Theorem ovnval2 40759

Description: Value of the Lebesgue outer measure of a subset A of the space of multidimensional real numbers. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
ovnval2.1	|- ( ph -> X e. Fin )
ovnval2.2	|- ( ph -> A C_ ( RR ^m X ) )
ovnval2.3	|- M = { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( A C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) }

Assertion

Ref	Expression
ovnval2	|- ( ph -> ( (voln* ` X ) ` A ) = if ( X = (/) , 0 , inf ( M , RR* , < ) ) )

Distinct variable groups:   A, i, z   i, X, j, k, z

Allowed substitution hints:   ph( z, i, j, k)   A( j, k)   M( z, i, j, k)

Proof of Theorem ovnval2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovnval2.1	. . 3 |- ( ph -> X e. Fin )
2	1	ovnval 40755	. 2 |- ( ph -> (voln* ` X ) = ( y e. ~P ( RR ^m X ) |-> if ( X = (/) , 0 , inf ( { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( y C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) } , RR* , < ) ) ) )
3		biidd 252	. . . 4 |- ( y = A -> ( X = (/) <-> X = (/) ) )
4		sseq1 3626	. . . . . . . . 9 |- ( y = A -> ( y C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) <-> A C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) ) )
5	4	anbi1d 741	. . . . . . . 8 |- ( y = A -> ( ( y C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) <-> ( A C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) ) )
6	5	rexbidv 3052	. . . . . . 7 |- ( y = A -> ( E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( y C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) <-> E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( A C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) ) )
7	6	rabbidv 3189	. . . . . 6 |- ( y = A -> { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( y C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) } = { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( A C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) } )
8		ovnval2.3	. . . . . 6 |- M = { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( A C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) }
9	7, 8	syl6eqr 2674	. . . . 5 |- ( y = A -> { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( y C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) } = M )
10	9	infeq1d 8383	. . . 4 |- ( y = A -> inf ( { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( y C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) } , RR* , < ) = inf ( M , RR* , < ) )
11	3, 10	ifbieq2d 4111	. . 3 |- ( y = A -> if ( X = (/) , 0 , inf ( { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( y C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) } , RR* , < ) ) = if ( X = (/) , 0 , inf ( M , RR* , < ) ) )
12	11	adantl 482	. 2 |- ( ( ph /\ y = A ) -> if ( X = (/) , 0 , inf ( { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( y C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) } , RR* , < ) ) = if ( X = (/) , 0 , inf ( M , RR* , < ) ) )
13		ovnval2.2	. . 3 |- ( ph -> A C_ ( RR ^m X ) )
14		ovexd 6680	. . . . 5 |- ( ph -> ( RR ^m X ) e. _V )
15	14, 13	ssexd 4805	. . . 4 |- ( ph -> A e. _V )
16		elpwg 4166	. . . 4 |- ( A e. _V -> ( A e. ~P ( RR ^m X ) <-> A C_ ( RR ^m X ) ) )
17	15, 16	syl 17	. . 3 |- ( ph -> ( A e. ~P ( RR ^m X ) <-> A C_ ( RR ^m X ) ) )
18	13, 17	mpbird 247	. 2 |- ( ph -> A e. ~P ( RR ^m X ) )
19		c0ex 10034	. . . 4 |- 0 e. _V
20	19	a1i 11	. . 3 |- ( ph -> 0 e. _V )
21	8	infeq1i 8384	. . . 4 |- inf ( M , RR* , < ) = inf ( { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( A C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) } , RR* , < )
22		xrltso 11974	. . . . . 6 |- < Or RR*
23	22	infex 8399	. . . . 5 |- inf ( { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( A C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) } , RR* , < ) e. _V
24	23	a1i 11	. . . 4 |- ( ph -> inf ( { z e. RR* | E. i e. ( ( ( RR X. RR ) ^m X ) ^m NN ) ( A C_ U_ j e. NN X_ k e. X ( ( [,) o. ( i ` j ) ) ` k ) /\ z = (Σ^ ` ( j e. NN |-> prod_ k e. X ( vol ` ( ( [,) o. ( i ` j ) ) ` k ) ) ) ) ) } , RR* , < ) e. _V )
25	21, 24	syl5eqel 2705	. . 3 |- ( ph -> inf ( M , RR* , < ) e. _V )
26	20, 25	ifcld 4131	. 2 |- ( ph -> if ( X = (/) , 0 , inf ( M , RR* , < ) ) e. _V )
27	2, 12, 18, 26	fvmptd 6288	1 |- ( ph -> ( (voln* ` X ) ` A ) = if ( X = (/) , 0 , inf ( M , RR* , < ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   U_ciun 4520   |-> cmpt 4729   X. cxp 5112   o. ccom 5118   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   X_cixp 7908   Fincfn 7955  infcinf 8347   RRcr 9935   0cc0 9936   RR*cxr 10073   < clt 10074   NNcn 11020   [,)cico 12177   prod_cprod 14635   volcvol 23232  Σ^{^}csumge0 40579  voln^*covoln 40750

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-mulcl 9998  ax-i2m1 10004  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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-pred 5680  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-ixp 7909  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-seq 12802  df-prod 14636  df-ovoln 40751

This theorem is referenced by:  ovn0val  40764  ovnn0val  40765  ovnval2b  40766  ovn0  40780  ovnhoilem1  40815  ovnovollem3  40872