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Theorem limsupbnd1OLD 13544
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.) Obsolete version of limsupbnd1 13543 as of 12-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
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
limsupbnd1OLD  |-  ( ph  ->  ( limsup `  F )  <_  A )
Distinct variable groups:    j, k, A    B, j, k    j, F, k    ph, j, k

Proof of Theorem limsupbnd1OLD
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 466 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  B  C_  RR )
4 limsupbnd.2 . . . . . 6  |-  ( ph  ->  F : B --> RR* )
54adantr 466 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  F : B
--> RR* )
6 simpr 462 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  k  e.  RR )
7 limsupbnd.3 . . . . . 6  |-  ( ph  ->  A  e.  RR* )
87adantr 466 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  A  e. 
RR* )
9 eqid 2422 . . . . . 6  |-  ( n  e.  RR  |->  sup (
( ( F "
( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )  =  ( n  e.  RR  |->  sup (
( ( F "
( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
109limsupgle 13534 . . . . 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 1270 . . . 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 9637 . . . . . . . . . . . 12  |-  RR  e.  _V
1312ssex 4568 . . . . . . . . . . 11  |-  ( B 
C_  RR  ->  B  e. 
_V )
142, 13syl 17 . . . . . . . . . 10  |-  ( ph  ->  B  e.  _V )
15 xrex 11306 . . . . . . . . . . 11  |-  RR*  e.  _V
1615a1i 11 . . . . . . . . . 10  |-  ( ph  -> 
RR*  e.  _V )
17 fex2 6762 . . . . . . . . . 10  |-  ( ( F : B --> RR*  /\  B  e.  _V  /\  RR*  e.  _V )  ->  F  e. 
_V )
184, 14, 16, 17syl3anc 1264 . . . . . . . . 9  |-  ( ph  ->  F  e.  _V )
19 limsupclOLD 13529 . . . . . . . . 9  |-  ( F  e.  _V  ->  ( limsup `
 F )  e. 
RR* )
2018, 19syl 17 . . . . . . . 8  |-  ( ph  ->  ( limsup `  F )  e.  RR* )
21 xrleid 11456 . . . . . . . 8  |-  ( (
limsup `  F )  e. 
RR*  ->  ( limsup `  F
)  <_  ( limsup `  F ) )
2220, 21syl 17 . . . . . . 7  |-  ( ph  ->  ( limsup `  F )  <_  ( limsup `  F )
)
239limsupleOLD 13536 . . . . . . . 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 1264 . . . . . . 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 213 . . . . . 6  |-  ( ph  ->  A. k  e.  RR  ( limsup `  F )  <_  ( ( n  e.  RR  |->  sup ( ( ( F " ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k ) )
2625r19.21bi 2791 . . . . 5  |-  ( (
ph  /\  k  e.  RR )  ->  ( limsup `  F )  <_  (
( n  e.  RR  |->  sup ( ( ( F
" ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k ) )
2720adantr 466 . . . . . 6  |-  ( (
ph  /\  k  e.  RR )  ->  ( limsup `  F )  e.  RR* )
289limsupgf 13532 . . . . . . . 8  |-  ( n  e.  RR  |->  sup (
( ( F "
( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) : RR --> RR*
2928a1i 11 . . . . . . 7  |-  ( ph  ->  ( n  e.  RR  |->  sup ( ( ( F
" ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) : RR --> RR* )
3029ffvelrnda 6037 . . . . . 6  |-  ( (
ph  /\  k  e.  RR )  ->  ( ( n  e.  RR  |->  sup ( ( ( F
" ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k )  e.  RR* )
31 xrletr 11462 . . . . . 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 ) )
3227, 30, 8, 31syl3anc 1264 . . . . 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 ) )
3326, 32mpand 679 . . . 4  |-  ( (
ph  /\  k  e.  RR )  ->  ( ( ( n  e.  RR  |->  sup ( ( ( F
" ( n [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) `  k )  <_  A  ->  ( limsup `
 F )  <_  A ) )
3411, 33sylbird 238 . . 3  |-  ( (
ph  /\  k  e.  RR )  ->  ( A. j  e.  B  (
k  <_  j  ->  ( F `  j )  <_  A )  -> 
( limsup `  F )  <_  A ) )
3534rexlimdva 2914 . 2  |-  ( ph  ->  ( E. k  e.  RR  A. j  e.  B  ( k  <_ 
j  ->  ( F `  j )  <_  A
)  ->  ( limsup `  F )  <_  A
) )
361, 35mpd 15 1  |-  ( ph  ->  ( limsup `  F )  <_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    e. wcel 1872   A.wral 2771   E.wrex 2772   _Vcvv 3080    i^i cin 3435    C_ wss 3436   class class class wbr 4423    |-> cmpt 4482   "cima 4856   -->wf 5597   ` cfv 5601  (class class class)co 6305   supcsup 7963   RRcr 9545   +oocpnf 9679   RR*cxr 9681    < clt 9682    <_ cle 9683   [,)cico 11644   limsupclspold 13524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-sup 7965  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-ico 11648  df-limsupOLD 13526
This theorem is referenced by:  caucvgrlemOLD  13736  limsupreOLD  37662
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