Theorem smflimsuplem6 41031

Description: The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
smflimsuplem6.a	|- F/ n ph
smflimsuplem6.b	|- F/ m ph
smflimsuplem6.m	|- ( ph -> M e. ZZ )
smflimsuplem6.z	|- Z = ( ZZ>= ` M )
smflimsuplem6.s	|- ( ph -> S e. SAlg )
smflimsuplem6.f	|- ( ph -> F : Z --> (SMblFn ` S ) )
smflimsuplem6.e	|- E = ( n e. Z |-> { x e. |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | sup ( ran ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) , RR* , < ) e. RR } )
smflimsuplem6.h	|- H = ( n e. Z |-> ( x e. ( E ` n ) |-> sup ( ran ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) , RR* , < ) ) )
smflimsuplem6.r	|- ( ph -> ( limsup ` ( m e. Z |-> ( ( F ` m ) ` X ) ) ) e. RR )
smflimsuplem6.n	|- ( ph -> N e. Z )
smflimsuplem6.x	|- ( ph -> X e. |^|_ m e. ( ZZ>= ` N ) dom ( F ` m ) )

Assertion

Ref	Expression
smflimsuplem6	|- ( ph -> ( n e. Z |-> ( ( H ` n ) ` X ) ) e. dom ~~> )

Distinct variable groups:   n, F, x   m, M   m, N, n   m, X, n   m, Z, n, x

Allowed substitution hints:   ph( x, m, n)   S( x, m, n)   E( x, m, n)   F( m)   H( x, m, n)   M( x, n)   N( x)   X( x)

Proof of Theorem smflimsuplem6

Step	Hyp	Ref	Expression
1		smflimsuplem6.z	. . . . 5 |- Z = ( ZZ>= ` M )
2	1	fvexi 6202	. . . 4 |- Z e. _V
3	2	a1i 11	. . 3 |- ( ph -> Z e. _V )
4	3	mptexd 6487	. 2 |- ( ph -> ( n e. Z |-> ( ( H ` n ) ` X ) ) e. _V )
5		fvexd 6203	. 2 |- ( ph -> ( limsup ` ( m e. ( ZZ>= ` N ) |-> ( ( F ` m ) ` X ) ) ) e. _V )
6		smflimsuplem6.a	. . . 4 |- F/ n ph
7		smflimsuplem6.b	. . . 4 |- F/ m ph
8		smflimsuplem6.m	. . . 4 |- ( ph -> M e. ZZ )
9		smflimsuplem6.s	. . . 4 |- ( ph -> S e. SAlg )
10		smflimsuplem6.f	. . . 4 |- ( ph -> F : Z --> (SMblFn ` S ) )
11		smflimsuplem6.e	. . . 4 |- E = ( n e. Z |-> { x e. |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | sup ( ran ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) , RR* , < ) e. RR } )
12		smflimsuplem6.h	. . . 4 |- H = ( n e. Z |-> ( x e. ( E ` n ) |-> sup ( ran ( m e. ( ZZ>= ` n ) |-> ( ( F ` m ) ` x ) ) , RR* , < ) ) )
13		smflimsuplem6.r	. . . 4 |- ( ph -> ( limsup ` ( m e. Z |-> ( ( F ` m ) ` X ) ) ) e. RR )
14		smflimsuplem6.n	. . . 4 |- ( ph -> N e. Z )
15		smflimsuplem6.x	. . . 4 |- ( ph -> X e. |^|_ m e. ( ZZ>= ` N ) dom ( F ` m ) )
16	6, 7, 8, 1, 9, 10, 11, 12, 13, 14, 15	smflimsuplem5 41030	. . 3 |- ( ph -> ( n e. ( ZZ>= ` N ) |-> ( ( H ` n ) ` X ) ) ~~> ( limsup ` ( m e. ( ZZ>= ` N ) |-> ( ( F ` m ) ` X ) ) ) )
17		fvexd 6203	. . . 4 |- ( ph -> ( ZZ>= ` N ) e. _V )
18	1	eluzelz2 39627	. . . . 5 |- ( N e. Z -> N e. ZZ )
19	14, 18	syl 17	. . . 4 |- ( ph -> N e. ZZ )
20		eqid 2622	. . . 4 |- ( ZZ>= ` N ) = ( ZZ>= ` N )
21	1	eleq2i 2693	. . . . . . . 8 |- ( N e. Z <-> N e. ( ZZ>= ` M ) )
22	21	biimpi 206	. . . . . . 7 |- ( N e. Z -> N e. ( ZZ>= ` M ) )
23		uzss 11708	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> ( ZZ>= ` N ) C_ ( ZZ>= ` M ) )
24	22, 23	syl 17	. . . . . 6 |- ( N e. Z -> ( ZZ>= ` N ) C_ ( ZZ>= ` M ) )
25	24, 1	syl6sseqr 3652	. . . . 5 |- ( N e. Z -> ( ZZ>= ` N ) C_ Z )
26	14, 25	syl 17	. . . 4 |- ( ph -> ( ZZ>= ` N ) C_ Z )
27		ssid 3624	. . . . 5 |- ( ZZ>= ` N ) C_ ( ZZ>= ` N )
28	27	a1i 11	. . . 4 |- ( ph -> ( ZZ>= ` N ) C_ ( ZZ>= ` N ) )
29		fvexd 6203	. . . 4 |- ( ( ph /\ n e. ( ZZ>= ` N ) ) -> ( ( H ` n ) ` X ) e. _V )
30	6, 3, 17, 19, 20, 26, 28, 29	climeqmpt 39929	. . 3 |- ( ph -> ( ( n e. Z |-> ( ( H ` n ) ` X ) ) ~~> ( limsup ` ( m e. ( ZZ>= ` N ) |-> ( ( F ` m ) ` X ) ) ) <-> ( n e. ( ZZ>= ` N ) |-> ( ( H ` n ) ` X ) ) ~~> ( limsup ` ( m e. ( ZZ>= ` N ) |-> ( ( F ` m ) ` X ) ) ) ) )
31	16, 30	mpbird 247	. 2 |- ( ph -> ( n e. Z |-> ( ( H ` n ) ` X ) ) ~~> ( limsup ` ( m e. ( ZZ>= ` N ) |-> ( ( F ` m ) ` X ) ) ) )
32		breldmg 5330	. 2 |- ( ( ( n e. Z |-> ( ( H ` n ) ` X ) ) e. _V /\ ( limsup ` ( m e. ( ZZ>= ` N ) |-> ( ( F ` m ) ` X ) ) ) e. _V /\ ( n e. Z |-> ( ( H ` n ) ` X ) ) ~~> ( limsup ` ( m e. ( ZZ>= ` N ) |-> ( ( F ` m ) ` X ) ) ) ) -> ( n e. Z |-> ( ( H ` n ) ` X ) ) e. dom ~~> )
33	4, 5, 31, 32	syl3anc 1326	1 |- ( ph -> ( n e. Z |-> ( ( H ` n ) ` X ) ) e. dom ~~> )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   |^|_ciin 4521   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888   supcsup 8346   RRcr 9935   RR*cxr 10073   < clt 10074   ZZcz 11377   ZZ>=cuz 11687   limsupclsp 14201   ~~> cli 14215  SAlgcsalg 40528  SMblFncsmblfn 40909

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-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-int 4476  df-iun 4522  df-iin 4523  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-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-ioo 12179  df-ico 12181  df-fz 12327  df-fl 12593  df-ceil 12594  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-smblfn 40910

This theorem is referenced by:  smflimsuplem7  41032