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Theorem limsupgle 12976
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 12975 . . . 4  |-  ( C  e.  RR  ->  ( G `  C )  =  sup ( ( ( F " ( C [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
323ad2ant2 1010 . . 3  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( G `  C
)  =  sup (
( ( F "
( C [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
43breq1d 4323 . 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 3592 . . 3  |-  ( ( F " ( C [,) +oo ) )  i^i  RR* )  C_  RR*
6 simp3 990 . . 3  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  A  e.  RR* )
7 supxrleub 11310 . . 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 663 . 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 5201 . . . . . . 7  |-  ( F
" ( C [,) +oo ) )  C_  ran  F
10 simp1r 1013 . . . . . . . 8  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  F : B --> RR* )
11 frn 5586 . . . . . . . 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 3388 . . . . . 6  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( F " ( C [,) +oo ) ) 
C_  RR* )
14 df-ss 3363 . . . . . 6  |-  ( ( F " ( C [,) +oo ) ) 
C_  RR*  <->  ( ( F
" ( C [,) +oo ) )  i^i  RR* )  =  ( F " ( C [,) +oo ) ) )
1513, 14sylib 196 . . . . 5  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( ( F "
( C [,) +oo ) )  i^i  RR* )  =  ( F " ( C [,) +oo ) ) )
16 imadmres 5351 . . . . 5  |-  ( F
" dom  ( F  |`  ( C [,) +oo ) ) )  =  ( F " ( C [,) +oo ) )
1715, 16syl6eqr 2493 . . . 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 2944 . . 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 5580 . . . . 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 5584 . . . . . . . 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 3573 . . . . . 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 5152 . . . . . 6  |-  dom  ( F  |`  ( C [,) +oo ) )  =  ( ( C [,) +oo )  i^i  dom  F )
25 incom 3564 . . . . . 6  |-  ( B  i^i  ( C [,) +oo ) )  =  ( ( C [,) +oo )  i^i  B )
2623, 24, 253eqtr4g 2500 . . . . 5  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  dom  ( F  |`  ( C [,) +oo )
)  =  ( B  i^i  ( C [,) +oo ) ) )
27 inss1 3591 . . . . 5  |-  ( B  i^i  ( C [,) +oo ) )  C_  B
2826, 27syl6eqss 3427 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  dom  ( F  |`  ( C [,) +oo )
)  C_  B )
29 breq1 4316 . . . . 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 661 . . 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 2510 . . . . . . . 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 3560 . . . . . . . 8  |-  ( j  e.  ( B  i^i  ( C [,) +oo )
)  <->  ( j  e.  B  /\  j  e.  ( C [,) +oo ) ) )
3432, 33syl6bb 261 . . . . . . 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 992 . . . . . . . . 9  |-  ( ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e. 
RR* )  /\  j  e.  B )  ->  C  e.  RR )
36 simp1l 1012 . . . . . . . . . 10  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  B  C_  RR )
3736sselda 3377 . . . . . . . . 9  |-  ( ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e. 
RR* )  /\  j  e.  B )  ->  j  e.  RR )
38 elicopnf 11406 . . . . . . . . . 10  |-  ( C  e.  RR  ->  (
j  e.  ( C [,) +oo )  <->  ( j  e.  RR  /\  C  <_ 
j ) ) )
3938baibd 900 . . . . . . . . 9  |-  ( ( C  e.  RR  /\  j  e.  RR )  ->  ( j  e.  ( C [,) +oo )  <->  C  <_  j ) )
4035, 37, 39syl2anc 661 . . . . . . . 8  |-  ( ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e. 
RR* )  /\  j  e.  B )  ->  (
j  e.  ( C [,) +oo )  <->  C  <_  j ) )
4140pm5.32da 641 . . . . . . 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 253 . . . . . 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 317 . . . . 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 446 . . . . 5  |-  ( ( ( j  e.  B  /\  C  <_  j )  ->  ( F `  j )  <_  A
)  <->  ( j  e.  B  ->  ( C  <_  j  ->  ( F `  j )  <_  A
) ) )
4543, 44syl6bb 261 . . . 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 2758 . . 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 279 . 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 279 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2736    i^i cin 3348    C_ wss 3349   class class class wbr 4313    e. cmpt 4371   dom cdm 4861   ran crn 4862    |` cres 4863   "cima 4864    Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112   supcsup 7711   RRcr 9302   +oocpnf 9436   RR*cxr 9438    < clt 9439    <_ cle 9440   [,)cico 11323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-po 4662  df-so 4663  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-ico 11327
This theorem is referenced by:  limsupgre  12980  limsupbnd1  12981  limsupbnd2  12982  mbflimsup  21166
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