Theorem limsupvaluzmpt 39949

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
limsupvaluzmpt.j	|- F/ j ph
limsupvaluzmpt.m	|- ( ph -> M e. ZZ )
limsupvaluzmpt.z	|- Z = ( ZZ>= ` M )
limsupvaluzmpt.b	|- ( ( ph /\ j e. Z ) -> B e. RR* )

Assertion

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

Distinct variable groups:   B, k   j, Z, k

Allowed substitution hints:   ph( j, k)   B( j)   M( j, k)

Proof of Theorem limsupvaluzmpt

Step	Hyp	Ref	Expression
1		limsupvaluzmpt.m	. . 3 |- ( ph -> M e. ZZ )
2		limsupvaluzmpt.z	. . 3 |- Z = ( ZZ>= ` M )
3		limsupvaluzmpt.j	. . . 4 |- F/ j ph
4		limsupvaluzmpt.b	. . . 4 |- ( ( ph /\ j e. Z ) -> B e. RR* )
5	3, 4	fmptd2f 39442	. . 3 |- ( ph -> ( j e. Z |-> B ) : Z --> RR* )
6	1, 2, 5	limsupvaluz 39940	. 2 |- ( ph -> ( limsup ` ( j e. Z |-> B ) ) = inf ( ran ( k e. Z |-> sup ( ran ( ( j e. Z |-> B ) |` ( ZZ>= ` k ) ) , RR* , < ) ) , RR* , < ) )
7	2	uzssd3 39653	. . . . . . . . 9 |- ( k e. Z -> ( ZZ>= ` k ) C_ Z )
8	7	resmptd 5452	. . . . . . . 8 |- ( k e. Z -> ( ( j e. Z |-> B ) |` ( ZZ>= ` k ) ) = ( j e. ( ZZ>= ` k ) |-> B ) )
9	8	rneqd 5353	. . . . . . 7 |- ( k e. Z -> ran ( ( j e. Z |-> B ) |` ( ZZ>= ` k ) ) = ran ( j e. ( ZZ>= ` k ) |-> B ) )
10	9	supeq1d 8352	. . . . . 6 |- ( k e. Z -> sup ( ran ( ( j e. Z |-> B ) |` ( ZZ>= ` k ) ) , RR* , < ) = sup ( ran ( j e. ( ZZ>= ` k ) |-> B ) , RR* , < ) )
11	10	mpteq2ia 4740	. . . . 5 |- ( k e. Z |-> sup ( ran ( ( j e. Z |-> B ) |` ( ZZ>= ` k ) ) , RR* , < ) ) = ( k e. Z |-> sup ( ran ( j e. ( ZZ>= ` k ) |-> B ) , RR* , < ) )
12	11	a1i 11	. . . 4 |- ( ph -> ( k e. Z |-> sup ( ran ( ( j e. Z |-> B ) |` ( ZZ>= ` k ) ) , RR* , < ) ) = ( k e. Z |-> sup ( ran ( j e. ( ZZ>= ` k ) |-> B ) , RR* , < ) ) )
13	12	rneqd 5353	. . 3 |- ( ph -> ran ( k e. Z |-> sup ( ran ( ( j e. Z |-> B ) |` ( ZZ>= ` k ) ) , RR* , < ) ) = ran ( k e. Z |-> sup ( ran ( j e. ( ZZ>= ` k ) |-> B ) , RR* , < ) ) )
14	13	infeq1d 8383	. 2 |- ( ph -> inf ( ran ( k e. Z |-> sup ( ran ( ( j e. Z |-> B ) |` ( ZZ>= ` k ) ) , RR* , < ) ) , RR* , < ) = inf ( ran ( k e. Z |-> sup ( ran ( j e. ( ZZ>= ` k ) |-> B ) , RR* , < ) ) , RR* , < ) )
15	6, 14	eqtrd 2656	1 |- ( ph -> ( limsup ` ( j e. Z |-> B ) ) = inf ( ran ( k e. Z |-> sup ( ran ( j e. ( ZZ>= ` k ) |-> B ) , RR* , < ) ) , RR* , < ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   |-> cmpt 4729   ran crn 5115   |` cres 5116   ` cfv 5888   supcsup 8346  infcinf 8347   RR*cxr 10073   < clt 10074   ZZcz 11377   ZZ>=cuz 11687   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:  smflimsuplem4  41029