Theorem omssubaddlemOLD 29131

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.) Obsolete version of omssubaddlem 29127 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
omssubaddlemOLD	|- ( 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 omssubaddlemOLD

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

Step	Hyp	Ref	Expression
1		omssubaddlemOLD.m	. . . . . . . 8 |- ( ph -> ( M ` A ) e. RR )
2		omssubaddlemOLD.e	. . . . . . . . 9 |- ( ph -> E e. RR+ )
3	2	rpred 11341	. . . . . . . 8 |- ( ph -> E e. RR )
4	1, 3	readdcld 9670	. . . . . . 7 |- ( ph -> ( ( M ` A ) + E ) e. RR )
5	4	rexrd 9690	. . . . . 6 |- ( ph -> ( ( M ` A ) + E ) e. RR* )
6		omsOLD.o	. . . . . . . . . . 11 |- ( ph -> Q e. V )
7		omsOLD.r	. . . . . . . . . . 11 |- ( ph -> R : Q --> ( 0 [,] +oo ) )
8		omsfOLD 29124	. . . . . . . . . . 11 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) ) -> (toOMeas ` R ) : ~P U. dom R --> ( 0 [,] +oo ) )
9	6, 7, 8	syl2anc 667	. . . . . . . . . 10 |- ( ph -> (toOMeas ` R ) : ~P U. dom R --> ( 0 [,] +oo ) )
10		omsOLD.m	. . . . . . . . . . 11 |- M = (toOMeas ` R )
11	10	feq1i 5720	. . . . . . . . . 10 |- ( M : ~P U. dom R --> ( 0 [,] +oo ) <-> (toOMeas ` R ) : ~P U. dom R --> ( 0 [,] +oo ) )
12	9, 11	sylibr 216	. . . . . . . . 9 |- ( ph -> M : ~P U. dom R --> ( 0 [,] +oo ) )
13		omssubaddlemOLD.a	. . . . . . . . . . 11 |- ( ph -> A C_ U. Q )
14		fdm 5733	. . . . . . . . . . . . 13 |- ( R : Q --> ( 0 [,] +oo ) -> dom R = Q )
15	7, 14	syl 17	. . . . . . . . . . . 12 |- ( ph -> dom R = Q )
16	15	unieqd 4208	. . . . . . . . . . 11 |- ( ph -> U. dom R = U. Q )
17	13, 16	sseqtr4d 3469	. . . . . . . . . 10 |- ( ph -> A C_ U. dom R )
18		uniexg 6588	. . . . . . . . . . . . 13 |- ( Q e. V -> U. Q e. _V )
19	6, 18	syl 17	. . . . . . . . . . . 12 |- ( ph -> U. Q e. _V )
20	13, 19	jca 535	. . . . . . . . . . 11 |- ( ph -> ( A C_ U. Q /\ U. Q e. _V ) )
21		ssexg 4549	. . . . . . . . . . 11 |- ( ( A C_ U. Q /\ U. Q e. _V ) -> A e. _V )
22		elpwg 3959	. . . . . . . . . . 11 |- ( A e. _V -> ( A e. ~P U. dom R <-> A C_ U. dom R ) )
23	20, 21, 22	3syl 18	. . . . . . . . . 10 |- ( ph -> ( A e. ~P U. dom R <-> A C_ U. dom R ) )
24	17, 23	mpbird 236	. . . . . . . . 9 |- ( ph -> A e. ~P U. dom R )
25	12, 24	ffvelrnd 6023	. . . . . . . 8 |- ( ph -> ( M ` A ) e. ( 0 [,] +oo ) )
26		elxrge0 11741	. . . . . . . . 9 |- ( ( M ` A ) e. ( 0 [,] +oo ) <-> ( ( M ` A ) e. RR* /\ 0 <_ ( M ` A ) ) )
27	26	simprbi 466	. . . . . . . 8 |- ( ( M ` A ) e. ( 0 [,] +oo ) -> 0 <_ ( M ` A ) )
28	25, 27	syl 17	. . . . . . 7 |- ( ph -> 0 <_ ( M ` A ) )
29	2	rpge0d 11345	. . . . . . 7 |- ( ph -> 0 <_ E )
30	1, 3, 28, 29	addge0d 10189	. . . . . 6 |- ( ph -> 0 <_ ( ( M ` A ) + E ) )
31	5, 30	jca 535	. . . . 5 |- ( ph -> ( ( ( M ` A ) + E ) e. RR* /\ 0 <_ ( ( M ` A ) + E ) ) )
32		elxrge0 11741	. . . . 5 |- ( ( ( M ` A ) + E ) e. ( 0 [,] +oo ) <-> ( ( ( M ` A ) + E ) e. RR* /\ 0 <_ ( ( M ` A ) + E ) ) )
33	31, 32	sylibr 216	. . . 4 |- ( ph -> ( ( M ` A ) + E ) e. ( 0 [,] +oo ) )
34	1, 2	ltaddrpd 11371	. . . . . 6 |- ( ph -> ( M ` A ) < ( ( M ` A ) + E ) )
35		ovex 6318	. . . . . . 7 |- ( ( M ` A ) + E ) e. _V
36		fvex 5875	. . . . . . 7 |- ( M ` A ) e. _V
37	35, 36	brcnv 5017	. . . . . 6 |- ( ( ( M ` A ) + E ) `' < ( M ` A ) <-> ( M ` A ) < ( ( M ` A ) + E ) )
38	34, 37	sylibr 216	. . . . 5 |- ( ph -> ( ( M ` A ) + E ) `' < ( M ` A ) )
39	10	fveq1i 5866	. . . . . 6 |- ( M ` A ) = ( (toOMeas ` R ) ` A )
40		omsfvalOLD 29122	. . . . . . 7 |- ( ( Q e. V /\ R : Q --> ( 0 [,] +oo ) /\ A C_ U. Q ) -> ( (toOMeas ` R ) ` A ) = sup ( ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) , ( 0 [,] +oo ) , `' < ) )
41	6, 7, 13, 40	syl3anc 1268	. . . . . 6 |- ( ph -> ( (toOMeas ` R ) ` A ) = sup ( ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) , ( 0 [,] +oo ) , `' < ) )
42	39, 41	syl5eq 2497	. . . . 5 |- ( ph -> ( M ` A ) = sup ( ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) , ( 0 [,] +oo ) , `' < ) )
43	38, 42	breqtrd 4427	. . . 4 |- ( ph -> ( ( M ` A ) + E ) `' < sup ( ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) , ( 0 [,] +oo ) , `' < ) )
44	33, 43	jca 535	. . 3 |- ( ph -> ( ( ( M ` A ) + E ) e. ( 0 [,] +oo ) /\ ( ( M ` A ) + E ) `' < sup ( ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) , ( 0 [,] +oo ) , `' < ) ) )
45		iccssxr 11717	. . . . . . 7 |- ( 0 [,] +oo ) C_ RR*
46		xrltso 11440	. . . . . . 7 |- < Or RR*
47		soss 4773	. . . . . . 7 |- ( ( 0 [,] +oo ) C_ RR* -> ( < Or RR* -> < Or ( 0 [,] +oo ) ) )
48	45, 46, 47	mp2 9	. . . . . 6 |- < Or ( 0 [,] +oo )
49		cnvso 5375	. . . . . 6 |- ( < Or ( 0 [,] +oo ) <-> `' < Or ( 0 [,] +oo ) )
50	48, 49	mpbi 212	. . . . 5 |- `' < Or ( 0 [,] +oo )
51	50	a1i 11	. . . 4 |- ( ph -> `' < Or ( 0 [,] +oo ) )
52		omsclOLD 29123	. . . . . 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 ) )
53	6, 7, 24, 52	syl3anc 1268	. . . . 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 ) )
54		xrge0infssOLD 28341	. . . . 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 ) ) -. e `' < t /\ A. t e. ( 0 [,] +oo ) ( t `' < e -> E. u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) t `' < u ) ) )
55	53, 54	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 ) ) -. e `' < t /\ A. t e. ( 0 [,] +oo ) ( t `' < e -> E. u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) t `' < u ) ) )
56	51, 55	suplub 7974	. . 3 |- ( ph -> ( ( ( ( M ` A ) + E ) e. ( 0 [,] +oo ) /\ ( ( M ` A ) + E ) `' < sup ( ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) , ( 0 [,] +oo ) , `' < ) ) -> E. u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) ( ( M ` A ) + E ) `' < u ) )
57	44, 56	mpd 15	. 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 ) ) ( ( M ` A ) + E ) `' < u )
58		eqid 2451	. . . . . . . 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 ) )
59		esumex 28850	. . . . . . . 8 |- Σ* w e. x ( R ` w ) e. _V
60	58, 59	elrnmpti 5085	. . . . . . 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 ) )
61	60	anbi1i 701	. . . . . 6 |- ( ( u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) /\ ( ( M ` A ) + E ) `' < u ) <-> ( E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } u = Σ* w e. x ( R ` w ) /\ ( ( M ` A ) + E ) `' < u ) )
62		r19.41v 2942	. . . . . 6 |- ( E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } ( u = Σ* w e. x ( R ` w ) /\ ( ( M ` A ) + E ) `' < u ) <-> ( E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } u = Σ* w e. x ( R ` w ) /\ ( ( M ` A ) + E ) `' < u ) )
63	61, 62	bitr4i 256	. . . . 5 |- ( ( u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) /\ ( ( M ` A ) + E ) `' < u ) <-> E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } ( u = Σ* w e. x ( R ` w ) /\ ( ( M ` A ) + E ) `' < u ) )
64	63	exbii 1718	. . . 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 ) ) /\ ( ( M ` A ) + E ) `' < u ) <-> E. u E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } ( u = Σ* w e. x ( R ` w ) /\ ( ( M ` A ) + E ) `' < u ) )
65		df-rex 2743	. . . 4 |- ( E. u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) ( ( M ` A ) + E ) `' < u <-> E. u ( u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) /\ ( ( M ` A ) + E ) `' < u ) )
66		rexcom4 3067	. . . 4 |- ( E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } E. u ( u = Σ* w e. x ( R ` w ) /\ ( ( M ` A ) + E ) `' < u ) <-> E. u E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } ( u = Σ* w e. x ( R ` w ) /\ ( ( M ` A ) + E ) `' < u ) )
67	64, 65, 66	3bitr4i 281	. . 3 |- ( E. u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) ( ( M ` A ) + E ) `' < u <-> E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } E. u ( u = Σ* w e. x ( R ` w ) /\ ( ( M ` A ) + E ) `' < u ) )
68		simpr 463	. . . . . . 7 |- ( ( u = Σ* w e. x ( R ` w ) /\ ( ( M ` A ) + E ) `' < u ) -> ( ( M ` A ) + E ) `' < u )
69		simpl 459	. . . . . . 7 |- ( ( u = Σ* w e. x ( R ` w ) /\ ( ( M ` A ) + E ) `' < u ) -> u = Σ* w e. x ( R ` w ) )
70	68, 69	breqtrd 4427	. . . . . 6 |- ( ( u = Σ* w e. x ( R ` w ) /\ ( ( M ` A ) + E ) `' < u ) -> ( ( M ` A ) + E ) `' < Σ* w e. x ( R ` w ) )
71	35, 59	brcnv 5017	. . . . . 6 |- ( ( ( M ` A ) + E ) `' < Σ* w e. x ( R ` w ) <-> Σ* w e. x ( R ` w ) < ( ( M ` A ) + E ) )
72	70, 71	sylib 200	. . . . 5 |- ( ( u = Σ* w e. x ( R ` w ) /\ ( ( M ` A ) + E ) `' < u ) -> Σ* w e. x ( R ` w ) < ( ( M ` A ) + E ) )
73	72	exlimiv 1776	. . . 4 |- ( E. u ( u = Σ* w e. x ( R ` w ) /\ ( ( M ` A ) + E ) `' < u ) -> Σ* w e. x ( R ` w ) < ( ( M ` A ) + E ) )
74	73	reximi 2855	. . 3 |- ( E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } E. u ( u = Σ* w e. x ( R ` w ) /\ ( ( M ` A ) + E ) `' < u ) -> E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) }Σ* w e. x ( R ` w ) < ( ( M ` A ) + E ) )
75	67, 74	sylbi 199	. 2 |- ( E. u e. ran ( x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) } |-> Σ* w e. x ( R ` w ) ) ( ( M ` A ) + E ) `' < u -> E. x e. { z e. ~P dom R | ( A C_ U. z /\ z ~<_ om ) }Σ* w e. x ( R ` w ) < ( ( M ` A ) + E ) )
76	57, 75	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 188   /\ wa 371   = wceq 1444   E.wex 1663   e. wcel 1887   A.wral 2737   E.wrex 2738   {crab 2741   _Vcvv 3045   C_ wss 3404   ~Pcpw 3951   U.cuni 4198   class class class wbr 4402   |-> cmpt 4461   Or wor 4754   `'ccnv 4833   dom cdm 4834   ran crn 4835   -->wf 5578   ` cfv 5582  (class class class)co 6290   omcom 6692   ~<_ cdom 7567   supcsup 7954   RRcr 9538   0cc0 9539   + caddc 9542   +oocpnf 9672   RR*cxr 9674   < clt 9675   <_ cle 9676   RR+crp 11302   [,]cicc 11638  Σ^*cesum 28848  toOMeascomsold 29114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xadd 11410  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12214  df-hash 12516  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-tset 15209  df-ple 15210  df-ds 15212  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-ordt 15399  df-xrs 15400  df-mre 15492  df-mrc 15493  df-acs 15495  df-ps 16446  df-tsr 16447  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-cntz 16971  df-cmn 17432  df-fbas 18967  df-fg 18968  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-ntr 20035  df-nei 20114  df-cn 20243  df-haus 20331  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-tsms 21141  df-esum 28849  df-omsOLD 29116

This theorem is referenced by: (None)