Theorem ovolval 23242

Description: The value of the outer measure. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 17-Sep-2020.)

Hypothesis

Ref	Expression
ovolval.1	|- M = { 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* , < ) ) }

Assertion

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

Distinct variable group:   y, f, A

Allowed substitution hints:   M( y, f)

Proof of Theorem ovolval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reex 10027	. . 3 |- RR e. _V
2	1	elpw2 4828	. 2 |- ( A e. ~P RR <-> A C_ RR )
3		sseq1 3626	. . . . . . . 8 |- ( x = A -> ( x C_ U. ran ( (,) o. f ) <-> A C_ U. ran ( (,) o. f ) ) )
4	3	anbi1d 741	. . . . . . 7 |- ( x = A -> ( ( x C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) <-> ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) ) )
5	4	rexbidv 3052	. . . . . 6 |- ( x = A -> ( E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( x 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 = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) ) )
6	5	rabbidv 3189	. . . . 5 |- ( x = A -> { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( x 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* , < ) ) } )
7		ovolval.1	. . . . 5 |- M = { 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* , < ) ) }
8	6, 7	syl6eqr 2674	. . . 4 |- ( x = A -> { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( x C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } = M )
9	8	infeq1d 8383	. . 3 |- ( x = A -> inf ( { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( x C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } , RR* , < ) = inf ( M , RR* , < ) )
10		df-ovol 23233	. . 3 |- vol* = ( x e. ~P RR |-> inf ( { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( x C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } , RR* , < ) )
11		xrltso 11974	. . . 4 |- < Or RR*
12	11	infex 8399	. . 3 |- inf ( M , RR* , < ) e. _V
13	9, 10, 12	fvmpt 6282	. 2 |- ( A e. ~P RR -> ( vol* ` A ) = inf ( M , RR* , < ) )
14	2, 13	sylbir 225	1 |- ( A C_ RR -> ( vol* ` A ) = inf ( M , RR* , < ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   X. cxp 5112   ran crn 5115   o. ccom 5118   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   supcsup 8346  infcinf 8347   RRcr 9935   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   (,)cioo 12175   seqcseq 12801   abscabs 13974   vol*covol 23231

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-ovol 23233

This theorem is referenced by:  ovolcl  23246  ovollb  23247  ovolgelb  23248  ovolge0  23249  ovolsslem  23252  ovolshft  23279  ovolicc2  23290  ismblfin  33450  ovolval2  40858