Theorem smfliminf 41037

Description: The inferior limit of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (e) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
smfliminf.n	|- F/_ m F
smfliminf.x	|- F/_ x F
smfliminf.m	|- ( ph -> M e. ZZ )
smfliminf.z	|- Z = ( ZZ>= ` M )
smfliminf.s	|- ( ph -> S e. SAlg )
smfliminf.f	|- ( ph -> F : Z --> (SMblFn ` S ) )
smfliminf.d	|- D = { x e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | (liminf ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) e. RR }
smfliminf.g	|- G = ( x e. D |-> (liminf ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) )

Assertion

Ref	Expression
smfliminf	|- ( ph -> G e. (SMblFn ` S ) )

Distinct variable groups:   n, F   m, Z, n, x

Allowed substitution hints:   ph( x, m, n)   D( x, m, n)   S( x, m, n)   F( x, m)   G( x, m, n)   M( x, m, n)

Proof of Theorem smfliminf

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

Step	Hyp	Ref	Expression
1		smfliminf.m	. 2 |- ( ph -> M e. ZZ )
2		smfliminf.z	. 2 |- Z = ( ZZ>= ` M )
3		smfliminf.s	. 2 |- ( ph -> S e. SAlg )
4		smfliminf.f	. 2 |- ( ph -> F : Z --> (SMblFn ` S ) )
5		smfliminf.d	. . 3 |- D = { x e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | (liminf ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) e. RR }
6		nfcv 2764	. . . . 5 |- F/_ x Z
7		nfcv 2764	. . . . . 6 |- F/_ x ( ZZ>= ` n )
8		smfliminf.x	. . . . . . . 8 |- F/_ x F
9		nfcv 2764	. . . . . . . 8 |- F/_ x m
10	8, 9	nffv 6198	. . . . . . 7 |- F/_ x ( F ` m )
11	10	nfdm 5367	. . . . . 6 |- F/_ x dom ( F ` m )
12	7, 11	nfiin 4549	. . . . 5 |- F/_ x |^|_ m e. ( ZZ>= ` n ) dom ( F ` m )
13	6, 12	nfiun 4548	. . . 4 |- F/_ x U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m )
14		nfcv 2764	. . . . . 6 |- F/_ x ( ZZ>= ` i )
15		nfcv 2764	. . . . . . . 8 |- F/_ x k
16	8, 15	nffv 6198	. . . . . . 7 |- F/_ x ( F ` k )
17	16	nfdm 5367	. . . . . 6 |- F/_ x dom ( F ` k )
18	14, 17	nfiin 4549	. . . . 5 |- F/_ x |^|_ k e. ( ZZ>= ` i ) dom ( F ` k )
19	6, 18	nfiun 4548	. . . 4 |- F/_ x U_ i e. Z |^|_ k e. ( ZZ>= ` i ) dom ( F ` k )
20		nfcv 2764	. . . . 5 |- F/_ i |^|_ m e. ( ZZ>= ` n ) dom ( F ` m )
21		nfcv 2764	. . . . 5 |- F/_ n |^|_ k e. ( ZZ>= ` i ) dom ( F ` k )
22		fveq2 6191	. . . . . . 7 |- ( n = i -> ( ZZ>= ` n ) = ( ZZ>= ` i ) )
23	22	iineq1d 39267	. . . . . 6 |- ( n = i -> |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) = |^|_ m e. ( ZZ>= ` i ) dom ( F ` m ) )
24		nfcv 2764	. . . . . . . . 9 |- F/_ k ( F ` m )
25	24	nfdm 5367	. . . . . . . 8 |- F/_ k dom ( F ` m )
26		smfliminf.n	. . . . . . . . . 10 |- F/_ m F
27		nfcv 2764	. . . . . . . . . 10 |- F/_ m k
28	26, 27	nffv 6198	. . . . . . . . 9 |- F/_ m ( F ` k )
29	28	nfdm 5367	. . . . . . . 8 |- F/_ m dom ( F ` k )
30		fveq2 6191	. . . . . . . . 9 |- ( m = k -> ( F ` m ) = ( F ` k ) )
31	30	dmeqd 5326	. . . . . . . 8 |- ( m = k -> dom ( F ` m ) = dom ( F ` k ) )
32	25, 29, 31	cbviin 4558	. . . . . . 7 |- |^|_ m e. ( ZZ>= ` i ) dom ( F ` m ) = |^|_ k e. ( ZZ>= ` i ) dom ( F ` k )
33	32	a1i 11	. . . . . 6 |- ( n = i -> |^|_ m e. ( ZZ>= ` i ) dom ( F ` m ) = |^|_ k e. ( ZZ>= ` i ) dom ( F ` k ) )
34	23, 33	eqtrd 2656	. . . . 5 |- ( n = i -> |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) = |^|_ k e. ( ZZ>= ` i ) dom ( F ` k ) )
35	20, 21, 34	cbviun 4557	. . . 4 |- U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) = U_ i e. Z |^|_ k e. ( ZZ>= ` i ) dom ( F ` k )
36	13, 19, 35	rabeqif 3191	. . 3 |- { x e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | (liminf ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) e. RR } = { x e. U_ i e. Z |^|_ k e. ( ZZ>= ` i ) dom ( F ` k ) | (liminf ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) e. RR }
37		nfcv 2764	. . . 4 |- F/_ y U_ i e. Z |^|_ k e. ( ZZ>= ` i ) dom ( F ` k )
38		nfv 1843	. . . 4 |- F/ y (liminf ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) e. RR
39		nfcv 2764	. . . . . 6 |- F/_ xliminf
40		nfcv 2764	. . . . . . . 8 |- F/_ x y
41	16, 40	nffv 6198	. . . . . . 7 |- F/_ x ( ( F ` k ) ` y )
42	6, 41	nfmpt 4746	. . . . . 6 |- F/_ x ( k e. Z |-> ( ( F ` k ) ` y ) )
43	39, 42	nffv 6198	. . . . 5 |- F/_ x (liminf ` ( k e. Z |-> ( ( F ` k ) ` y ) ) )
44		nfcv 2764	. . . . 5 |- F/_ x RR
45	43, 44	nfel 2777	. . . 4 |- F/ x (liminf ` ( k e. Z |-> ( ( F ` k ) ` y ) ) ) e. RR
46		nfv 1843	. . . . . . . 8 |- F/ m x = y
47		fveq2 6191	. . . . . . . . 9 |- ( x = y -> ( ( F ` m ) ` x ) = ( ( F ` m ) ` y ) )
48	47	adantr 481	. . . . . . . 8 |- ( ( x = y /\ m e. Z ) -> ( ( F ` m ) ` x ) = ( ( F ` m ) ` y ) )
49	46, 48	mpteq2da 4743	. . . . . . 7 |- ( x = y -> ( m e. Z |-> ( ( F ` m ) ` x ) ) = ( m e. Z |-> ( ( F ` m ) ` y ) ) )
50		nfcv 2764	. . . . . . . . 9 |- F/_ k ( ( F ` m ) ` y )
51		nfcv 2764	. . . . . . . . . 10 |- F/_ m y
52	28, 51	nffv 6198	. . . . . . . . 9 |- F/_ m ( ( F ` k ) ` y )
53	30	fveq1d 6193	. . . . . . . . 9 |- ( m = k -> ( ( F ` m ) ` y ) = ( ( F ` k ) ` y ) )
54	50, 52, 53	cbvmpt 4749	. . . . . . . 8 |- ( m e. Z |-> ( ( F ` m ) ` y ) ) = ( k e. Z |-> ( ( F ` k ) ` y ) )
55	54	a1i 11	. . . . . . 7 |- ( x = y -> ( m e. Z |-> ( ( F ` m ) ` y ) ) = ( k e. Z |-> ( ( F ` k ) ` y ) ) )
56	49, 55	eqtrd 2656	. . . . . 6 |- ( x = y -> ( m e. Z |-> ( ( F ` m ) ` x ) ) = ( k e. Z |-> ( ( F ` k ) ` y ) ) )
57	56	fveq2d 6195	. . . . 5 |- ( x = y -> (liminf ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) = (liminf ` ( k e. Z |-> ( ( F ` k ) ` y ) ) ) )
58	57	eleq1d 2686	. . . 4 |- ( x = y -> ( (liminf ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) e. RR <-> (liminf ` ( k e. Z |-> ( ( F ` k ) ` y ) ) ) e. RR ) )
59	19, 37, 38, 45, 58	cbvrab 3198	. . 3 |- { x e. U_ i e. Z |^|_ k e. ( ZZ>= ` i ) dom ( F ` k ) | (liminf ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) e. RR } = { y e. U_ i e. Z |^|_ k e. ( ZZ>= ` i ) dom ( F ` k ) | (liminf ` ( k e. Z |-> ( ( F ` k ) ` y ) ) ) e. RR }
60	5, 36, 59	3eqtri 2648	. 2 |- D = { y e. U_ i e. Z |^|_ k e. ( ZZ>= ` i ) dom ( F ` k ) | (liminf ` ( k e. Z |-> ( ( F ` k ) ` y ) ) ) e. RR }
61		smfliminf.g	. . 3 |- G = ( x e. D |-> (liminf ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) )
62		nfrab1 3122	. . . . 5 |- F/_ x { x e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | (liminf ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) e. RR }
63	5, 62	nfcxfr 2762	. . . 4 |- F/_ x D
64		nfcv 2764	. . . 4 |- F/_ y D
65		nfcv 2764	. . . 4 |- F/_ y (liminf ` ( m e. Z |-> ( ( F ` m ) ` x ) ) )
66	63, 64, 65, 43, 57	cbvmptf 4748	. . 3 |- ( x e. D |-> (liminf ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) ) = ( y e. D |-> (liminf ` ( k e. Z |-> ( ( F ` k ) ` y ) ) ) )
67	61, 66	eqtri 2644	. 2 |- G = ( y e. D |-> (liminf ` ( k e. Z |-> ( ( F ` k ) ` y ) ) ) )
68	1, 2, 3, 4, 60, 67	smfliminflem 41036	1 |- ( ph -> G e. (SMblFn ` S ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   F/_wnfc 2751   {crab 2916   U_ciun 4520   |^|_ciin 4521   |-> cmpt 4729   dom cdm 5114   -->wf 5884   ` cfv 5888   RRcr 9935   ZZcz 11377   ZZ>=cuz 11687  liminfclsi 39983  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-inf2 8538  ax-cc 9257  ax-ac2 9285  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-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-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  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-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-ceil 12594  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-s4 13595  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-rest 16083  df-topgen 16104  df-top 20699  df-bases 20750  df-liminf 39984  df-salg 40529  df-salgen 40533  df-smblfn 40910

This theorem is referenced by:  smfliminfmpt  41038