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Theorem limsuple 13303
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 995 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  F : B --> RR* )
2 reex 9494 . . . . . . 7  |-  RR  e.  _V
32ssex 4509 . . . . . 6  |-  ( B 
C_  RR  ->  B  e. 
_V )
433ad2ant1 1015 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  B  e.  _V )
5 xrex 11136 . . . . . 6  |-  RR*  e.  _V
65a1i 11 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  RR*  e.  _V )
7 fex2 6654 . . . . 5  |-  ( ( F : B --> RR*  /\  B  e.  _V  /\  RR*  e.  _V )  ->  F  e. 
_V )
81, 4, 6, 7syl3anc 1226 . . . 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 13299 . . . 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 4379 . 2  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( A  <_  ( limsup `  F
)  <->  A  <_  sup ( ran  G ,  RR* ,  `'  <  ) ) )
139limsupgf 13300 . . . . 5  |-  G : RR
--> RR*
14 frn 5645 . . . . 5  |-  ( G : RR --> RR*  ->  ran 
G  C_  RR* )
1513, 14ax-mp 5 . . . 4  |-  ran  G  C_ 
RR*
16 simp3 996 . . . 4  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  A  e.  RR* )
17 infmxrgelb 11447 . . . 4  |-  ( ( ran  G  C_  RR*  /\  A  e.  RR* )  ->  ( A  <_  sup ( ran  G ,  RR* ,  `'  <  )  <->  A. x  e.  ran  G  A  <_  x )
)
1815, 16, 17sylancr 661 . . 3  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( A  <_  sup ( ran  G ,  RR* ,  `'  <  )  <->  A. x  e.  ran  G  A  <_  x )
)
19 ffn 5639 . . . . 5  |-  ( G : RR --> RR*  ->  G  Fn  RR )
2013, 19ax-mp 5 . . . 4  |-  G  Fn  RR
21 breq2 4371 . . . . 5  |-  ( x  =  ( G `  j )  ->  ( A  <_  x  <->  A  <_  ( G `  j ) ) )
2221ralrn 5936 . . . 4  |-  ( G  Fn  RR  ->  ( A. x  e.  ran  G  A  <_  x  <->  A. j  e.  RR  A  <_  ( G `  j )
) )
2320, 22ax-mp 5 . . 3  |-  ( A. x  e.  ran  G  A  <_  x  <->  A. j  e.  RR  A  <_  ( G `  j ) )
2418, 23syl6bb 261 . 2  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( A  <_  sup ( ran  G ,  RR* ,  `'  <  )  <->  A. j  e.  RR  A  <_  ( G `  j ) ) )
2512, 24bitrd 253 1  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( A  <_  ( limsup `  F
)  <->  A. j  e.  RR  A  <_  ( G `  j ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 971    = wceq 1399    e. wcel 1826   A.wral 2732   _Vcvv 3034    i^i cin 3388    C_ wss 3389   class class class wbr 4367    |-> cmpt 4425   `'ccnv 4912   ran crn 4914   "cima 4916    Fn wfn 5491   -->wf 5492   ` cfv 5496  (class class class)co 6196   supcsup 7815   RRcr 9402   +oocpnf 9536   RR*cxr 9538    < clt 9539    <_ cle 9540   [,)cico 11452   limsupclsp 13295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-limsup 13296
This theorem is referenced by:  limsuplt  13304  limsupbnd1  13307  limsupbnd2  13308  mbflimsup  22158
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