Theorem limsupbnd1 14213

Description: If a sequence is eventually at most A, then the limsup is also at most A. (The converse is only true if the less or equal is replaced by strictly less than; consider the sequence 1 / n which is never less or equal to zero even though the limsup is.) (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)

Hypotheses

Ref	Expression
limsupbnd.1	|- ( ph -> B C_ RR )
limsupbnd.2	|- ( ph -> F : B --> RR* )
limsupbnd.3	|- ( ph -> A e. RR* )
limsupbnd1.4	|- ( ph -> E. k e. RR A. j e. B ( k <_ j -> ( F ` j ) <_ A ) )

Assertion

Ref	Expression
limsupbnd1	|- ( ph -> ( limsup ` F ) <_ A )

Distinct variable groups:   j, k, A   B, j, k   j, F, k   ph, j, k

Proof of Theorem limsupbnd1

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		limsupbnd1.4	. 2 |- ( ph -> E. k e. RR A. j e. B ( k <_ j -> ( F ` j ) <_ A ) )
2		limsupbnd.1	. . . . . 6 |- ( ph -> B C_ RR )
3	2	adantr 481	. . . . 5 |- ( ( ph /\ k e. RR ) -> B C_ RR )
4		limsupbnd.2	. . . . . 6 |- ( ph -> F : B --> RR* )
5	4	adantr 481	. . . . 5 |- ( ( ph /\ k e. RR ) -> F : B --> RR* )
6		simpr 477	. . . . 5 |- ( ( ph /\ k e. RR ) -> k e. RR )
7		limsupbnd.3	. . . . . 6 |- ( ph -> A e. RR* )
8	7	adantr 481	. . . . 5 |- ( ( ph /\ k e. RR ) -> A e. RR* )
9		eqid 2622	. . . . . 6 |- ( n e. RR |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( n e. RR |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) )
10	9	limsupgle 14208	. . . . 5 |- ( ( ( B C_ RR /\ F : B --> RR* ) /\ k e. RR /\ A e. RR* ) -> ( ( ( n e. RR |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) ) ` k ) <_ A <-> A. j e. B ( k <_ j -> ( F ` j ) <_ A ) ) )
11	3, 5, 6, 8, 10	syl211anc 1332	. . . 4 |- ( ( ph /\ k e. RR ) -> ( ( ( n e. RR |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) ) ` k ) <_ A <-> A. j e. B ( k <_ j -> ( F ` j ) <_ A ) ) )
12		reex 10027	. . . . . . . . . . . 12 |- RR e. _V
13	12	ssex 4802	. . . . . . . . . . 11 |- ( B C_ RR -> B e. _V )
14	2, 13	syl 17	. . . . . . . . . 10 |- ( ph -> B e. _V )
15		xrex 11829	. . . . . . . . . . 11 |- RR* e. _V
16	15	a1i 11	. . . . . . . . . 10 |- ( ph -> RR* e. _V )
17		fex2 7121	. . . . . . . . . 10 |- ( ( F : B --> RR* /\ B e. _V /\ RR* e. _V ) -> F e. _V )
18	4, 14, 16, 17	syl3anc 1326	. . . . . . . . 9 |- ( ph -> F e. _V )
19		limsupcl 14204	. . . . . . . . 9 |- ( F e. _V -> ( limsup ` F ) e. RR* )
20	18, 19	syl 17	. . . . . . . 8 |- ( ph -> ( limsup ` F ) e. RR* )
21		xrleid 11983	. . . . . . . 8 |- ( ( limsup ` F ) e. RR* -> ( limsup ` F ) <_ ( limsup ` F ) )
22	20, 21	syl 17	. . . . . . 7 |- ( ph -> ( limsup ` F ) <_ ( limsup ` F ) )
23	9	limsuple 14209	. . . . . . . 8 |- ( ( B C_ RR /\ F : B --> RR* /\ ( limsup ` F ) e. RR* ) -> ( ( limsup ` F ) <_ ( limsup ` F ) <-> A. k e. RR ( limsup ` F ) <_ ( ( n e. RR |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) ) ` k ) ) )
24	2, 4, 20, 23	syl3anc 1326	. . . . . . 7 |- ( ph -> ( ( limsup ` F ) <_ ( limsup ` F ) <-> A. k e. RR ( limsup ` F ) <_ ( ( n e. RR |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) ) ` k ) ) )
25	22, 24	mpbid 222	. . . . . 6 |- ( ph -> A. k e. RR ( limsup ` F ) <_ ( ( n e. RR |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) ) ` k ) )
26	25	r19.21bi 2932	. . . . 5 |- ( ( ph /\ k e. RR ) -> ( limsup ` F ) <_ ( ( n e. RR |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) ) ` k ) )
27	20	adantr 481	. . . . . 6 |- ( ( ph /\ k e. RR ) -> ( limsup ` F ) e. RR* )
28	9	limsupgf 14206	. . . . . . . 8 |- ( n e. RR |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) ) : RR --> RR*
29	28	a1i 11	. . . . . . 7 |- ( ph -> ( n e. RR |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) ) : RR --> RR* )
30	29	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ k e. RR ) -> ( ( n e. RR |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) ) ` k ) e. RR* )
31		xrletr 11989	. . . . . 6 |- ( ( ( limsup ` F ) e. RR* /\ ( ( n e. RR |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) ) ` k ) e. RR* /\ A e. RR* ) -> ( ( ( limsup ` F ) <_ ( ( n e. RR |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) ) ` k ) /\ ( ( n e. RR |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) ) ` k ) <_ A ) -> ( limsup ` F ) <_ A ) )
32	27, 30, 8, 31	syl3anc 1326	. . . . 5 |- ( ( ph /\ k e. RR ) -> ( ( ( limsup ` F ) <_ ( ( n e. RR |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) ) ` k ) /\ ( ( n e. RR |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) ) ` k ) <_ A ) -> ( limsup ` F ) <_ A ) )
33	26, 32	mpand 711	. . . 4 |- ( ( ph /\ k e. RR ) -> ( ( ( n e. RR |-> sup ( ( ( F " ( n [,) +oo ) ) i^i RR* ) , RR* , < ) ) ` k ) <_ A -> ( limsup ` F ) <_ A ) )
34	11, 33	sylbird 250	. . 3 |- ( ( ph /\ k e. RR ) -> ( A. j e. B ( k <_ j -> ( F ` j ) <_ A ) -> ( limsup ` F ) <_ A ) )
35	34	rexlimdva 3031	. 2 |- ( ph -> ( E. k e. RR A. j e. B ( k <_ j -> ( F ` j ) <_ A ) -> ( limsup ` F ) <_ A ) )
36	1, 35	mpd 15	1 |- ( ph -> ( limsup ` F ) <_ A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   supcsup 8346   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-ico 12181  df-limsup 14202

This theorem is referenced by:  caucvgrlem  14403  limsupre  39873  limsupbnd1f  39918