Theorem smfsupxr 41022

Description: The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

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

Assertion

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

Distinct variable groups:   n, Z, x   ph, n, x

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

Proof of Theorem smfsupxr

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		smfsupxr.g	. . . 4 |- G = ( x e. D |-> sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR* , < ) )
2	1	a1i 11	. . 3 |- ( ph -> G = ( x e. D |-> sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR* , < ) ) )
3		smfsupxr.d	. . . . . 6 |- D = { x e. |^|_ n e. Z dom ( F ` n ) | sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR* , < ) e. RR }
4	3	a1i 11	. . . . 5 |- ( ph -> D = { x e. |^|_ n e. Z dom ( F ` n ) | sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR* , < ) e. RR } )
5		nfv 1843	. . . . . . . 8 |- F/ n ph
6		nfcv 2764	. . . . . . . . 9 |- F/_ n x
7		nfii1 4551	. . . . . . . . 9 |- F/_ n |^|_ n e. Z dom ( F ` n )
8	6, 7	nfel 2777	. . . . . . . 8 |- F/ n x e. |^|_ n e. Z dom ( F ` n )
9	5, 8	nfan 1828	. . . . . . 7 |- F/ n ( ph /\ x e. |^|_ n e. Z dom ( F ` n ) )
10		smfsupxr.m	. . . . . . . . 9 |- ( ph -> M e. ZZ )
11		smfsupxr.z	. . . . . . . . 9 |- Z = ( ZZ>= ` M )
12	10, 11	uzn0d 39652	. . . . . . . 8 |- ( ph -> Z =/= (/) )
13	12	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. |^|_ n e. Z dom ( F ` n ) ) -> Z =/= (/) )
14		smfsupxr.s	. . . . . . . . . . 11 |- ( ph -> S e. SAlg )
15	14	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ n e. Z ) -> S e. SAlg )
16		smfsupxr.f	. . . . . . . . . . 11 |- ( ph -> F : Z --> (SMblFn ` S ) )
17	16	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ph /\ n e. Z ) -> ( F ` n ) e. (SMblFn ` S ) )
18		eqid 2622	. . . . . . . . . 10 |- dom ( F ` n ) = dom ( F ` n )
19	15, 17, 18	smff 40941	. . . . . . . . 9 |- ( ( ph /\ n e. Z ) -> ( F ` n ) : dom ( F ` n ) --> RR )
20	19	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ x e. |^|_ n e. Z dom ( F ` n ) ) /\ n e. Z ) -> ( F ` n ) : dom ( F ` n ) --> RR )
21		eliinid 39294	. . . . . . . . 9 |- ( ( x e. |^|_ n e. Z dom ( F ` n ) /\ n e. Z ) -> x e. dom ( F ` n ) )
22	21	adantll 750	. . . . . . . 8 |- ( ( ( ph /\ x e. |^|_ n e. Z dom ( F ` n ) ) /\ n e. Z ) -> x e. dom ( F ` n ) )
23	20, 22	ffvelrnd 6360	. . . . . . 7 |- ( ( ( ph /\ x e. |^|_ n e. Z dom ( F ` n ) ) /\ n e. Z ) -> ( ( F ` n ) ` x ) e. RR )
24	9, 13, 23	supxrre3rnmpt 39656	. . . . . 6 |- ( ( ph /\ x e. |^|_ n e. Z dom ( F ` n ) ) -> ( sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR* , < ) e. RR <-> E. y e. RR A. n e. Z ( ( F ` n ) ` x ) <_ y ) )
25	24	rabbidva 3188	. . . . 5 |- ( ph -> { x e. |^|_ n e. Z dom ( F ` n ) | sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR* , < ) e. RR } = { x e. |^|_ n e. Z dom ( F ` n ) | E. y e. RR A. n e. Z ( ( F ` n ) ` x ) <_ y } )
26	4, 25	eqtrd 2656	. . . 4 |- ( ph -> D = { x e. |^|_ n e. Z dom ( F ` n ) | E. y e. RR A. n e. Z ( ( F ` n ) ` x ) <_ y } )
27		nfmpt1 4747	. . . . . . . . . . . 12 |- F/_ n ( n e. Z |-> ( ( F ` n ) ` x ) )
28	27	nfrn 5368	. . . . . . . . . . 11 |- F/_ n ran ( n e. Z |-> ( ( F ` n ) ` x ) )
29		nfcv 2764	. . . . . . . . . . 11 |- F/_ n RR*
30		nfcv 2764	. . . . . . . . . . 11 |- F/_ n <
31	28, 29, 30	nfsup 8357	. . . . . . . . . 10 |- F/_ n sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR* , < )
32		nfcv 2764	. . . . . . . . . 10 |- F/_ n RR
33	31, 32	nfel 2777	. . . . . . . . 9 |- F/ n sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR* , < ) e. RR
34	33, 7	nfrab 3123	. . . . . . . 8 |- F/_ n { x e. |^|_ n e. Z dom ( F ` n ) | sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR* , < ) e. RR }
35	3, 34	nfcxfr 2762	. . . . . . 7 |- F/_ n D
36	6, 35	nfel 2777	. . . . . 6 |- F/ n x e. D
37	5, 36	nfan 1828	. . . . 5 |- F/ n ( ph /\ x e. D )
38	12	adantr 481	. . . . 5 |- ( ( ph /\ x e. D ) -> Z =/= (/) )
39	19	adantlr 751	. . . . . 6 |- ( ( ( ph /\ x e. D ) /\ n e. Z ) -> ( F ` n ) : dom ( F ` n ) --> RR )
40		nfcv 2764	. . . . . . . . . . . 12 |- F/_ x Z
41		smfsupxr.x	. . . . . . . . . . . . . 14 |- F/_ x F
42		nfcv 2764	. . . . . . . . . . . . . 14 |- F/_ x n
43	41, 42	nffv 6198	. . . . . . . . . . . . 13 |- F/_ x ( F ` n )
44	43	nfdm 5367	. . . . . . . . . . . 12 |- F/_ x dom ( F ` n )
45	40, 44	nfiin 4549	. . . . . . . . . . 11 |- F/_ x |^|_ n e. Z dom ( F ` n )
46	45	ssrab2f 39300	. . . . . . . . . 10 |- { x e. |^|_ n e. Z dom ( F ` n ) | sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR* , < ) e. RR } C_ |^|_ n e. Z dom ( F ` n )
47	3, 46	eqsstri 3635	. . . . . . . . 9 |- D C_ |^|_ n e. Z dom ( F ` n )
48		id 22	. . . . . . . . 9 |- ( x e. D -> x e. D )
49	47, 48	sseldi 3601	. . . . . . . 8 |- ( x e. D -> x e. |^|_ n e. Z dom ( F ` n ) )
50	49, 21	sylan 488	. . . . . . 7 |- ( ( x e. D /\ n e. Z ) -> x e. dom ( F ` n ) )
51	50	adantll 750	. . . . . 6 |- ( ( ( ph /\ x e. D ) /\ n e. Z ) -> x e. dom ( F ` n ) )
52	39, 51	ffvelrnd 6360	. . . . 5 |- ( ( ( ph /\ x e. D ) /\ n e. Z ) -> ( ( F ` n ) ` x ) e. RR )
53	48, 3	syl6eleq 2711	. . . . . . . 8 |- ( x e. D -> x e. { x e. |^|_ n e. Z dom ( F ` n ) | sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR* , < ) e. RR } )
54		rabidim2 39284	. . . . . . . 8 |- ( x e. { x e. |^|_ n e. Z dom ( F ` n ) | sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR* , < ) e. RR } -> sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR* , < ) e. RR )
55	53, 54	syl 17	. . . . . . 7 |- ( x e. D -> sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR* , < ) e. RR )
56	55	adantl 482	. . . . . 6 |- ( ( ph /\ x e. D ) -> sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR* , < ) e. RR )
57	49	adantl 482	. . . . . . 7 |- ( ( ph /\ x e. D ) -> x e. |^|_ n e. Z dom ( F ` n ) )
58	57, 24	syldan 487	. . . . . 6 |- ( ( ph /\ x e. D ) -> ( sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR* , < ) e. RR <-> E. y e. RR A. n e. Z ( ( F ` n ) ` x ) <_ y ) )
59	56, 58	mpbid 222	. . . . 5 |- ( ( ph /\ x e. D ) -> E. y e. RR A. n e. Z ( ( F ` n ) ` x ) <_ y )
60	37, 38, 52, 59	supxrrernmpt 39648	. . . 4 |- ( ( ph /\ x e. D ) -> sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR* , < ) = sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR , < ) )
61	26, 60	mpteq12dva 4732	. . 3 |- ( ph -> ( x e. D |-> sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR* , < ) ) = ( x e. { x e. |^|_ n e. Z dom ( F ` n ) | E. y e. RR A. n e. Z ( ( F ` n ) ` x ) <_ y } |-> sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR , < ) ) )
62	2, 61	eqtrd 2656	. 2 |- ( ph -> G = ( x e. { x e. |^|_ n e. Z dom ( F ` n ) | E. y e. RR A. n e. Z ( ( F ` n ) ` x ) <_ y } |-> sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR , < ) ) )
63		smfsupxr.n	. . 3 |- F/_ n F
64		eqid 2622	. . 3 |- { x e. |^|_ n e. Z dom ( F ` n ) | E. y e. RR A. n e. Z ( ( F ` n ) ` x ) <_ y } = { x e. |^|_ n e. Z dom ( F ` n ) | E. y e. RR A. n e. Z ( ( F ` n ) ` x ) <_ y }
65		eqid 2622	. . 3 |- ( x e. { x e. |^|_ n e. Z dom ( F ` n ) | E. y e. RR A. n e. Z ( ( F ` n ) ` x ) <_ y } |-> sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR , < ) ) = ( x e. { x e. |^|_ n e. Z dom ( F ` n ) | E. y e. RR A. n e. Z ( ( F ` n ) ` x ) <_ y } |-> sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR , < ) )
66	63, 41, 10, 11, 14, 16, 64, 65	smfsup 41020	. 2 |- ( ph -> ( x e. { x e. |^|_ n e. Z dom ( F ` n ) | E. y e. RR A. n e. Z ( ( F ` n ) ` x ) <_ y } |-> sup ( ran ( n e. Z |-> ( ( F ` n ) ` x ) ) , RR , < ) ) e. (SMblFn ` S ) )
67	62, 66	eqeltrd 2701	1 |- ( ph -> G e. (SMblFn ` S ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   (/)c0 3915   |^|_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   <_ cle 10075   ZZcz 11377   ZZ>=cuz 11687  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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-ioo 12179  df-ioc 12180  df-ico 12181  df-fl 12593  df-rest 16083  df-topgen 16104  df-top 20699  df-bases 20750  df-salg 40529  df-salgen 40533  df-smblfn 40910

This theorem is referenced by:  smflimsuplem3  41028