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Theorem limsupgle 12271
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 12270 . . . 4  |-  ( C  e.  RR  ->  ( G `  C )  =  sup ( ( ( F " ( C [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
323ad2ant2 979 . . 3  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( G `  C
)  =  sup (
( ( F "
( C [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
43breq1d 4222 . 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 3562 . . 3  |-  ( ( F " ( C [,)  +oo ) )  i^i  RR* )  C_  RR*
6 simp3 959 . . 3  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  A  e.  RR* )
7 supxrleub 10905 . . 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 645 . 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 5216 . . . . . . 7  |-  ( F
" ( C [,)  +oo ) )  C_  ran  F
10 simp1r 982 . . . . . . . 8  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  F : B --> RR* )
11 frn 5597 . . . . . . . 8  |-  ( F : B --> RR*  ->  ran 
F  C_  RR* )
1210, 11syl 16 . . . . . . 7  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ran  F  C_  RR* )
139, 12syl5ss 3359 . . . . . 6  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( F " ( C [,)  +oo ) )  C_  RR* )
14 df-ss 3334 . . . . . 6  |-  ( ( F " ( C [,)  +oo ) )  C_  RR*  <->  ( ( F " ( C [,)  +oo ) )  i^i  RR* )  =  ( F " ( C [,)  +oo ) ) )
1513, 14sylib 189 . . . . 5  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( ( F "
( C [,)  +oo ) )  i^i  RR* )  =  ( F " ( C [,)  +oo ) ) )
16 imadmres 5362 . . . . 5  |-  ( F
" dom  ( F  |`  ( C [,)  +oo ) ) )  =  ( F " ( C [,)  +oo ) )
1715, 16syl6eqr 2486 . . . 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 2910 . . 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 5591 . . . . 5  |-  ( F : B --> RR*  ->  F  Fn  B )
2010, 19syl 16 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  F  Fn  B )
21 fdm 5595 . . . . . . . 8  |-  ( F : B --> RR*  ->  dom 
F  =  B )
2210, 21syl 16 . . . . . . 7  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  dom  F  =  B )
2322ineq2d 3542 . . . . . 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 5167 . . . . . 6  |-  dom  ( F  |`  ( C [,)  +oo ) )  =  ( ( C [,)  +oo )  i^i  dom  F )
25 incom 3533 . . . . . 6  |-  ( B  i^i  ( C [,)  +oo ) )  =  ( ( C [,)  +oo )  i^i  B )
2623, 24, 253eqtr4g 2493 . . . . 5  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  dom  ( F  |`  ( C [,)  +oo )
)  =  ( B  i^i  ( C [,)  +oo ) ) )
27 inss1 3561 . . . . 5  |-  ( B  i^i  ( C [,)  +oo ) )  C_  B
2826, 27syl6eqss 3398 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  dom  ( F  |`  ( C [,)  +oo )
)  C_  B )
29 breq1 4215 . . . . 5  |-  ( x  =  ( F `  j )  ->  (
x  <_  A  <->  ( F `  j )  <_  A
) )
3029ralima 5978 . . . 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
) )
3120, 28, 30syl2anc 643 . . 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
) )
3226eleq2d 2503 . . . . . . . 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 ) ) ) )
33 elin 3530 . . . . . . . 8  |-  ( j  e.  ( B  i^i  ( C [,)  +oo )
)  <->  ( j  e.  B  /\  j  e.  ( C [,)  +oo ) ) )
3432, 33syl6bb 253 . . . . . . 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 ) ) ) )
35 simpl2 961 . . . . . . . . 9  |-  ( ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e. 
RR* )  /\  j  e.  B )  ->  C  e.  RR )
36 simp1l 981 . . . . . . . . . 10  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  B  C_  RR )
3736sselda 3348 . . . . . . . . 9  |-  ( ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e. 
RR* )  /\  j  e.  B )  ->  j  e.  RR )
38 elicopnf 11000 . . . . . . . . . 10  |-  ( C  e.  RR  ->  (
j  e.  ( C [,)  +oo )  <->  ( j  e.  RR  /\  C  <_ 
j ) ) )
3938baibd 876 . . . . . . . . 9  |-  ( ( C  e.  RR  /\  j  e.  RR )  ->  ( j  e.  ( C [,)  +oo )  <->  C  <_  j ) )
4035, 37, 39syl2anc 643 . . . . . . . 8  |-  ( ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e. 
RR* )  /\  j  e.  B )  ->  (
j  e.  ( C [,)  +oo )  <->  C  <_  j ) )
4140pm5.32da 623 . . . . . . 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 ) ) )
4234, 41bitrd 245 . . . . . 6  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( j  e.  dom  ( F  |`  ( C [,)  +oo ) )  <->  ( j  e.  B  /\  C  <_ 
j ) ) )
4342imbi1d 309 . . . . 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 ) ) )
44 impexp 434 . . . . 5  |-  ( ( ( j  e.  B  /\  C  <_  j )  ->  ( F `  j )  <_  A
)  <->  ( j  e.  B  ->  ( C  <_  j  ->  ( F `  j )  <_  A
) ) )
4543, 44syl6bb 253 . . . 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
) ) ) )
4645ralbidv2 2727 . . 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 ) ) )
4718, 31, 463bitrd 271 . 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
) ) )
484, 8, 473bitrd 271 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705    i^i cin 3319    C_ wss 3320   class class class wbr 4212    e. cmpt 4266   dom cdm 4878   ran crn 4879    |` cres 4880   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   supcsup 7445   RRcr 8989    +oocpnf 9117   RR*cxr 9119    < clt 9120    <_ cle 9121   [,)cico 10918
This theorem is referenced by:  limsupgre  12275  limsupbnd1  12276  limsupbnd2  12277  mbflimsup  19558
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-ico 10922
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