Theorem smfco 41009

Description: The composition of a Borel sigma-measurable function with a sigma-measurable function, is sigma-measurable. Proposition 121E (g) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
smfco.s	|- ( ph -> S e. SAlg )
smfco.f	|- ( ph -> F e. (SMblFn ` S ) )
smfco.j	|- J = ( topGen ` ran (,) )
smfco.b	|- B = (SalGen ` J )
smfco.h	|- ( ph -> H e. (SMblFn ` B ) )

Assertion

Ref	Expression
smfco	|- ( ph -> ( H o. F ) e. (SMblFn ` S ) )

Proof of Theorem smfco

Dummy variables x e a are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ a ph
2		smfco.s	. 2 |- ( ph -> S e. SAlg )
3		cnvimass 5485	. . . 4 |- ( `' F " dom H ) C_ dom F
4	3	a1i 11	. . 3 |- ( ph -> ( `' F " dom H ) C_ dom F )
5		smfco.f	. . . 4 |- ( ph -> F e. (SMblFn ` S ) )
6		eqid 2622	. . . 4 |- dom F = dom F
7	2, 5, 6	smfdmss 40942	. . 3 |- ( ph -> dom F C_ U. S )
8	4, 7	sstrd 3613	. 2 |- ( ph -> ( `' F " dom H ) C_ U. S )
9		smfco.j	. . . . . . . . 9 |- J = ( topGen ` ran (,) )
10		retop 22565	. . . . . . . . 9 |- ( topGen ` ran (,) ) e. Top
11	9, 10	eqeltri 2697	. . . . . . . 8 |- J e. Top
12	11	a1i 11	. . . . . . 7 |- ( ph -> J e. Top )
13		smfco.b	. . . . . . 7 |- B = (SalGen ` J )
14	12, 13	salgencld 40567	. . . . . 6 |- ( ph -> B e. SAlg )
15		smfco.h	. . . . . 6 |- ( ph -> H e. (SMblFn ` B ) )
16		eqid 2622	. . . . . 6 |- dom H = dom H
17	14, 15, 16	smff 40941	. . . . 5 |- ( ph -> H : dom H --> RR )
18	17	ffund 6049	. . . 4 |- ( ph -> Fun H )
19	2, 5, 6	smff 40941	. . . . 5 |- ( ph -> F : dom F --> RR )
20	19	ffund 6049	. . . 4 |- ( ph -> Fun F )
21	18, 20	fco3 39421	. . 3 |- ( ph -> ( H o. F ) : ( `' F " dom H ) --> ran H )
22	17	frnd 39426	. . 3 |- ( ph -> ran H C_ RR )
23	21, 22	fssd 6057	. 2 |- ( ph -> ( H o. F ) : ( `' F " dom H ) --> RR )
24	23	adantr 481	. . . . . 6 |- ( ( ph /\ a e. RR ) -> ( H o. F ) : ( `' F " dom H ) --> RR )
25		rexr 10085	. . . . . . 7 |- ( a e. RR -> a e. RR* )
26	25	adantl 482	. . . . . 6 |- ( ( ph /\ a e. RR ) -> a e. RR* )
27	24, 26	preimaioomnf 40929	. . . . 5 |- ( ( ph /\ a e. RR ) -> ( `' ( H o. F ) " ( -oo (,) a ) ) = { x e. ( `' F " dom H ) | ( ( H o. F ) ` x ) < a } )
28	27	eqcomd 2628	. . . 4 |- ( ( ph /\ a e. RR ) -> { x e. ( `' F " dom H ) | ( ( H o. F ) ` x ) < a } = ( `' ( H o. F ) " ( -oo (,) a ) ) )
29		cnvco 5308	. . . . . 6 |- `' ( H o. F ) = ( `' F o. `' H )
30	29	imaeq1i 5463	. . . . 5 |- ( `' ( H o. F ) " ( -oo (,) a ) ) = ( ( `' F o. `' H ) " ( -oo (,) a ) )
31	30	a1i 11	. . . 4 |- ( ( ph /\ a e. RR ) -> ( `' ( H o. F ) " ( -oo (,) a ) ) = ( ( `' F o. `' H ) " ( -oo (,) a ) ) )
32		imaco 5640	. . . . 5 |- ( ( `' F o. `' H ) " ( -oo (,) a ) ) = ( `' F " ( `' H " ( -oo (,) a ) ) )
33	32	a1i 11	. . . 4 |- ( ( ph /\ a e. RR ) -> ( ( `' F o. `' H ) " ( -oo (,) a ) ) = ( `' F " ( `' H " ( -oo (,) a ) ) ) )
34	28, 31, 33	3eqtrd 2660	. . 3 |- ( ( ph /\ a e. RR ) -> { x e. ( `' F " dom H ) | ( ( H o. F ) ` x ) < a } = ( `' F " ( `' H " ( -oo (,) a ) ) ) )
35	17	adantr 481	. . . . . . . . 9 |- ( ( ph /\ a e. RR ) -> H : dom H --> RR )
36	35, 26	preimaioomnf 40929	. . . . . . . 8 |- ( ( ph /\ a e. RR ) -> ( `' H " ( -oo (,) a ) ) = { x e. dom H | ( H ` x ) < a } )
37	36	eqcomd 2628	. . . . . . 7 |- ( ( ph /\ a e. RR ) -> { x e. dom H | ( H ` x ) < a } = ( `' H " ( -oo (,) a ) ) )
38	37	eqcomd 2628	. . . . . 6 |- ( ( ph /\ a e. RR ) -> ( `' H " ( -oo (,) a ) ) = { x e. dom H | ( H ` x ) < a } )
39	14	adantr 481	. . . . . . 7 |- ( ( ph /\ a e. RR ) -> B e. SAlg )
40	15	adantr 481	. . . . . . 7 |- ( ( ph /\ a e. RR ) -> H e. (SMblFn ` B ) )
41		simpr 477	. . . . . . 7 |- ( ( ph /\ a e. RR ) -> a e. RR )
42	39, 40, 16, 41	smfpreimalt 40940	. . . . . 6 |- ( ( ph /\ a e. RR ) -> { x e. dom H | ( H ` x ) < a } e. ( B ↾t dom H ) )
43	38, 42	eqeltrd 2701	. . . . 5 |- ( ( ph /\ a e. RR ) -> ( `' H " ( -oo (,) a ) ) e. ( B ↾t dom H ) )
44	14	elexd 3214	. . . . . . 7 |- ( ph -> B e. _V )
45	15	dmexd 39422	. . . . . . 7 |- ( ph -> dom H e. _V )
46		elrest 16088	. . . . . . 7 |- ( ( B e. _V /\ dom H e. _V ) -> ( ( `' H " ( -oo (,) a ) ) e. ( B ↾t dom H ) <-> E. e e. B ( `' H " ( -oo (,) a ) ) = ( e i^i dom H ) ) )
47	44, 45, 46	syl2anc 693	. . . . . 6 |- ( ph -> ( ( `' H " ( -oo (,) a ) ) e. ( B ↾t dom H ) <-> E. e e. B ( `' H " ( -oo (,) a ) ) = ( e i^i dom H ) ) )
48	47	adantr 481	. . . . 5 |- ( ( ph /\ a e. RR ) -> ( ( `' H " ( -oo (,) a ) ) e. ( B ↾t dom H ) <-> E. e e. B ( `' H " ( -oo (,) a ) ) = ( e i^i dom H ) ) )
49	43, 48	mpbid 222	. . . 4 |- ( ( ph /\ a e. RR ) -> E. e e. B ( `' H " ( -oo (,) a ) ) = ( e i^i dom H ) )
50		imaeq2 5462	. . . . . . . . 9 |- ( ( `' H " ( -oo (,) a ) ) = ( e i^i dom H ) -> ( `' F " ( `' H " ( -oo (,) a ) ) ) = ( `' F " ( e i^i dom H ) ) )
51	50	3ad2ant3 1084	. . . . . . . 8 |- ( ( ph /\ e e. B /\ ( `' H " ( -oo (,) a ) ) = ( e i^i dom H ) ) -> ( `' F " ( `' H " ( -oo (,) a ) ) ) = ( `' F " ( e i^i dom H ) ) )
52		ovexd 6680	. . . . . . . . . . 11 |- ( ( ph /\ e e. B ) -> ( S ↾t dom F ) e. _V )
53	5	elexd 3214	. . . . . . . . . . . . . 14 |- ( ph -> F e. _V )
54		cnvexg 7112	. . . . . . . . . . . . . 14 |- ( F e. _V -> `' F e. _V )
55	53, 54	syl 17	. . . . . . . . . . . . 13 |- ( ph -> `' F e. _V )
56		imaexg 7103	. . . . . . . . . . . . 13 |- ( `' F e. _V -> ( `' F " dom H ) e. _V )
57	55, 56	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( `' F " dom H ) e. _V )
58	57	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ e e. B ) -> ( `' F " dom H ) e. _V )
59	2	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ e e. B ) -> S e. SAlg )
60	5	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ e e. B ) -> F e. (SMblFn ` S ) )
61		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ e e. B ) -> e e. B )
62		eqid 2622	. . . . . . . . . . . 12 |- ( `' F " e ) = ( `' F " e )
63	59, 60, 6, 9, 13, 61, 62	smfpimbor1 41007	. . . . . . . . . . 11 |- ( ( ph /\ e e. B ) -> ( `' F " e ) e. ( S ↾t dom F ) )
64		eqid 2622	. . . . . . . . . . 11 |- ( ( `' F " e ) i^i ( `' F " dom H ) ) = ( ( `' F " e ) i^i ( `' F " dom H ) )
65	52, 58, 63, 64	elrestd 39291	. . . . . . . . . 10 |- ( ( ph /\ e e. B ) -> ( ( `' F " e ) i^i ( `' F " dom H ) ) e. ( ( S ↾t dom F ) ↾t ( `' F " dom H ) ) )
66		inpreima 6342	. . . . . . . . . . . . 13 |- ( Fun F -> ( `' F " ( e i^i dom H ) ) = ( ( `' F " e ) i^i ( `' F " dom H ) ) )
67	20, 66	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( `' F " ( e i^i dom H ) ) = ( ( `' F " e ) i^i ( `' F " dom H ) ) )
68	67	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ e e. B ) -> ( `' F " ( e i^i dom H ) ) = ( ( `' F " e ) i^i ( `' F " dom H ) ) )
69	5	dmexd 39422	. . . . . . . . . . . . . 14 |- ( ph -> dom F e. _V )
70		restabs 20969	. . . . . . . . . . . . . 14 |- ( ( S e. SAlg /\ ( `' F " dom H ) C_ dom F /\ dom F e. _V ) -> ( ( S ↾t dom F ) ↾t ( `' F " dom H ) ) = ( S ↾t ( `' F " dom H ) ) )
71	2, 4, 69, 70	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ph -> ( ( S ↾t dom F ) ↾t ( `' F " dom H ) ) = ( S ↾t ( `' F " dom H ) ) )
72	71	eqcomd 2628	. . . . . . . . . . . 12 |- ( ph -> ( S ↾t ( `' F " dom H ) ) = ( ( S ↾t dom F ) ↾t ( `' F " dom H ) ) )
73	72	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ e e. B ) -> ( S ↾t ( `' F " dom H ) ) = ( ( S ↾t dom F ) ↾t ( `' F " dom H ) ) )
74	68, 73	eleq12d 2695	. . . . . . . . . 10 |- ( ( ph /\ e e. B ) -> ( ( `' F " ( e i^i dom H ) ) e. ( S ↾t ( `' F " dom H ) ) <-> ( ( `' F " e ) i^i ( `' F " dom H ) ) e. ( ( S ↾t dom F ) ↾t ( `' F " dom H ) ) ) )
75	65, 74	mpbird 247	. . . . . . . . 9 |- ( ( ph /\ e e. B ) -> ( `' F " ( e i^i dom H ) ) e. ( S ↾t ( `' F " dom H ) ) )
76	75	3adant3 1081	. . . . . . . 8 |- ( ( ph /\ e e. B /\ ( `' H " ( -oo (,) a ) ) = ( e i^i dom H ) ) -> ( `' F " ( e i^i dom H ) ) e. ( S ↾t ( `' F " dom H ) ) )
77	51, 76	eqeltrd 2701	. . . . . . 7 |- ( ( ph /\ e e. B /\ ( `' H " ( -oo (,) a ) ) = ( e i^i dom H ) ) -> ( `' F " ( `' H " ( -oo (,) a ) ) ) e. ( S ↾t ( `' F " dom H ) ) )
78	77	3exp 1264	. . . . . 6 |- ( ph -> ( e e. B -> ( ( `' H " ( -oo (,) a ) ) = ( e i^i dom H ) -> ( `' F " ( `' H " ( -oo (,) a ) ) ) e. ( S ↾t ( `' F " dom H ) ) ) ) )
79	78	adantr 481	. . . . 5 |- ( ( ph /\ a e. RR ) -> ( e e. B -> ( ( `' H " ( -oo (,) a ) ) = ( e i^i dom H ) -> ( `' F " ( `' H " ( -oo (,) a ) ) ) e. ( S ↾t ( `' F " dom H ) ) ) ) )
80	79	rexlimdv 3030	. . . 4 |- ( ( ph /\ a e. RR ) -> ( E. e e. B ( `' H " ( -oo (,) a ) ) = ( e i^i dom H ) -> ( `' F " ( `' H " ( -oo (,) a ) ) ) e. ( S ↾t ( `' F " dom H ) ) ) )
81	49, 80	mpd 15	. . 3 |- ( ( ph /\ a e. RR ) -> ( `' F " ( `' H " ( -oo (,) a ) ) ) e. ( S ↾t ( `' F " dom H ) ) )
82	34, 81	eqeltrd 2701	. 2 |- ( ( ph /\ a e. RR ) -> { x e. ( `' F " dom H ) | ( ( H o. F ) ` x ) < a } e. ( S ↾t ( `' F " dom H ) ) )
83	1, 2, 8, 23, 82	issmfd 40944	1 |- ( ph -> ( H o. F ) e. (SMblFn ` S ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   U.cuni 4436   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   -oocmnf 10072   RR*cxr 10073   < clt 10074   (,)cioo 12175   ↾_t crest 16081   topGenctg 16098   Topctop 20698  SAlgcsalg 40528  SalGencsalgen 40532  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-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: (None)