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Theorem limsupgle 13528
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:    k, F    A, j    B, j    C, j, k    j, F
Allowed substitution hints:    A( k)    B( k)    G( j, 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 13527 . . . 4  |-  ( C  e.  RR  ->  ( G `  C )  =  sup ( ( ( F " ( C [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
323ad2ant2 1029 . . 3  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( G `  C
)  =  sup (
( ( F "
( C [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
43breq1d 4411 . 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 3652 . . 3  |-  ( ( F " ( C [,) +oo ) )  i^i  RR* )  C_  RR*
6 simp3 1009 . . 3  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  A  e.  RR* )
7 supxrleub 11609 . . 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 668 . 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 5178 . . . . . . 7  |-  ( F
" ( C [,) +oo ) )  C_  ran  F
10 simp1r 1032 . . . . . . . 8  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  F : B --> RR* )
11 frn 5733 . . . . . . . 8  |-  ( F : B --> RR*  ->  ran 
F  C_  RR* )
1210, 11syl 17 . . . . . . 7  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ran  F  C_  RR* )
139, 12syl5ss 3442 . . . . . 6  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( F " ( C [,) +oo ) ) 
C_  RR* )
14 df-ss 3417 . . . . . 6  |-  ( ( F " ( C [,) +oo ) ) 
C_  RR*  <->  ( ( F
" ( C [,) +oo ) )  i^i  RR* )  =  ( F " ( C [,) +oo ) ) )
1513, 14sylib 200 . . . . 5  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( ( F "
( C [,) +oo ) )  i^i  RR* )  =  ( F " ( C [,) +oo ) ) )
16 imadmres 5326 . . . . 5  |-  ( F
" dom  ( F  |`  ( C [,) +oo ) ) )  =  ( F " ( C [,) +oo ) )
1715, 16syl6eqr 2502 . . . 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 2992 . . 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 5726 . . . . 5  |-  ( F : B --> RR*  ->  F  Fn  B )
2010, 19syl 17 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  F  Fn  B )
21 fdm 5731 . . . . . . . 8  |-  ( F : B --> RR*  ->  dom 
F  =  B )
2210, 21syl 17 . . . . . . 7  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  dom  F  =  B )
2322ineq2d 3633 . . . . . 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 5124 . . . . . 6  |-  dom  ( F  |`  ( C [,) +oo ) )  =  ( ( C [,) +oo )  i^i  dom  F )
25 incom 3624 . . . . . 6  |-  ( B  i^i  ( C [,) +oo ) )  =  ( ( C [,) +oo )  i^i  B )
2623, 24, 253eqtr4g 2509 . . . . 5  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  dom  ( F  |`  ( C [,) +oo )
)  =  ( B  i^i  ( C [,) +oo ) ) )
27 inss1 3651 . . . . 5  |-  ( B  i^i  ( C [,) +oo ) )  C_  B
2826, 27syl6eqss 3481 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  dom  ( F  |`  ( C [,) +oo )
)  C_  B )
29 breq1 4404 . . . . 5  |-  ( x  =  ( F `  j )  ->  (
x  <_  A  <->  ( F `  j )  <_  A
) )
3029ralima 6143 . . . 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 666 . . 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 2513 . . . . . . . 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 3616 . . . . . . . 8  |-  ( j  e.  ( B  i^i  ( C [,) +oo )
)  <->  ( j  e.  B  /\  j  e.  ( C [,) +oo ) ) )
3432, 33syl6bb 265 . . . . . . 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 1011 . . . . . . . . 9  |-  ( ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e. 
RR* )  /\  j  e.  B )  ->  C  e.  RR )
36 simp1l 1031 . . . . . . . . . 10  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  B  C_  RR )
3736sselda 3431 . . . . . . . . 9  |-  ( ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e. 
RR* )  /\  j  e.  B )  ->  j  e.  RR )
38 elicopnf 11727 . . . . . . . . . 10  |-  ( C  e.  RR  ->  (
j  e.  ( C [,) +oo )  <->  ( j  e.  RR  /\  C  <_ 
j ) ) )
3938baibd 919 . . . . . . . . 9  |-  ( ( C  e.  RR  /\  j  e.  RR )  ->  ( j  e.  ( C [,) +oo )  <->  C  <_  j ) )
4035, 37, 39syl2anc 666 . . . . . . . 8  |-  ( ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e. 
RR* )  /\  j  e.  B )  ->  (
j  e.  ( C [,) +oo )  <->  C  <_  j ) )
4140pm5.32da 646 . . . . . . 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 257 . . . . . 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 319 . . . . 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 448 . . . . 5  |-  ( ( ( j  e.  B  /\  C  <_  j )  ->  ( F `  j )  <_  A
)  <->  ( j  e.  B  ->  ( C  <_  j  ->  ( F `  j )  <_  A
) ) )
4543, 44syl6bb 265 . . . 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 2822 . . 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 283 . 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 283 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 setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886   A.wral 2736    i^i cin 3402    C_ wss 3403   class class class wbr 4401    |-> cmpt 4460   dom cdm 4833   ran crn 4834    |` cres 4835   "cima 4836    Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6288   supcsup 7951   RRcr 9535   +oocpnf 9669   RR*cxr 9671    < clt 9672    <_ cle 9673   [,)cico 11634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-po 4754  df-so 4755  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-sup 7953  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-ico 11638
This theorem is referenced by:  limsupgre  13535  limsupgreOLD  13536  limsupbnd1  13537  limsupbnd1OLD  13538  limsupbnd2  13539  limsupbnd2OLD  13540  mbflimsup  22616  mbflimsupOLD  22617
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