Theorem meaiuninc 40698

Description: Measures are continuous from below (bounded case): if E is a sequence of non-decreasing measurable sets (with bounded measure) then the measure of the union is the limit of the measures. This is Proposition 112C (e) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Hypotheses

Ref	Expression
meaiuninc.m	|- ( ph -> M e. Meas )
meaiuninc.n	|- ( ph -> N e. ZZ )
meaiuninc.z	|- Z = ( ZZ>= ` N )
meaiuninc.e	|- ( ph -> E : Z --> dom M )
meaiuninc.i	|- ( ( ph /\ n e. Z ) -> ( E ` n ) C_ ( E ` ( n + 1 ) ) )
meaiuninc.x	|- ( ph -> E. x e. RR A. n e. Z ( M ` ( E ` n ) ) <_ x )
meaiuninc.s	|- S = ( n e. Z |-> ( M ` ( E ` n ) ) )

Assertion

Ref	Expression
meaiuninc	|- ( ph -> S ~~> ( M ` U_ n e. Z ( E ` n ) ) )

Distinct variable groups:   n, E, x   n, M, x   n, N, x   n, Z, x   ph, n, x

Allowed substitution hints:   S( x, n)

Proof of Theorem meaiuninc

Dummy variables k i m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		meaiuninc.s	. . . 4 |- S = ( n e. Z |-> ( M ` ( E ` n ) ) )
2		fveq2 6191	. . . . . 6 |- ( n = m -> ( E ` n ) = ( E ` m ) )
3	2	fveq2d 6195	. . . . 5 |- ( n = m -> ( M ` ( E ` n ) ) = ( M ` ( E ` m ) ) )
4	3	cbvmptv 4750	. . . 4 |- ( n e. Z |-> ( M ` ( E ` n ) ) ) = ( m e. Z |-> ( M ` ( E ` m ) ) )
5	1, 4	eqtri 2644	. . 3 |- S = ( m e. Z |-> ( M ` ( E ` m ) ) )
6	5	a1i 11	. 2 |- ( ph -> S = ( m e. Z |-> ( M ` ( E ` m ) ) ) )
7		meaiuninc.m	. . 3 |- ( ph -> M e. Meas )
8		meaiuninc.n	. . 3 |- ( ph -> N e. ZZ )
9		meaiuninc.z	. . 3 |- Z = ( ZZ>= ` N )
10		meaiuninc.e	. . 3 |- ( ph -> E : Z --> dom M )
11		meaiuninc.i	. . 3 |- ( ( ph /\ n e. Z ) -> ( E ` n ) C_ ( E ` ( n + 1 ) ) )
12		meaiuninc.x	. . 3 |- ( ph -> E. x e. RR A. n e. Z ( M ` ( E ` n ) ) <_ x )
13	5, 1	eqtr3i 2646	. . 3 |- ( m e. Z |-> ( M ` ( E ` m ) ) ) = ( n e. Z |-> ( M ` ( E ` n ) ) )
14		fveq2 6191	. . . . . . 7 |- ( k = i -> ( E ` k ) = ( E ` i ) )
15	14	cbviunv 4559	. . . . . 6 |- U_ k e. ( N..^ m ) ( E ` k ) = U_ i e. ( N..^ m ) ( E ` i )
16	15	difeq2i 3725	. . . . 5 |- ( ( E ` m ) \ U_ k e. ( N..^ m ) ( E ` k ) ) = ( ( E ` m ) \ U_ i e. ( N..^ m ) ( E ` i ) )
17	16	mpteq2i 4741	. . . 4 |- ( m e. Z |-> ( ( E ` m ) \ U_ k e. ( N..^ m ) ( E ` k ) ) ) = ( m e. Z |-> ( ( E ` m ) \ U_ i e. ( N..^ m ) ( E ` i ) ) )
18		fveq2 6191	. . . . . 6 |- ( m = n -> ( E ` m ) = ( E ` n ) )
19		oveq2 6658	. . . . . . 7 |- ( m = n -> ( N..^ m ) = ( N..^ n ) )
20	19	iuneq1d 4545	. . . . . 6 |- ( m = n -> U_ i e. ( N..^ m ) ( E ` i ) = U_ i e. ( N..^ n ) ( E ` i ) )
21	18, 20	difeq12d 3729	. . . . 5 |- ( m = n -> ( ( E ` m ) \ U_ i e. ( N..^ m ) ( E ` i ) ) = ( ( E ` n ) \ U_ i e. ( N..^ n ) ( E ` i ) ) )
22	21	cbvmptv 4750	. . . 4 |- ( m e. Z |-> ( ( E ` m ) \ U_ i e. ( N..^ m ) ( E ` i ) ) ) = ( n e. Z |-> ( ( E ` n ) \ U_ i e. ( N..^ n ) ( E ` i ) ) )
23	17, 22	eqtri 2644	. . 3 |- ( m e. Z |-> ( ( E ` m ) \ U_ k e. ( N..^ m ) ( E ` k ) ) ) = ( n e. Z |-> ( ( E ` n ) \ U_ i e. ( N..^ n ) ( E ` i ) ) )
24	7, 8, 9, 10, 11, 12, 13, 23	meaiuninclem 40697	. 2 |- ( ph -> ( m e. Z |-> ( M ` ( E ` m ) ) ) ~~> ( M ` U_ n e. Z ( E ` n ) ) )
25	6, 24	eqbrtrd 4675	1 |- ( ph -> S ~~> ( M ` U_ n e. Z ( E ` n ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   \ cdif 3571   C_ wss 3574   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   + caddc 9939   <_ cle 10075   ZZcz 11377   ZZ>=cuz 11687  ..^cfzo 12465   ~~> cli 14215  Meascmea 40666

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-disj 4621  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-omul 7565  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-acn 8768  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-xadd 11947  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-salg 40529  df-sumge0 40580  df-mea 40667

This theorem is referenced by:  meaiuninc2  40699