Theorem limsuplesup 39931

Description: An upper bound for the superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
limsuplesup.1	|- ( ph -> F e. V )
limsuplesup.2	|- ( ph -> K e. RR )

Assertion

Ref	Expression
limsuplesup	|- ( ph -> ( limsup ` F ) <_ sup ( ( ( F " ( K [,) +oo ) ) i^i RR* ) , RR* , < ) )

Proof of Theorem limsuplesup

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limsuplesup.1	. . 3 |- ( ph -> F e. V )
2		eqid 2622	. . . 4 |- ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
3	2	limsupval 14205	. . 3 |- ( F e. V -> ( limsup ` F ) = inf ( ran ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
4	1, 3	syl 17	. 2 |- ( ph -> ( limsup ` F ) = inf ( ran ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
5		nfv 1843	. . 3 |- F/ k ph
6		inss2 3834	. . . . 5 |- ( ( F " ( k [,) +oo ) ) i^i RR* ) C_ RR*
7	6	a1i 11	. . . 4 |- ( ( ph /\ k e. RR ) -> ( ( F " ( k [,) +oo ) ) i^i RR* ) C_ RR* )
8	7	supxrcld 39290	. . 3 |- ( ( ph /\ k e. RR ) -> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR* )
9		limsuplesup.2	. . 3 |- ( ph -> K e. RR )
10		inss2 3834	. . . . 5 |- ( ( F " ( K [,) +oo ) ) i^i RR* ) C_ RR*
11	10	a1i 11	. . . 4 |- ( ph -> ( ( F " ( K [,) +oo ) ) i^i RR* ) C_ RR* )
12	11	supxrcld 39290	. . 3 |- ( ph -> sup ( ( ( F " ( K [,) +oo ) ) i^i RR* ) , RR* , < ) e. RR* )
13		oveq1 6657	. . . . . 6 |- ( k = K -> ( k [,) +oo ) = ( K [,) +oo ) )
14	13	imaeq2d 5466	. . . . 5 |- ( k = K -> ( F " ( k [,) +oo ) ) = ( F " ( K [,) +oo ) ) )
15	14	ineq1d 3813	. . . 4 |- ( k = K -> ( ( F " ( k [,) +oo ) ) i^i RR* ) = ( ( F " ( K [,) +oo ) ) i^i RR* ) )
16	15	supeq1d 8352	. . 3 |- ( k = K -> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) = sup ( ( ( F " ( K [,) +oo ) ) i^i RR* ) , RR* , < ) )
17	5, 8, 9, 12, 16	infxrlbrnmpt2 39637	. 2 |- ( ph -> inf ( ran ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) <_ sup ( ( ( F " ( K [,) +oo ) ) i^i RR* ) , RR* , < ) )
18	4, 17	eqbrtrd 4675	1 |- ( ph -> ( limsup ` F ) <_ sup ( ( ( F " ( K [,) +oo ) ) i^i RR* ) , RR* , < ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   "cima 5117   ` cfv 5888  (class class class)co 6650   supcsup 8346  infcinf 8347   RRcr 9935   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   [,)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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-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-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-limsup 14202

This theorem is referenced by: (None)