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Theorem limsuple 13058
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 989 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  F : B --> RR* )
2 reex 9474 . . . . . . 7  |-  RR  e.  _V
32ssex 4534 . . . . . 6  |-  ( B 
C_  RR  ->  B  e. 
_V )
433ad2ant1 1009 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  B  e.  _V )
5 xrex 11089 . . . . . 6  |-  RR*  e.  _V
65a1i 11 . . . . 5  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  RR*  e.  _V )
7 fex2 6632 . . . . 5  |-  ( ( F : B --> RR*  /\  B  e.  _V  /\  RR*  e.  _V )  ->  F  e. 
_V )
81, 4, 6, 7syl3anc 1219 . . . 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 13054 . . . 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 4402 . 2  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( A  <_  ( limsup `  F
)  <->  A  <_  sup ( ran  G ,  RR* ,  `'  <  ) ) )
139limsupgf 13055 . . . . 5  |-  G : RR
--> RR*
14 frn 5663 . . . . 5  |-  ( G : RR --> RR*  ->  ran 
G  C_  RR* )
1513, 14ax-mp 5 . . . 4  |-  ran  G  C_ 
RR*
16 simp3 990 . . . 4  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  A  e.  RR* )
17 infmxrgelb 11398 . . . 4  |-  ( ( ran  G  C_  RR*  /\  A  e.  RR* )  ->  ( A  <_  sup ( ran  G ,  RR* ,  `'  <  )  <->  A. x  e.  ran  G  A  <_  x )
)
1815, 16, 17sylancr 663 . . 3  |-  ( ( B  C_  RR  /\  F : B --> RR*  /\  A  e. 
RR* )  ->  ( A  <_  sup ( ran  G ,  RR* ,  `'  <  )  <->  A. x  e.  ran  G  A  <_  x )
)
19 ffn 5657 . . . . 5  |-  ( G : RR --> RR*  ->  G  Fn  RR )
2013, 19ax-mp 5 . . . 4  |-  G  Fn  RR
21 breq2 4394 . . . . 5  |-  ( x  =  ( G `  j )  ->  ( A  <_  x  <->  A  <_  ( G `  j ) ) )
2221ralrn 5945 . . . 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 965    = wceq 1370    e. wcel 1758   A.wral 2795   _Vcvv 3068    i^i cin 3425    C_ wss 3426   class class class wbr 4390    |-> cmpt 4448   `'ccnv 4937   ran crn 4939   "cima 4941    Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190   supcsup 7791   RRcr 9382   +oocpnf 9516   RR*cxr 9518    < clt 9519    <_ cle 9520   [,)cico 11403   limsupclsp 13050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-po 4739  df-so 4740  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-limsup 13051
This theorem is referenced by:  limsuplt  13059  limsupbnd1  13062  limsupbnd2  13063  mbflimsup  21260
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