ) ) x. E ) ) /\ ( ( g ` t ) < ( ( j + ( 1 / 3 ) ) x. E ) /\ ( ( j - ( 4 / 3 ) ) x. E ) < ( g ` t ) ) ) ) -> ( F ` t ) <_ ( ( j - ( 1 / 3 ) ) x. E ) )
35		simprrr 805	. . . . . . . 8 |- ( ( ( ( E e. RR+ /\ ( F ` t ) e. RR /\ ( g ` t ) e. RR ) /\ j e. RR ) /\ ( ( ( ( j - ( 4 / 3 ) ) x. E ) < ( F ` t ) /\ ( F ` t ) <_ ( ( j - ( 1 / 3 ) ) x. E ) ) /\ ( ( g ` t ) < ( ( j + ( 1 / 3 ) ) x. E ) /\ ( ( j - ( 4 / 3 ) ) x. E ) < ( g ` t ) ) ) ) -> ( ( j - ( 4 / 3 ) ) x. E ) < ( g ` t ) )
36		simprrl 804	. . . . . . . 8 |- ( ( ( ( E e. RR+ /\ ( F ` t ) e. RR /\ ( g ` t ) e. RR ) /\ j e. RR ) /\ ( ( ( ( j - ( 4 / 3 ) ) x. E ) < ( F ` t ) /\ ( F ` t ) <_ ( ( j - ( 1 / 3 ) ) x. E ) ) /\ ( ( g ` t ) < ( ( j + ( 1 / 3 ) ) x. E ) /\ ( ( j - ( 4 / 3 ) ) x. E ) < ( g ` t ) ) ) ) -> ( g ` t ) < ( ( j + ( 1 / 3 ) ) x. E ) )
37	29, 30, 31, 32, 33, 34, 35, 36	stoweidlem13 40230	. . . . . . 7 |- ( ( ( ( E e. RR+ /\ ( F ` t ) e. RR /\ ( g ` t ) e. RR ) /\ j e. RR ) /\ ( ( ( ( j - ( 4 / 3 ) ) x. E ) < ( F ` t ) /\ ( F ` t ) <_ ( ( j - ( 1 / 3 ) ) x. E ) ) /\ ( ( g ` t ) < ( ( j + ( 1 / 3 ) ) x. E ) /\ ( ( j - ( 4 / 3 ) ) x. E ) < ( g ` t ) ) ) ) -> ( abs ` ( ( g ` t ) - ( F ` t ) ) ) < ( 2 x. E ) )
38	37	ex 450	. . . . . 6 |- ( ( ( E e. RR+ /\ ( F ` t ) e. RR /\ ( g ` t ) e. RR ) /\ j e. RR ) -> ( ( ( ( ( j - ( 4 / 3 ) ) x. E ) < ( F ` t ) /\ ( F ` t ) <_ ( ( j - ( 1 / 3 ) ) x. E ) ) /\ ( ( g ` t ) < ( ( j + ( 1 / 3 ) ) x. E ) /\ ( ( j - ( 4 / 3 ) ) x. E ) < ( g ` t ) ) ) -> ( abs ` ( ( g ` t ) - ( F ` t ) ) ) < ( 2 x. E ) ) )
39	38	rexlimdva 3031	. . . . 5 |- ( ( E e. RR+ /\ ( F ` t ) e. RR /\ ( g ` t ) e. RR ) -> ( E. j e. RR ( ( ( ( j - ( 4 / 3 ) ) x. E ) < ( F ` t ) /\ ( F ` t ) <_ ( ( j - ( 1 / 3 ) ) x. E ) ) /\ ( ( g ` t ) < ( ( j + ( 1 / 3 ) ) x. E ) /\ ( ( j - ( 4 / 3 ) ) x. E ) < ( g ` t ) ) ) -> ( abs ` ( ( g ` t ) - ( F ` t ) ) ) < ( 2 x. E ) ) )
40	22, 25, 28, 39	syl3anc 1326	. . . 4 |- ( ( ( ph /\ g e. A ) /\ t e. T ) -> ( E. j e. RR ( ( ( ( j - ( 4 / 3 ) ) x. E ) < ( F ` t ) /\ ( F ` t ) <_ ( ( j - ( 1 / 3 ) ) x. E ) ) /\ ( ( g ` t ) < ( ( j + ( 1 / 3 ) ) x. E ) /\ ( ( j - ( 4 / 3 ) ) x. E ) < ( g ` t ) ) ) -> ( abs ` ( ( g ` t ) - ( F ` t ) ) ) < ( 2 x. E ) ) )
41	21, 40	ralimdaa 2958	. . 3 |- ( ( ph /\ g e. A ) -> ( A. t e. T E. j e. RR ( ( ( ( j - ( 4 / 3 ) ) x. E ) < ( F ` t ) /\ ( F ` t ) <_ ( ( j - ( 1 / 3 ) ) x. E ) ) /\ ( ( g ` t ) < ( ( j + ( 1 / 3 ) ) x. E ) /\ ( ( j - ( 4 / 3 ) ) x. E ) < ( g ` t ) ) ) -> A. t e. T ( abs ` ( ( g ` t ) - ( F ` t ) ) ) < ( 2 x. E ) ) )
42	41	reximdva 3017	. 2 |- ( ph -> ( E. g e. A A. t e. T E. j e. RR ( ( ( ( j - ( 4 / 3 ) ) x. E ) < ( F ` t ) /\ ( F ` t ) <_ ( ( j - ( 1 / 3 ) ) x. E ) ) /\ ( ( g ` t ) < ( ( j + ( 1 / 3 ) ) x. E ) /\ ( ( j - ( 4 / 3 ) ) x. E ) < ( g ` t ) ) ) -> E. g e. A A. t e. T ( abs ` ( ( g ` t ) - ( F ` t ) ) ) < ( 2 x. E ) ) )
43	19, 42	mpd 15	1 |- ( ph -> E. g e. A A. t e. T ( abs ` ( ( g ` t ) - ( F ` t ) ) ) < ( 2 x. E ) )

Syntax hints:   -> 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   {crab 2916   C_ wss 3574   (/)c0 3915   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   2c2 11070   3c3 11071   4c4 11072   RR+crp 11832   (,)cioo 12175   ...cfz 12326   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-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:  stoweidlem62  40279