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Theorem limsupgle 11967
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 11966 . . . 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 4049 . 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 3403 . . 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 10661 . . 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 5041 . . . . . . 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 5411 . . . . . . . 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 3203 . . . . . 6  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( F " ( C [,)  +oo ) )  C_  RR* )
14 df-ss 3179 . . . . . 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 5181 . . . . 5  |-  ( F
" dom  ( F  |`  ( C [,)  +oo ) ) )  =  ( F " ( C [,)  +oo ) )
1715, 16syl6eqr 2346 . . . 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 2755 . . 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 5405 . . . . 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 5409 . . . . . . . 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 3383 . . . . . 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 4992 . . . . . 6  |-  dom  ( F  |`  ( C [,)  +oo ) )  =  ( ( C [,)  +oo )  i^i  dom  F )
25 incom 3374 . . . . . 6  |-  ( B  i^i  ( C [,)  +oo ) )  =  ( ( C [,)  +oo )  i^i  B )
2623, 24, 253eqtr4g 2353 . . . . 5  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  dom  ( F  |`  ( C [,)  +oo )
)  =  ( B  i^i  ( C [,)  +oo ) ) )
27 inss1 3402 . . . . . 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 3225 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  dom  ( F  |`  ( C [,)  +oo )
)  C_  B )
30 breq1 4042 . . . . 5  |-  ( x  =  ( F `  j )  ->  (
x  <_  A  <->  ( F `  j )  <_  A
) )
3130ralima 5774 . . . 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 2363 . . . . . . . 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 3371 . . . . . . . 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 3193 . . . . . . . . 9  |-  ( ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e. 
RR* )  /\  j  e.  B )  ->  j  e.  RR )
39 elicopnf 10755 . . . . . . . . . 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 2578 . . 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 1632    e. wcel 1696   A.wral 2556    i^i cin 3164    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752    +oocpnf 8880   RR*cxr 8882    < clt 8883    <_ cle 8884   [,)cico 10674
This theorem is referenced by:  limsupgre  11971  limsupbnd1  11972  limsupbnd2  11973  mbflimsup  19037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-ico 10678
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