Theorem ismbl3 40203

Description: The predicate " A is Lebesgue-measurable". Similar to ismbl2 23295, but here +e is used, and the precondition ( vol* ` x ) e. RR can be dropped. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Assertion

Ref	Expression
ismbl3	|- ( A e. dom vol <-> ( A C_ RR /\ A. x e. ~P RR ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) )

Distinct variable group:   x, A

Proof of Theorem ismbl3

Step	Hyp	Ref	Expression
1		ismbl2 23295	. 2 |- ( A e. dom vol <-> ( A C_ RR /\ A. x e. ~P RR ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) ) )
2		inss1 3833	. . . . . . . . . . . 12 |- ( x i^i A ) C_ x
3	2	a1i 11	. . . . . . . . . . 11 |- ( ( x e. ~P RR /\ ( vol* ` x ) e. RR ) -> ( x i^i A ) C_ x )
4		elpwi 4168	. . . . . . . . . . . 12 |- ( x e. ~P RR -> x C_ RR )
5	4	adantr 481	. . . . . . . . . . 11 |- ( ( x e. ~P RR /\ ( vol* ` x ) e. RR ) -> x C_ RR )
6		simpr 477	. . . . . . . . . . 11 |- ( ( x e. ~P RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` x ) e. RR )
7		ovolsscl 23254	. . . . . . . . . . 11 |- ( ( ( x i^i A ) C_ x /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i A ) ) e. RR )
8	3, 5, 6, 7	syl3anc 1326	. . . . . . . . . 10 |- ( ( x e. ~P RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x i^i A ) ) e. RR )
9		difssd 3738	. . . . . . . . . . 11 |- ( ( x e. ~P RR /\ ( vol* ` x ) e. RR ) -> ( x \ A ) C_ x )
10		ovolsscl 23254	. . . . . . . . . . 11 |- ( ( ( x \ A ) C_ x /\ x C_ RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x \ A ) ) e. RR )
11	9, 5, 6, 10	syl3anc 1326	. . . . . . . . . 10 |- ( ( x e. ~P RR /\ ( vol* ` x ) e. RR ) -> ( vol* ` ( x \ A ) ) e. RR )
12	8, 11	rexaddd 12065	. . . . . . . . 9 |- ( ( x e. ~P RR /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) )
13	12	adantlr 751	. . . . . . . 8 |- ( ( ( x e. ~P RR /\ ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) ) /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) = ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) )
14		id 22	. . . . . . . . . 10 |- ( ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) -> ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) )
15	14	imp 445	. . . . . . . . 9 |- ( ( ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) )
16	15	adantll 750	. . . . . . . 8 |- ( ( ( x e. ~P RR /\ ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) ) /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) )
17	13, 16	eqbrtrd 4675	. . . . . . 7 |- ( ( ( x e. ~P RR /\ ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) ) /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) )
18	2, 4	syl5ss 3614	. . . . . . . . . . . . 13 |- ( x e. ~P RR -> ( x i^i A ) C_ RR )
19		ovolcl 23246	. . . . . . . . . . . . 13 |- ( ( x i^i A ) C_ RR -> ( vol* ` ( x i^i A ) ) e. RR* )
20	18, 19	syl 17	. . . . . . . . . . . 12 |- ( x e. ~P RR -> ( vol* ` ( x i^i A ) ) e. RR* )
21	4	ssdifssd 3748	. . . . . . . . . . . . 13 |- ( x e. ~P RR -> ( x \ A ) C_ RR )
22		ovolcl 23246	. . . . . . . . . . . . 13 |- ( ( x \ A ) C_ RR -> ( vol* ` ( x \ A ) ) e. RR* )
23	21, 22	syl 17	. . . . . . . . . . . 12 |- ( x e. ~P RR -> ( vol* ` ( x \ A ) ) e. RR* )
24	20, 23	xaddcld 12131	. . . . . . . . . . 11 |- ( x e. ~P RR -> ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) e. RR* )
25		pnfge 11964	. . . . . . . . . . 11 |- ( ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) e. RR* -> ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) <_ +oo )
26	24, 25	syl 17	. . . . . . . . . 10 |- ( x e. ~P RR -> ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) <_ +oo )
27	26	adantr 481	. . . . . . . . 9 |- ( ( x e. ~P RR /\ -. ( vol* ` x ) e. RR ) -> ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) <_ +oo )
28		ovolf 23250	. . . . . . . . . . . . 13 |- vol* : ~P RR --> ( 0 [,] +oo )
29	28	ffvelrni 6358	. . . . . . . . . . . 12 |- ( x e. ~P RR -> ( vol* ` x ) e. ( 0 [,] +oo ) )
30	29	adantr 481	. . . . . . . . . . 11 |- ( ( x e. ~P RR /\ -. ( vol* ` x ) e. RR ) -> ( vol* ` x ) e. ( 0 [,] +oo ) )
31		simpr 477	. . . . . . . . . . 11 |- ( ( x e. ~P RR /\ -. ( vol* ` x ) e. RR ) -> -. ( vol* ` x ) e. RR )
32		xrge0nre 12277	. . . . . . . . . . 11 |- ( ( ( vol* ` x ) e. ( 0 [,] +oo ) /\ -. ( vol* ` x ) e. RR ) -> ( vol* ` x ) = +oo )
33	30, 31, 32	syl2anc 693	. . . . . . . . . 10 |- ( ( x e. ~P RR /\ -. ( vol* ` x ) e. RR ) -> ( vol* ` x ) = +oo )
34	33	eqcomd 2628	. . . . . . . . 9 |- ( ( x e. ~P RR /\ -. ( vol* ` x ) e. RR ) -> +oo = ( vol* ` x ) )
35	27, 34	breqtrd 4679	. . . . . . . 8 |- ( ( x e. ~P RR /\ -. ( vol* ` x ) e. RR ) -> ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) )
36	35	adantlr 751	. . . . . . 7 |- ( ( ( x e. ~P RR /\ ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) ) /\ -. ( vol* ` x ) e. RR ) -> ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) )
37	17, 36	pm2.61dan 832	. . . . . 6 |- ( ( x e. ~P RR /\ ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) ) -> ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) )
38	37	ex 450	. . . . 5 |- ( x e. ~P RR -> ( ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) -> ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) )
39	12	eqcomd 2628	. . . . . . . 8 |- ( ( x e. ~P RR /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) = ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) )
40	39	3adant2 1080	. . . . . . 7 |- ( ( x e. ~P RR /\ ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) = ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) )
41		simp2 1062	. . . . . . 7 |- ( ( x e. ~P RR /\ ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) )
42	40, 41	eqbrtrd 4675	. . . . . 6 |- ( ( x e. ~P RR /\ ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) /\ ( vol* ` x ) e. RR ) -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) )
43	42	3exp 1264	. . . . 5 |- ( x e. ~P RR -> ( ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) -> ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) ) )
44	38, 43	impbid 202	. . . 4 |- ( x e. ~P RR -> ( ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) <-> ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) )
45	44	ralbiia 2979	. . 3 |- ( A. x e. ~P RR ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) <-> A. x e. ~P RR ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) )
46	45	anbi2i 730	. 2 |- ( ( A C_ RR /\ A. x e. ~P RR ( ( vol* ` x ) e. RR -> ( ( vol* ` ( x i^i A ) ) + ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) ) <-> ( A C_ RR /\ A. x e. ~P RR ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) )
47	1, 46	bitri 264	1 |- ( A e. dom vol <-> ( A C_ RR /\ A. x e. ~P RR ( ( vol* ` ( x i^i A ) ) +e ( vol* ` ( x \ A ) ) ) <_ ( vol* ` x ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   +oocpnf 10071   RR*cxr 10073   <_ cle 10075   +ecxad 11944   [,]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-q 11789  df-rp 11833  df-xadd 11947  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fl 12593  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:  ismbl4  40210