Theorem limsupvaluz 39940

Description: The superior limit, when the domain of the function is a set of upper integers (the first condition is needed, otherwise the l.h.s. would be -oo and the r.h.s. would be +oo). (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
limsupvaluz.m	|- ( ph -> M e. ZZ )
limsupvaluz.z	|- Z = ( ZZ>= ` M )
limsupvaluz.f	|- ( ph -> F : Z --> RR* )

Assertion

Ref	Expression
limsupvaluz	|- ( ph -> ( limsup ` F ) = inf ( ran ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) ) , RR* , < ) )

Distinct variable groups:   k, F   k, Z

Allowed substitution hints:   ph( k)   M( k)

Proof of Theorem limsupvaluz

Dummy variables i n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) )
2		limsupvaluz.f	. . . . 5 |- ( ph -> F : Z --> RR* )
3		limsupvaluz.z	. . . . . . 7 |- Z = ( ZZ>= ` M )
4		fvex 6201	. . . . . . 7 |- ( ZZ>= ` M ) e. _V
5	3, 4	eqeltri 2697	. . . . . 6 |- Z e. _V
6	5	a1i 11	. . . . 5 |- ( ph -> Z e. _V )
7	2, 6	fexd 39296	. . . 4 |- ( ph -> F e. _V )
8	7	elexd 3214	. . 3 |- ( ph -> F e. _V )
9		uzssre 39620	. . . . 5 |- ( ZZ>= ` M ) C_ RR
10	3, 9	eqsstri 3635	. . . 4 |- Z C_ RR
11	10	a1i 11	. . 3 |- ( ph -> Z C_ RR )
12		limsupvaluz.m	. . . 4 |- ( ph -> M e. ZZ )
13	3	uzsup 12662	. . . 4 |- ( M e. ZZ -> sup ( Z , RR* , < ) = +oo )
14	12, 13	syl 17	. . 3 |- ( ph -> sup ( Z , RR* , < ) = +oo )
15	1, 8, 11, 14	limsupval2 14211	. 2 |- ( ph -> ( limsup ` F ) = inf ( ( ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) " Z ) , RR* , < ) )
16	11	mptima2 39457	. . . 4 |- ( ph -> ( ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) " Z ) = ran ( i e. Z |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) )
17		oveq1 6657	. . . . . . . . . . 11 |- ( i = n -> ( i [,) +oo ) = ( n [,) +oo ) )
18	17	imaeq2d 5466	. . . . . . . . . 10 |- ( i = n -> ( F " ( i [,) +oo ) ) = ( F " ( n [,) +oo ) ) )
19	18	ineq1d 3813	. . . . . . . . 9 |- ( i = n -> ( ( F " ( i [,) +oo ) ) i^i RR* ) = ( ( F " ( n [,) +oo ) ) i^i RR* ) )
20	19	supeq1d 8352	. . . . . . . 8 |- ( i = n -> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) = sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) )
21	20	cbvmptv 4750	. . . . . . 7 |- ( i e. Z |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( n e. Z |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) )
22	21	a1i 11	. . . . . 6 |- ( ph -> ( i e. Z |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( n e. Z |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) ) )
23		fimass 6081	. . . . . . . . . . . 12 |- ( F : Z --> RR* -> ( F " ( n [,) +oo ) ) C_ RR* )
24	2, 23	syl 17	. . . . . . . . . . 11 |- ( ph -> ( F " ( n [,) +oo ) ) C_ RR* )
25		df-ss 3588	. . . . . . . . . . . 12 |- ( ( F " ( n [,) +oo ) ) C_ RR* <-> ( ( F " ( n [,) +oo ) ) i^i RR* ) = ( F " ( n [,) +oo ) ) )
26	25	biimpi 206	. . . . . . . . . . 11 |- ( ( F " ( n [,) +oo ) ) C_ RR* -> ( ( F " ( n [,) +oo ) ) i^i RR* ) = ( F " ( n [,) +oo ) ) )
27	24, 26	syl 17	. . . . . . . . . 10 |- ( ph -> ( ( F " ( n [,) +oo ) ) i^i RR* ) = ( F " ( n [,) +oo ) ) )
28	27	adantr 481	. . . . . . . . 9 |- ( ( ph /\ n e. Z ) -> ( ( F " ( n [,) +oo ) ) i^i RR* ) = ( F " ( n [,) +oo ) ) )
29		df-ima 5127	. . . . . . . . . 10 |- ( F " ( n [,) +oo ) ) = ran ( F |` ( n [,) +oo ) )
30	29	a1i 11	. . . . . . . . 9 |- ( ( ph /\ n e. Z ) -> ( F " ( n [,) +oo ) ) = ran ( F |` ( n [,) +oo ) ) )
31	2	freld 39425	. . . . . . . . . . . . 13 |- ( ph -> Rel F )
32		resindm 5444	. . . . . . . . . . . . 13 |- ( Rel F -> ( F |` ( ( n [,) +oo ) i^i dom F ) ) = ( F |` ( n [,) +oo ) ) )
33	31, 32	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( F |` ( ( n [,) +oo ) i^i dom F ) ) = ( F |` ( n [,) +oo ) ) )
34	33	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ n e. Z ) -> ( F |` ( ( n [,) +oo ) i^i dom F ) ) = ( F |` ( n [,) +oo ) ) )
35		incom 3805	. . . . . . . . . . . . . . 15 |- ( ( n [,) +oo ) i^i Z ) = ( Z i^i ( n [,) +oo ) )
36	3	ineq1i 3810	. . . . . . . . . . . . . . 15 |- ( Z i^i ( n [,) +oo ) ) = ( ( ZZ>= ` M ) i^i ( n [,) +oo ) )
37	35, 36	eqtri 2644	. . . . . . . . . . . . . 14 |- ( ( n [,) +oo ) i^i Z ) = ( ( ZZ>= ` M ) i^i ( n [,) +oo ) )
38	37	a1i 11	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. Z ) -> ( ( n [,) +oo ) i^i Z ) = ( ( ZZ>= ` M ) i^i ( n [,) +oo ) ) )
39	2	fdmd 39420	. . . . . . . . . . . . . . 15 |- ( ph -> dom F = Z )
40	39	ineq2d 3814	. . . . . . . . . . . . . 14 |- ( ph -> ( ( n [,) +oo ) i^i dom F ) = ( ( n [,) +oo ) i^i Z ) )
41	40	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. Z ) -> ( ( n [,) +oo ) i^i dom F ) = ( ( n [,) +oo ) i^i Z ) )
42	3	eleq2i 2693	. . . . . . . . . . . . . . . 16 |- ( n e. Z <-> n e. ( ZZ>= ` M ) )
43	42	biimpi 206	. . . . . . . . . . . . . . 15 |- ( n e. Z -> n e. ( ZZ>= ` M ) )
44	43	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ph /\ n e. Z ) -> n e. ( ZZ>= ` M ) )
45	44	uzinico2 39789	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. Z ) -> ( ZZ>= ` n ) = ( ( ZZ>= ` M ) i^i ( n [,) +oo ) ) )
46	38, 41, 45	3eqtr4d 2666	. . . . . . . . . . . 12 |- ( ( ph /\ n e. Z ) -> ( ( n [,) +oo ) i^i dom F ) = ( ZZ>= ` n ) )
47	46	reseq2d 5396	. . . . . . . . . . 11 |- ( ( ph /\ n e. Z ) -> ( F |` ( ( n [,) +oo ) i^i dom F ) ) = ( F |` ( ZZ>= ` n ) ) )
48	34, 47	eqtr3d 2658	. . . . . . . . . 10 |- ( ( ph /\ n e. Z ) -> ( F |` ( n [,) +oo ) ) = ( F |` ( ZZ>= ` n ) ) )
49	48	rneqd 5353	. . . . . . . . 9 |- ( ( ph /\ n e. Z ) -> ran ( F |` ( n [,) +oo ) ) = ran ( F |` ( ZZ>= ` n ) ) )
50	28, 30, 49	3eqtrd 2660	. . . . . . . 8 |- ( ( ph /\ n e. Z ) -> ( ( F " ( n [,) +oo ) ) i^i RR* ) = ran ( F |` ( ZZ>= ` n ) ) )
51	50	supeq1d 8352	. . . . . . 7 |- ( ( ph /\ n e. Z ) -> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) )
52	51	mpteq2dva 4744	. . . . . 6 |- ( ph -> ( n e. Z |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) )
53	22, 52	eqtrd 2656	. . . . 5 |- ( ph -> ( i e. Z |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) )
54	53	rneqd 5353	. . . 4 |- ( ph -> ran ( i e. Z |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) )
55	16, 54	eqtrd 2656	. . 3 |- ( ph -> ( ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) " Z ) = ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) )
56	55	infeq1d 8383	. 2 |- ( ph -> inf ( ( ( i e. RR |-> sup ( ( ( F " ( i [,) +oo ) ) i^i RR* ) , RR* , < ) ) " Z ) , RR* , < ) = inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR* , < ) )
57		fveq2 6191	. . . . . . . . 9 |- ( n = k -> ( ZZ>= ` n ) = ( ZZ>= ` k ) )
58	57	reseq2d 5396	. . . . . . . 8 |- ( n = k -> ( F |` ( ZZ>= ` n ) ) = ( F |` ( ZZ>= ` k ) ) )
59	58	rneqd 5353	. . . . . . 7 |- ( n = k -> ran ( F |` ( ZZ>= ` n ) ) = ran ( F |` ( ZZ>= ` k ) ) )
60	59	supeq1d 8352	. . . . . 6 |- ( n = k -> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) = sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) )
61	60	cbvmptv 4750	. . . . 5 |- ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) = ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) )
62	61	rneqi 5352	. . . 4 |- ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) = ran ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) )
63	62	infeq1i 8384	. . 3 |- inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR* , < ) = inf ( ran ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) ) , RR* , < )
64	63	a1i 11	. 2 |- ( ph -> inf ( ran ( n e. Z |-> sup ( ran ( F |` ( ZZ>= ` n ) ) , RR* , < ) ) , RR* , < ) = inf ( ran ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) ) , RR* , < ) )
65	15, 56, 64	3eqtrd 2660	1 |- ( ph -> ( limsup ` F ) = inf ( ran ( k e. Z |-> sup ( ran ( F |` ( ZZ>= ` k ) ) , RR* , < ) ) , RR* , < ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346  infcinf 8347   RRcr 9935   +oocpnf 10071   RR*cxr 10073   < clt 10074   ZZcz 11377   ZZ>=cuz 11687   [,)cico 12177   limsupclsp 14201

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-iun 4522  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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-ico 12181  df-fl 12593  df-limsup 14202

This theorem is referenced by:  limsupvaluzmpt  39949  limsupvaluz2  39970  limsupgtlem  40009