Theorem omssubaddlem 30361

Description: For any small margin E, we can find a covering approaching the outer measure of a set A by that margin. (Contributed by Thierry Arnoux, 18-Sep-2019.) (Revised by AV, 4-Oct-2020.)

Hypotheses

Ref	Expression
oms.m	|- M = (toOMeas ` R )
oms.o	|- ( ph -> Q e. V )
oms.r	|- ( ph -> R : Q --> ( 0 [,] +oo ) )
omssubaddlem.a	|- ( ph -> A C_ U. Q )
omssubaddlem.m	|- ( ph -> ( M ` A ) e. RR )
omssubaddlem.e	|- ( ph -> E e. RR+ )

Assertion

Ref	Expression
omssubaddlem	|- ( ph -> E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) }Σ* w e. x ( R ` w ) < ( ( M ` A ) + E ) )

Distinct variable groups:   x, Q, z   x, R, z   x, V, z   ph, x, z   w, A, x, z   x, E   x, M   w, Q   w, R   w, V

Allowed substitution hints:   ph( w)   E( z, w)   M( z, w)

Proof of Theorem omssubaddlem

Dummy variables e t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omssubaddlem.m	. . . . . 6 |- ( ph -> ( M ` A ) e. RR )
2		omssubaddlem.e	. . . . . . 7 |- ( ph -> E e. RR+ )
3	2	rpred 11872	. . . . . 6 |- ( ph -> E e. RR )
4	1, 3	readdcld 10069	. . . . 5 |- ( ph -> ( ( M ` A ) + E ) e. RR )
5	4	rexrd 10089	. . . 4 |- ( ph -> ( ( M ` A ) + E ) e. RR* )
6		oms.o	. . . . . . . . 9 |- ( ph -> Q e. V )
7		oms.r	. . . . . . . . 9 |- ( ph -> R : Q --> ( 0 [,] +oo ) )
8		omsf 30358	. . . . . . . . 9 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) ) -> (toOMeas ` R ) : ~P U. dom R --> ( 0 [,] +oo ) )
9	6, 7, 8	syl2anc 693	. . . . . . . 8 |- ( ph -> (toOMeas ` R ) : ~P U. dom R --> ( 0 [,] +oo ) )
10		oms.m	. . . . . . . . 9 |- M = (toOMeas ` R )
11	10	feq1i 6036	. . . . . . . 8 |- ( M : ~P U. dom R --> ( 0 [,] +oo ) <-> (toOMeas ` R ) : ~P U. dom R --> ( 0 [,] +oo ) )
12	9, 11	sylibr 224	. . . . . . 7 |- ( ph -> M : ~P U. dom R --> ( 0 [,] +oo ) )
13		omssubaddlem.a	. . . . . . . . 9 |- ( ph -> A C_ U. Q )
14		fdm 6051	. . . . . . . . . . 11 |- ( R : Q --> ( 0 [,] +oo ) -> dom R = Q )
15	7, 14	syl 17	. . . . . . . . . 10 |- ( ph -> dom R = Q )
16	15	unieqd 4446	. . . . . . . . 9 |- ( ph -> U. dom R = U. Q )
17	13, 16	sseqtr4d 3642	. . . . . . . 8 |- ( ph -> A C_ U. dom R )
18		uniexg 6955	. . . . . . . . . . 11 |- ( Q e. V -> U. Q e. _V )
19	6, 18	syl 17	. . . . . . . . . 10 |- ( ph -> U. Q e. _V )
20	13, 19	jca 554	. . . . . . . . 9 |- ( ph -> ( A C_ U. Q /\ U. Q e. _V ) )
21		ssexg 4804	. . . . . . . . 9 |- ( ( A C_ U. Q /\ U. Q e. _V ) -> A e. _V )
22		elpwg 4166	. . . . . . . . 9 |- ( A e. _V -> ( A e. ~P U. dom R <-> A C_ U. dom R ) )
23	20, 21, 22	3syl 18	. . . . . . . 8 |- ( ph -> ( A e. ~P U. dom R <-> A C_ U. dom R ) )
24	17, 23	mpbird 247	. . . . . . 7 |- ( ph -> A e. ~P U. dom R )
25	12, 24	ffvelrnd 6360	. . . . . 6 |- ( ph -> ( M ` A ) e. ( 0 [,] +oo ) )
26		elxrge0 12281	. . . . . . 7 |- ( ( M ` A ) e. ( 0 [,] +oo ) <-> ( ( M ` A ) e. RR* /\ 0 <_ ( M ` A ) ) )
27	26	simprbi 480	. . . . . 6 |- ( ( M ` A ) e. ( 0 [,] +oo ) -> 0 <_ ( M ` A ) )
28	25, 27	syl 17	. . . . 5 |- ( ph -> 0 <_ ( M ` A ) )
29	2	rpge0d 11876	. . . . 5 |- ( ph -> 0 <_ E )
30	1, 3, 28, 29	addge0d 10603	. . . 4 |- ( ph -> 0 <_ ( ( M ` A ) + E ) )
31		elxrge0 12281	. . . 4 |- ( ( ( M ` A ) + E ) e. ( 0 [,] +oo ) <-> ( ( ( M ` A ) + E ) e. RR* /\ 0 <_ ( ( M ` A ) + E ) ) )
32	5, 30, 31	sylanbrc 698	. . 3 |- ( ph -> ( ( M ` A ) + E ) e. ( 0 [,] +oo ) )
33	10	fveq1i 6192	. . . . 5 |- ( M ` A ) = ( (toOMeas ` R ) ` A )
34		omsfval 30356	. . . . . 6 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> ( (toOMeas ` R ) ` A ) = inf ( ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) , ( 0 [,] +oo ) , < ) )
35	6, 7, 13, 34	syl3anc 1326	. . . . 5 |- ( ph -> ( (toOMeas ` R ) ` A ) = inf ( ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) , ( 0 [,] +oo ) , < ) )
36	33, 35	syl5req 2669	. . . 4 |- ( ph -> inf ( ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) , ( 0 [,] +oo ) , < ) = ( M ` A ) )
37	1, 2	ltaddrpd 11905	. . . 4 |- ( ph -> ( M ` A ) < ( ( M ` A ) + E ) )
38	36, 37	eqbrtrd 4675	. . 3 |- ( ph -> inf ( ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) , ( 0 [,] +oo ) , < ) < ( ( M ` A ) + E ) )
39		iccssxr 12256	. . . . . 6 |- ( 0 [,] +oo ) C_ RR*
40		xrltso 11974	. . . . . 6 |- < Or RR*
41		soss 5053	. . . . . 6 |- ( ( 0 [,] +oo ) C_ RR* -> ( < Or RR* -> < Or ( 0 [,] +oo ) ) )
42	39, 40, 41	mp2 9	. . . . 5 |- < Or ( 0 [,] +oo )
43	42	a1i 11	. . . 4 |- ( ph -> < Or ( 0 [,] +oo ) )
44		omscl 30357	. . . . . 6 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A e. ~P U. dom R ) -> ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) C_ ( 0 [,] +oo ) )
45	6, 7, 24, 44	syl3anc 1326	. . . . 5 |- ( ph -> ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) C_ ( 0 [,] +oo ) )
46		xrge0infss 29525	. . . . 5 |- ( ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) C_ ( 0 [,] +oo ) -> E. e e. ( 0 [,] +oo ) ( A. t e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) -. t < e /\ A. t e. ( 0 [,] +oo ) ( e < t -> E. u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) u < t ) ) )
47	45, 46	syl 17	. . . 4 |- ( ph -> E. e e. ( 0 [,] +oo ) ( A. t e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) -. t < e /\ A. t e. ( 0 [,] +oo ) ( e < t -> E. u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) u < t ) ) )
48	43, 47	infglb 8396	. . 3 |- ( ph -> ( ( ( ( M ` A ) + E ) e. ( 0 [,] +oo ) /\ inf ( ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) , ( 0 [,] +oo ) , < ) < ( ( M ` A ) + E ) ) -> E. u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) u < ( ( M ` A ) + E ) ) )
49	32, 38, 48	mp2and 715	. 2 |- ( ph -> E. u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) u < ( ( M ` A ) + E ) )
50		eqid 2622	. . . . . . . 8 |- ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) = ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) )
51		esumex 30091	. . . . . . . 8 |- Σ* w e. x ( R ` w ) e. _V
52	50, 51	elrnmpti 5376	. . . . . . 7 |- ( u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) <-> E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } u = Σ* w e. x ( R ` w ) )
53	52	anbi1i 731	. . . . . 6 |- ( ( u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) /\ u < ( ( M ` A ) + E ) ) <-> ( E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } u = Σ* w e. x ( R ` w ) /\ u < ( ( M ` A ) + E ) ) )
54		r19.41v 3089	. . . . . 6 |- ( E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } ( u = Σ* w e. x ( R ` w ) /\ u < ( ( M ` A ) + E ) ) <-> ( E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } u = Σ* w e. x ( R ` w ) /\ u < ( ( M ` A ) + E ) ) )
55	53, 54	bitr4i 267	. . . . 5 |- ( ( u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) /\ u < ( ( M ` A ) + E ) ) <-> E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } ( u = Σ* w e. x ( R ` w ) /\ u < ( ( M ` A ) + E ) ) )
56	55	exbii 1774	. . . 4 |- ( E. u ( u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) /\ u < ( ( M ` A ) + E ) ) <-> E. u E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } ( u = Σ* w e. x ( R ` w ) /\ u < ( ( M ` A ) + E ) ) )
57		df-rex 2918	. . . 4 |- ( E. u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) u < ( ( M ` A ) + E ) <-> E. u ( u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) /\ u < ( ( M ` A ) + E ) ) )
58		rexcom4 3225	. . . 4 |- ( E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } E. u ( u = Σ* w e. x ( R ` w ) /\ u < ( ( M ` A ) + E ) ) <-> E. u E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } ( u = Σ* w e. x ( R ` w ) /\ u < ( ( M ` A ) + E ) ) )
59	56, 57, 58	3bitr4i 292	. . 3 |- ( E. u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) u < ( ( M ` A ) + E ) <-> E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } E. u ( u = Σ* w e. x ( R ` w ) /\ u < ( ( M ` A ) + E ) ) )
60		breq1 4656	. . . . . 6 |- ( u = Σ* w e. x ( R ` w ) -> ( u < ( ( M ` A ) + E ) <-> Σ* w e. x ( R ` w ) < ( ( M ` A ) + E ) ) )
61	60	biimpa 501	. . . . 5 |- ( ( u = Σ* w e. x ( R ` w ) /\ u < ( ( M ` A ) + E ) ) -> Σ* w e. x ( R ` w ) < ( ( M ` A ) + E ) )
62	61	exlimiv 1858	. . . 4 |- ( E. u ( u = Σ* w e. x ( R ` w ) /\ u < ( ( M ` A ) + E ) ) -> Σ* w e. x ( R ` w ) < ( ( M ` A ) + E ) )
63	62	reximi 3011	. . 3 |- ( E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } E. u ( u = Σ* w e. x ( R ` w ) /\ u < ( ( M ` A ) + E ) ) -> E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) }Σ* w e. x ( R ` w ) < ( ( M ` A ) + E ) )
64	59, 63	sylbi 207	. 2 |- ( E. u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) u < ( ( M ` A ) + E ) -> E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) }Σ* w e. x ( R ` w ) < ( ( M ` A ) + E ) )
65	49, 64	syl 17	1 |- ( ph -> E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) }Σ* w e. x ( R ` w ) < ( ( M ` A ) + E ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   Or wor 5034   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   omcom 7065   ~<_ cdom 7953  infcinf 8347   RRcr 9935   0cc0 9936   + caddc 9939   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   RR+crp 11832   [,]cicc 12178  Σ^*cesum 30089  toOMeascoms 30353

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-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

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-oadd 7564  df-er 7742  df-map 7859  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-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-xadd 11947  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  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-tset 15960  df-ple 15961  df-ds 15964  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-ordt 16161  df-xrs 16162  df-mre 16246  df-mrc 16247  df-acs 16249  df-ps 17200  df-tsr 17201  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-cntz 17750  df-cmn 18195  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-ntr 20824  df-nei 20902  df-cn 21031  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-tsms 21930  df-esum 30090  df-oms 30354

This theorem is referenced by: (None)