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Theorem limsupgle 11951
Description: The defining property of the superior limit function. (Contributed by Mario Carneiro, 5-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
limsupgle  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( ( G `  C )  <_  A  <->  A. j  e.  B  ( C  <_  j  ->  ( F `  j )  <_  A ) ) )
Distinct variable groups:    A, j    B, j    j, G    j,
k, C    j, F, k
Allowed substitution hints:    A( k)    B( k)    G( k)

Proof of Theorem limsupgle
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 limsupval.1 . . . . 5  |-  G  =  ( k  e.  RR  |->  sup ( ( ( F
" ( k [,) 
+oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
21limsupgval 11950 . . . 4  |-  ( C  e.  RR  ->  ( G `  C )  =  sup ( ( ( F " ( C [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
323ad2ant2 977 . . 3  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( G `  C
)  =  sup (
( ( F "
( C [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
43breq1d 4033 . 2  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( ( G `  C )  <_  A  <->  sup ( ( ( F
" ( C [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )  <_  A ) )
5 inss2 3390 . . 3  |-  ( ( F " ( C [,)  +oo ) )  i^i  RR* )  C_  RR*
6 simp3 957 . . 3  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  A  e.  RR* )
7 supxrleub 10645 . . 3  |-  ( ( ( ( F "
( C [,)  +oo ) )  i^i  RR* )  C_  RR*  /\  A  e. 
RR* )  ->  ( sup ( ( ( F
" ( C [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )  <_  A  <->  A. x  e.  ( ( F "
( C [,)  +oo ) )  i^i  RR* ) x  <_  A ) )
85, 6, 7sylancr 644 . 2  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( sup ( ( ( F " ( C [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )  <_  A  <->  A. x  e.  ( ( F "
( C [,)  +oo ) )  i^i  RR* ) x  <_  A ) )
9 imassrn 5025 . . . . . . 7  |-  ( F
" ( C [,)  +oo ) )  C_  ran  F
10 simp1r 980 . . . . . . . 8  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  F : B --> RR* )
11 frn 5395 . . . . . . . 8  |-  ( F : B --> RR*  ->  ran 
F  C_  RR* )
1210, 11syl 15 . . . . . . 7  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ran  F  C_  RR* )
139, 12syl5ss 3190 . . . . . 6  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( F " ( C [,)  +oo ) )  C_  RR* )
14 df-ss 3166 . . . . . 6  |-  ( ( F " ( C [,)  +oo ) )  C_  RR*  <->  ( ( F " ( C [,)  +oo ) )  i^i  RR* )  =  ( F " ( C [,)  +oo ) ) )
1513, 14sylib 188 . . . . 5  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( ( F "
( C [,)  +oo ) )  i^i  RR* )  =  ( F " ( C [,)  +oo ) ) )
16 imadmres 5165 . . . . 5  |-  ( F
" dom  ( F  |`  ( C [,)  +oo ) ) )  =  ( F " ( C [,)  +oo ) )
1715, 16syl6eqr 2333 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( ( F "
( C [,)  +oo ) )  i^i  RR* )  =  ( F " dom  ( F  |`  ( C [,)  +oo )
) ) )
1817raleqdv 2742 . . 3  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( A. x  e.  ( ( F "
( C [,)  +oo ) )  i^i  RR* ) x  <_  A  <->  A. x  e.  ( F " dom  ( F  |`  ( C [,)  +oo ) ) ) x  <_  A )
)
19 ffn 5389 . . . . 5  |-  ( F : B --> RR*  ->  F  Fn  B )
2010, 19syl 15 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  F  Fn  B )
21 fdm 5393 . . . . . . . 8  |-  ( F : B --> RR*  ->  dom 
F  =  B )
2210, 21syl 15 . . . . . . 7  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  dom  F  =  B )
2322ineq2d 3370 . . . . . 6  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( ( C [,)  +oo )  i^i  dom  F
)  =  ( ( C [,)  +oo )  i^i  B ) )
24 dmres 4976 . . . . . 6  |-  dom  ( F  |`  ( C [,)  +oo ) )  =  ( ( C [,)  +oo )  i^i  dom  F )
25 incom 3361 . . . . . 6  |-  ( B  i^i  ( C [,)  +oo ) )  =  ( ( C [,)  +oo )  i^i  B )
2623, 24, 253eqtr4g 2340 . . . . 5  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  dom  ( F  |`  ( C [,)  +oo )
)  =  ( B  i^i  ( C [,)  +oo ) ) )
27 inss1 3389 . . . . . 6  |-  ( B  i^i  ( C [,)  +oo ) )  C_  B
2827a1i 10 . . . . 5  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( B  i^i  ( C [,)  +oo ) )  C_  B )
2926, 28eqsstrd 3212 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  dom  ( F  |`  ( C [,)  +oo )
)  C_  B )
30 breq1 4026 . . . . 5  |-  ( x  =  ( F `  j )  ->  (
x  <_  A  <->  ( F `  j )  <_  A
) )
3130ralima 5758 . . . 4  |-  ( ( F  Fn  B  /\  dom  ( F  |`  ( C [,)  +oo ) )  C_  B )  ->  ( A. x  e.  ( F " dom  ( F  |`  ( C [,)  +oo ) ) ) x  <_  A  <->  A. j  e.  dom  ( F  |`  ( C [,)  +oo )
) ( F `  j )  <_  A
) )
3220, 29, 31syl2anc 642 . . 3  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( A. x  e.  ( F " dom  ( F  |`  ( C [,)  +oo ) ) ) x  <_  A  <->  A. j  e.  dom  ( F  |`  ( C [,)  +oo )
) ( F `  j )  <_  A
) )
3326eleq2d 2350 . . . . . . . 8  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( j  e.  dom  ( F  |`  ( C [,)  +oo ) )  <->  j  e.  ( B  i^i  ( C [,)  +oo ) ) ) )
34 elin 3358 . . . . . . . 8  |-  ( j  e.  ( B  i^i  ( C [,)  +oo )
)  <->  ( j  e.  B  /\  j  e.  ( C [,)  +oo ) ) )
3533, 34syl6bb 252 . . . . . . 7  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( j  e.  dom  ( F  |`  ( C [,)  +oo ) )  <->  ( j  e.  B  /\  j  e.  ( C [,)  +oo ) ) ) )
36 simpl2 959 . . . . . . . . 9  |-  ( ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e. 
RR* )  /\  j  e.  B )  ->  C  e.  RR )
37 simp1l 979 . . . . . . . . . 10  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  B  C_  RR )
3837sselda 3180 . . . . . . . . 9  |-  ( ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e. 
RR* )  /\  j  e.  B )  ->  j  e.  RR )
39 elicopnf 10739 . . . . . . . . . 10  |-  ( C  e.  RR  ->  (
j  e.  ( C [,)  +oo )  <->  ( j  e.  RR  /\  C  <_ 
j ) ) )
4039baibd 875 . . . . . . . . 9  |-  ( ( C  e.  RR  /\  j  e.  RR )  ->  ( j  e.  ( C [,)  +oo )  <->  C  <_  j ) )
4136, 38, 40syl2anc 642 . . . . . . . 8  |-  ( ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e. 
RR* )  /\  j  e.  B )  ->  (
j  e.  ( C [,)  +oo )  <->  C  <_  j ) )
4241pm5.32da 622 . . . . . . 7  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( ( j  e.  B  /\  j  e.  ( C [,)  +oo ) )  <->  ( j  e.  B  /\  C  <_ 
j ) ) )
4335, 42bitrd 244 . . . . . 6  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( j  e.  dom  ( F  |`  ( C [,)  +oo ) )  <->  ( j  e.  B  /\  C  <_ 
j ) ) )
4443imbi1d 308 . . . . 5  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( ( j  e. 
dom  ( F  |`  ( C [,)  +oo )
)  ->  ( F `  j )  <_  A
)  <->  ( ( j  e.  B  /\  C  <_  j )  ->  ( F `  j )  <_  A ) ) )
45 impexp 433 . . . . 5  |-  ( ( ( j  e.  B  /\  C  <_  j )  ->  ( F `  j )  <_  A
)  <->  ( j  e.  B  ->  ( C  <_  j  ->  ( F `  j )  <_  A
) ) )
4644, 45syl6bb 252 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( ( j  e. 
dom  ( F  |`  ( C [,)  +oo )
)  ->  ( F `  j )  <_  A
)  <->  ( j  e.  B  ->  ( C  <_  j  ->  ( F `  j )  <_  A
) ) ) )
4746ralbidv2 2565 . . 3  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( A. j  e. 
dom  ( F  |`  ( C [,)  +oo )
) ( F `  j )  <_  A  <->  A. j  e.  B  ( C  <_  j  ->  ( F `  j )  <_  A ) ) )
4818, 32, 473bitrd 270 . 2  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( A. x  e.  ( ( F "
( C [,)  +oo ) )  i^i  RR* ) x  <_  A  <->  A. j  e.  B  ( C  <_  j  ->  ( F `  j )  <_  A
) ) )
494, 8, 483bitrd 270 1  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( ( G `  C )  <_  A  <->  A. j  e.  B  ( C  <_  j  ->  ( F `  j )  <_  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868   [,)cico 10658
This theorem is referenced by:  limsupgre  11955  limsupbnd1  11956  limsupbnd2  11957  mbflimsup  19021
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
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