Theorem ovolval2 40858

Description: The value of the Lebesgue outer measure for subsets of the reals, expressed using Σ^{^}. See ovolval 23242 for an alternative expression. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
ovolval2.a	|- ( ph -> A C_ RR )
ovolval2.m	|- M = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = (Σ^ ` ( ( abs o. - ) o. f ) ) ) }

Assertion

Ref	Expression
ovolval2	|- ( ph -> ( vol* ` A ) = inf ( M , RR* , < ) )

Distinct variable groups:   A, f, y   ph, f, y

Allowed substitution hints:   M( y, f)

Proof of Theorem ovolval2

Step	Hyp	Ref	Expression
1		ovolval2.a	. . 3 |- ( ph -> A C_ RR )
2		eqid 2622	. . . 4 |- { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) }
3	2	ovolval 23242	. . 3 |- ( A C_ RR -> ( vol* ` A ) = inf ( { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } , RR* , < ) )
4	1, 3	syl 17	. 2 |- ( ph -> ( vol* ` A ) = inf ( { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } , RR* , < ) )
5	2	a1i 11	. . . 4 |- ( ph -> { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } )
6		reex 10027	. . . . . . . . . . . . . . 15 |- RR e. _V
7	6, 6	xpex 6962	. . . . . . . . . . . . . 14 |- ( RR X. RR ) e. _V
8		inss2 3834	. . . . . . . . . . . . . 14 |- ( <_ i^i ( RR X. RR ) ) C_ ( RR X. RR )
9		mapss 7900	. . . . . . . . . . . . . 14 |- ( ( ( RR X. RR ) e. _V /\ ( <_ i^i ( RR X. RR ) ) C_ ( RR X. RR ) ) -> ( ( <_ i^i ( RR X. RR ) ) ^m NN ) C_ ( ( RR X. RR ) ^m NN ) )
10	7, 8, 9	mp2an 708	. . . . . . . . . . . . 13 |- ( ( <_ i^i ( RR X. RR ) ) ^m NN ) C_ ( ( RR X. RR ) ^m NN )
11	10	sseli 3599	. . . . . . . . . . . 12 |- ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) -> f e. ( ( RR X. RR ) ^m NN ) )
12		1zzd 11408	. . . . . . . . . . . 12 |- ( f e. ( ( RR X. RR ) ^m NN ) -> 1 e. ZZ )
13	11, 12	syl 17	. . . . . . . . . . 11 |- ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) -> 1 e. ZZ )
14	13	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ) -> 1 e. ZZ )
15		nnuz 11723	. . . . . . . . . 10 |- NN = ( ZZ>= ` 1 )
16		absfico 39410	. . . . . . . . . . . . . 14 |- abs : CC --> ( 0 [,) +oo )
17		subf 10283	. . . . . . . . . . . . . 14 |- - : ( CC X. CC ) --> CC
18		fco 6058	. . . . . . . . . . . . . 14 |- ( ( abs : CC --> ( 0 [,) +oo ) /\ - : ( CC X. CC ) --> CC ) -> ( abs o. - ) : ( CC X. CC ) --> ( 0 [,) +oo ) )
19	16, 17, 18	mp2an 708	. . . . . . . . . . . . 13 |- ( abs o. - ) : ( CC X. CC ) --> ( 0 [,) +oo )
20	19	a1i 11	. . . . . . . . . . . 12 |- ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) -> ( abs o. - ) : ( CC X. CC ) --> ( 0 [,) +oo ) )
21		rr2sscn2 39582	. . . . . . . . . . . . 13 |- ( RR X. RR ) C_ ( CC X. CC )
22	21	a1i 11	. . . . . . . . . . . 12 |- ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) -> ( RR X. RR ) C_ ( CC X. CC ) )
23		elmapi 7879	. . . . . . . . . . . . 13 |- ( f e. ( ( RR X. RR ) ^m NN ) -> f : NN --> ( RR X. RR ) )
24	11, 23	syl 17	. . . . . . . . . . . 12 |- ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) -> f : NN --> ( RR X. RR ) )
25	20, 22, 24	fcoss 39402	. . . . . . . . . . 11 |- ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) -> ( ( abs o. - ) o. f ) : NN --> ( 0 [,) +oo ) )
26	25	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ) -> ( ( abs o. - ) o. f ) : NN --> ( 0 [,) +oo ) )
27		eqid 2622	. . . . . . . . . 10 |- seq 1 ( + , ( ( abs o. - ) o. f ) ) = seq 1 ( + , ( ( abs o. - ) o. f ) )
28	14, 15, 26, 27	sge0seq 40663	. . . . . . . . 9 |- ( ( ph /\ f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ) -> (Σ^ ` ( ( abs o. - ) o. f ) ) = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) )
29	28	eqcomd 2628	. . . . . . . 8 |- ( ( ph /\ f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ) -> sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) = (Σ^ ` ( ( abs o. - ) o. f ) ) )
30	29	eqeq2d 2632	. . . . . . 7 |- ( ( ph /\ f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ) -> ( y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) <-> y = (Σ^ ` ( ( abs o. - ) o. f ) ) ) )
31	30	anbi2d 740	. . . . . 6 |- ( ( ph /\ f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ) -> ( ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) <-> ( A C_ U. ran ( (,) o. f ) /\ y = (Σ^ ` ( ( abs o. - ) o. f ) ) ) ) )
32	31	rexbidva 3049	. . . . 5 |- ( ph -> ( E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) <-> E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = (Σ^ ` ( ( abs o. - ) o. f ) ) ) ) )
33	32	rabbidv 3189	. . . 4 |- ( ph -> { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = (Σ^ ` ( ( abs o. - ) o. f ) ) ) } )
34		ovolval2.m	. . . . . 6 |- M = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = (Σ^ ` ( ( abs o. - ) o. f ) ) ) }
35	34	eqcomi 2631	. . . . 5 |- { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = (Σ^ ` ( ( abs o. - ) o. f ) ) ) } = M
36	35	a1i 11	. . . 4 |- ( ph -> { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = (Σ^ ` ( ( abs o. - ) o. f ) ) ) } = M )
37	5, 33, 36	3eqtrd 2660	. . 3 |- ( ph -> { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } = M )
38	37	infeq1d 8383	. 2 |- ( ph -> inf ( { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } , RR* , < ) = inf ( M , RR* , < ) )
39	4, 38	eqtrd 2656	1 |- ( ph -> ( vol* ` A ) = inf ( M , RR* , < ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   U.cuni 4436   X. cxp 5112   ran crn 5115   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   supcsup 8346  infcinf 8347   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   ZZcz 11377   (,)cioo 12175   [,)cico 12177   seqcseq 12801   abscabs 13974   vol*covol 23231  Σ^{^}csumge0 40579

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

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-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-om 7066  df-1st 7168  df-2nd 7169  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-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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  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-sum 14417  df-ovol 23233  df-sumge0 40580

This theorem is referenced by:  ovolval3  40861