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Theorem limsupbnd1 11956
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 Mario Carneiro, 7-May-2016.)
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.
StepHypRef 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 )
32adantr 451 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  B  C_  RR )
4 limsupbnd.2 . . . . . 6  |-  ( ph  ->  F : B --> RR* )
54adantr 451 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  F : B
--> RR* )
6 simpr 447 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  k  e.  RR )
7 limsupbnd.3 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
87adantr 451 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  A  e. 
RR* )
9 eqid 2283 . . . . . 6  |-  ( n  e.  RR  |->  sup (
( ( F "
( n [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  ( n  e.  RR  |->  sup (
( ( F "
( n [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
109limsupgle 11951 . . . . 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
) ) )
113, 5, 6, 8, 10syl211anc 1188 . . . 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 8828 . . . . . . . . . . . 12  |-  RR  e.  _V
1312ssex 4158 . . . . . . . . . . 11  |-  ( B 
C_  RR  ->  B  e. 
_V )
142, 13syl 15 . . . . . . . . . 10  |-  ( ph  ->  B  e.  _V )
15 xrex 10351 . . . . . . . . . . 11  |-  RR*  e.  _V
1615a1i 10 . . . . . . . . . 10  |-  ( ph  -> 
RR*  e.  _V )
17 fex2 5401 . . . . . . . . . 10  |-  ( ( F : B --> RR*  /\  B  e.  _V  /\  RR*  e.  _V )  ->  F  e. 
_V )
184, 14, 16, 17syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  F  e.  _V )
19 limsupcl 11947 . . . . . . . . 9  |-  ( F  e.  _V  ->  ( limsup `
 F )  e. 
RR* )
2018, 19syl 15 . . . . . . . 8  |-  ( ph  ->  ( limsup `  F )  e.  RR* )
21 xrleid 10484 . . . . . . . 8  |-  ( (
limsup `  F )  e. 
RR*  ->  ( limsup `  F
)  <_  ( limsup `  F ) )
2220, 21syl 15 . . . . . . 7  |-  ( ph  ->  ( limsup `  F )  <_  ( limsup `  F )
)
239limsuple 11952 . . . . . . . 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 ) ) )
242, 4, 20, 23syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( limsup `  F
)  <_  ( limsup `  F )  <->  A. k  e.  RR  ( limsup `  F
)  <_  ( (
n  e.  RR  |->  sup ( ( ( F
" ( n [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k ) ) )
2522, 24mpbid 201 . . . . . 6  |-  ( ph  ->  A. k  e.  RR  ( limsup `  F )  <_  ( ( n  e.  RR  |->  sup ( ( ( F " ( n [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k ) )
2625r19.21bi 2641 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  ( limsup `  F )  <_  (
( n  e.  RR  |->  sup ( ( ( F
" ( n [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k ) )
2720adantr 451 . . . . . 6  |-  ( (
ph  /\  k  e.  RR )  ->  ( limsup `  F )  e.  RR* )
289limsupgf 11949 . . . . . . . 8  |-  ( n  e.  RR  |->  sup (
( ( F "
( n [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) : RR --> RR*
2928a1i 10 . . . . . . 7  |-  ( ph  ->  ( n  e.  RR  |->  sup ( ( ( F
" ( n [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) : RR --> RR* )
30 ffvelrn 5663 . . . . . . 7  |-  ( ( ( n  e.  RR  |->  sup ( ( ( F
" ( n [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) : RR --> RR*  /\  k  e.  RR )  ->  (
( n  e.  RR  |->  sup ( ( ( F
" ( n [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k )  e.  RR* )
3129, 30sylan 457 . . . . . 6  |-  ( (
ph  /\  k  e.  RR )  ->  ( ( n  e.  RR  |->  sup ( ( ( F
" ( n [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k )  e.  RR* )
32 xrletr 10489 . . . . . 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 ) )
3327, 31, 8, 32syl3anc 1182 . . . . 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 ) )
3426, 33mpand 656 . . . 4  |-  ( (
ph  /\  k  e.  RR )  ->  ( ( ( n  e.  RR  |->  sup ( ( ( F
" ( n [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k )  <_  A  ->  ( limsup `
 F )  <_  A ) )
3511, 34sylbird 226 . . 3  |-  ( (
ph  /\  k  e.  RR )  ->  ( A. j  e.  B  (
k  <_  j  ->  ( F `  j )  <_  A )  -> 
( limsup `  F )  <_  A ) )
3635rexlimdva 2667 . 2  |-  ( ph  ->  ( E. k  e.  RR  A. j  e.  B  ( k  <_ 
j  ->  ( F `  j )  <_  A
)  ->  ( limsup `  F )  <_  A
) )
371, 36mpd 14 1  |-  ( ph  ->  ( limsup `  F )  <_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    i^i cin 3151    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868   [,)cico 10658   limsupclsp 11944
This theorem is referenced by:  caucvgrlem  12145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-ico 10662  df-limsup 11945
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