Theorem ismbl 23294

Description: The predicate " A is Lebesgue-measurable". A set is measurable if it splits every other set x in a "nice" way, that is, if the measure of the pieces x i^i A and x \ A sum up to the measure of x (assuming that the measure of x is a real number, so that this addition makes sense). (Contributed by Mario Carneiro, 17-Mar-2014.)

Assertion

Ref	Expression
ismbl	|- ( A e. dom vol <-> ( A C_ RR /\ A. x e. ~P RR ( ( vol* ` x ) e. RR -> ( vol* ` x ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) ) ) )

Distinct variable group:   x, A

Proof of Theorem ismbl

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ineq2 3808	. . . . . . 7 |- ( y = A -> ( x i^i y ) = ( x i^i A ) )
2	1	fveq2d 6195	. . . . . 6 |- ( y = A -> ( vol* ` ( x i^i y ) ) = ( vol* ` ( x i^i A ) ) )
3		difeq2 3722	. . . . . . 7 |- ( y = A -> ( x \ y ) = ( x \ A ) )
4	3	fveq2d 6195	. . . . . 6 |- ( y = A -> ( vol* ` ( x \ y ) ) = ( vol* ` ( x \ A ) ) )
5	2, 4	oveq12d 6668	. . . . 5 |- ( y = A -> ( ( vol* ` ( x i^i y ) ) + ( vol* ` ( x \ y ) ) ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) )
6	5	eqeq2d 2632	. . . 4 |- ( y = A -> ( ( vol* ` x ) = ( ( vol* ` ( x i^i y ) ) + ( vol* ` ( x \ y ) ) ) <-> ( vol* ` x ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) ) )
7	6	ralbidv 2986	. . 3 |- ( y = A -> ( A. x e. ( `' vol* " RR ) ( vol* ` x ) = ( ( vol* ` ( x i^i y ) ) + ( vol* ` ( x \ y ) ) ) <-> A. x e. ( `' vol* " RR ) ( vol* ` x ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) ) )
8		df-vol 23234	. . . . . 6 |- vol = ( vol* |` { y | A. x e. ( `' vol* " RR ) ( vol* ` x ) = ( ( vol* ` ( x i^i y ) ) + ( vol* ` ( x \ y ) ) ) } )
9	8	dmeqi 5325	. . . . 5 |- dom vol = dom ( vol* |` { y | A. x e. ( `' vol* " RR ) ( vol* ` x ) = ( ( vol* ` ( x i^i y ) ) + ( vol* ` ( x \ y ) ) ) } )
10		dmres 5419	. . . . 5 |- dom ( vol* |` { y | A. x e. ( `' vol* " RR ) ( vol* ` x ) = ( ( vol* ` ( x i^i y ) ) + ( vol* ` ( x \ y ) ) ) } ) = ( { y | A. x e. ( `' vol* " RR ) ( vol* ` x ) = ( ( vol* ` ( x i^i y ) ) + ( vol* ` ( x \ y ) ) ) } i^i dom vol* )
11		ovolf 23250	. . . . . . 7 |- vol* : ~P RR --> ( 0 [,] +oo )
12	11	fdmi 6052	. . . . . 6 |- dom vol* = ~P RR
13	12	ineq2i 3811	. . . . 5 |- ( { y | A. x e. ( `' vol* " RR ) ( vol* ` x ) = ( ( vol* ` ( x i^i y ) ) + ( vol* ` ( x \ y ) ) ) } i^i dom vol* ) = ( { y | A. x e. ( `' vol* " RR ) ( vol* ` x ) = ( ( vol* ` ( x i^i y ) ) + ( vol* ` ( x \ y ) ) ) } i^i ~P RR )
14	9, 10, 13	3eqtri 2648	. . . 4 |- dom vol = ( { y | A. x e. ( `' vol* " RR ) ( vol* ` x ) = ( ( vol* ` ( x i^i y ) ) + ( vol* ` ( x \ y ) ) ) } i^i ~P RR )
15		dfrab2 3903	. . . 4 |- { y e. ~P RR | A. x e. ( `' vol* " RR ) ( vol* ` x ) = ( ( vol* ` ( x i^i y ) ) + ( vol* ` ( x \ y ) ) ) } = ( { y | A. x e. ( `' vol* " RR ) ( vol* ` x ) = ( ( vol* ` ( x i^i y ) ) + ( vol* ` ( x \ y ) ) ) } i^i ~P RR )
16	14, 15	eqtr4i 2647	. . 3 |- dom vol = { y e. ~P RR | A. x e. ( `' vol* " RR ) ( vol* ` x ) = ( ( vol* ` ( x i^i y ) ) + ( vol* ` ( x \ y ) ) ) }
17	7, 16	elrab2 3366	. 2 |- ( A e. dom vol <-> ( A e. ~P RR /\ A. x e. ( `' vol* " RR ) ( vol* ` x ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) ) )
18		reex 10027	. . . 4 |- RR e. _V
19	18	elpw2 4828	. . 3 |- ( A e. ~P RR <-> A C_ RR )
20		ffn 6045	. . . . . . 7 |- ( vol* : ~P RR --> ( 0 [,] +oo ) -> vol* Fn ~P RR )
21		elpreima 6337	. . . . . . 7 |- ( vol* Fn ~P RR -> ( x e. ( `' vol* " RR ) <-> ( x e. ~P RR /\ ( vol* ` x ) e. RR ) ) )
22	11, 20, 21	mp2b 10	. . . . . 6 |- ( x e. ( `' vol* " RR ) <-> ( x e. ~P RR /\ ( vol* ` x ) e. RR ) )
23	22	imbi1i 339	. . . . 5 |- ( ( x e. ( `' vol* " RR ) -> ( vol* ` x ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) ) <-> ( ( x e. ~P RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` x ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) ) )
24		impexp 462	. . . . 5 |- ( ( ( x e. ~P RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` x ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) ) <-> ( x e. ~P RR -> ( ( vol* ` x ) e. RR -> ( vol* ` x ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) ) ) )
25	23, 24	bitri 264	. . . 4 |- ( ( x e. ( `' vol* " RR ) -> ( vol* ` x ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) ) <-> ( x e. ~P RR -> ( ( vol* ` x ) e. RR -> ( vol* ` x ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) ) ) )
26	25	ralbii2 2978	. . 3 |- ( A. x e. ( `' vol* " RR ) ( vol* ` x ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <-> A. x e. ~P RR ( ( vol* ` x ) e. RR -> ( vol* ` x ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) ) )
27	19, 26	anbi12i 733	. 2 |- ( ( A e. ~P RR /\ A. x e. ( `' vol* " RR ) ( vol* ` x ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) ) <-> ( A C_ RR /\ A. x e. ~P RR ( ( vol* ` x ) e. RR -> ( vol* ` x ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) ) ) )
28	17, 27	bitri 264	1 |- ( A e. dom vol <-> ( A C_ RR /\ A. x e. ~P RR ( ( vol* ` x ) e. RR -> ( vol* ` x ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   `'ccnv 5113   dom cdm 5114   |` cres 5116   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   +oocpnf 10071   [,]cicc 12178   vol*covol 23231   volcvol 23232

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-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-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-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-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-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-er 7742  df-map 7859  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-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-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-ovol 23233  df-vol 23234

This theorem is referenced by:  ismbl2  23295  mblss  23299  mblsplit  23300  cmmbl  23302  shftmbl  23306  voliunlem2  23319