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Theorem limsuple 12264
Description: The defining property of the superior limit. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
Hypothesis
Ref Expression
limsupval.1  |-  G  =  ( k  e.  RR  |->  sup ( ( ( F
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
Assertion
Ref Expression
limsuple  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( A  <_  ( limsup `  F
)  <->  A. j  e.  RR  A  <_  ( G `  j ) ) )
Distinct variable groups:    A, j    B, j    j, G    j,
k, F
Allowed substitution hints:    A( k)    B( k)    G( k)

Proof of Theorem limsuple
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp2 958 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  F : B --> RR* )
2 reex 9073 . . . . . . 7  |-  RR  e.  _V
32ssex 4339 . . . . . 6  |-  ( B 
C_  RR  ->  B  e. 
_V )
433ad2ant1 978 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  B  e.  _V )
5 xrex 10601 . . . . . 6  |-  RR*  e.  _V
65a1i 11 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  RR*  e.  _V )
7 fex2 5595 . . . . 5  |-  ( ( F : B --> RR*  /\  B  e.  _V  /\  RR*  e.  _V )  ->  F  e. 
_V )
81, 4, 6, 7syl3anc 1184 . . . 4  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  F  e.  _V )
9 limsupval.1 . . . . 5  |-  G  =  ( k  e.  RR  |->  sup ( ( ( F
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
109limsupval 12260 . . . 4  |-  ( F  e.  _V  ->  ( limsup `
 F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
118, 10syl 16 . . 3  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( limsup `
 F )  =  sup ( ran  G ,  RR* ,  `'  <  ) )
1211breq2d 4216 . 2  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( A  <_  ( limsup `  F
)  <->  A  <_  sup ( ran  G ,  RR* ,  `'  <  ) ) )
139limsupgf 12261 . . . . 5  |-  G : RR
--> RR*
14 frn 5589 . . . . 5  |-  ( G : RR --> RR*  ->  ran 
G  C_  RR* )
1513, 14ax-mp 8 . . . 4  |-  ran  G  C_ 
RR*
16 simp3 959 . . . 4  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  A  e.  RR* )
17 infmxrgelb 10905 . . . 4  |-  ( ( ran  G  C_  RR*  /\  A  e.  RR* )  ->  ( A  <_  sup ( ran  G ,  RR* ,  `'  <  )  <->  A. x  e.  ran  G  A  <_  x )
)
1815, 16, 17sylancr 645 . . 3  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( A  <_  sup ( ran  G ,  RR* ,  `'  <  )  <->  A. x  e.  ran  G  A  <_  x )
)
19 ffn 5583 . . . . 5  |-  ( G : RR --> RR*  ->  G  Fn  RR )
2013, 19ax-mp 8 . . . 4  |-  G  Fn  RR
21 breq2 4208 . . . . 5  |-  ( x  =  ( G `  j )  ->  ( A  <_  x  <->  A  <_  ( G `  j ) ) )
2221ralrn 5865 . . . 4  |-  ( G  Fn  RR  ->  ( A. x  e.  ran  G  A  <_  x  <->  A. j  e.  RR  A  <_  ( G `  j )
) )
2320, 22ax-mp 8 . . 3  |-  ( A. x  e.  ran  G  A  <_  x  <->  A. j  e.  RR  A  <_  ( G `  j ) )
2418, 23syl6bb 253 . 2  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( A  <_  sup ( ran  G ,  RR* ,  `'  <  )  <->  A. j  e.  RR  A  <_  ( G `  j ) ) )
2512, 24bitrd 245 1  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( A  <_  ( limsup `  F
)  <->  A. j  e.  RR  A  <_  ( G `  j ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    i^i cin 3311    C_ wss 3312   class class class wbr 4204    e. cmpt 4258   `'ccnv 4869   ran crn 4871   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   supcsup 7437   RRcr 8981    +oocpnf 9109   RR*cxr 9111    < clt 9112    <_ cle 9113   [,)cico 10910   limsupclsp 12256
This theorem is referenced by:  limsuplt  12265  limsupbnd1  12268  limsupbnd2  12269  mbflimsup  19550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-limsup 12257
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