MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  limsupgle Structured version   Unicode version

Theorem limsupgle 13451
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 13450 . . . 4  |-  ( C  e.  RR  ->  ( G `  C )  =  sup ( ( ( F " ( C [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
323ad2ant2 1021 . . 3  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( G `  C
)  =  sup (
( ( F "
( C [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
43breq1d 4407 . 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 3662 . . 3  |-  ( ( F " ( C [,) +oo ) )  i^i  RR* )  C_  RR*
6 simp3 1001 . . 3  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  A  e.  RR* )
7 supxrleub 11573 . . 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 5170 . . . . . . 7  |-  ( F
" ( C [,) +oo ) )  C_  ran  F
10 simp1r 1024 . . . . . . . 8  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  F : B --> RR* )
11 frn 5722 . . . . . . . 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 3455 . . . . . 6  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( F " ( C [,) +oo ) ) 
C_  RR* )
14 df-ss 3430 . . . . . 6  |-  ( ( F " ( C [,) +oo ) ) 
C_  RR*  <->  ( ( F
" ( C [,) +oo ) )  i^i  RR* )  =  ( F " ( C [,) +oo ) ) )
1513, 14sylib 198 . . . . 5  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  ( ( F "
( C [,) +oo ) )  i^i  RR* )  =  ( F " ( C [,) +oo ) ) )
16 imadmres 5317 . . . . 5  |-  ( F
" dom  ( F  |`  ( C [,) +oo ) ) )  =  ( F " ( C [,) +oo ) )
1715, 16syl6eqr 2463 . . . 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 3012 . . 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 5716 . . . . 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 5720 . . . . . . . 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 3643 . . . . . 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 5116 . . . . . 6  |-  dom  ( F  |`  ( C [,) +oo ) )  =  ( ( C [,) +oo )  i^i  dom  F )
25 incom 3634 . . . . . 6  |-  ( B  i^i  ( C [,) +oo ) )  =  ( ( C [,) +oo )  i^i  B )
2623, 24, 253eqtr4g 2470 . . . . 5  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  dom  ( F  |`  ( C [,) +oo )
)  =  ( B  i^i  ( C [,) +oo ) ) )
27 inss1 3661 . . . . 5  |-  ( B  i^i  ( C [,) +oo ) )  C_  B
2826, 27syl6eqss 3494 . . . 4  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  dom  ( F  |`  ( C [,) +oo )
)  C_  B )
29 breq1 4400 . . . . 5  |-  ( x  =  ( F `  j )  ->  (
x  <_  A  <->  ( F `  j )  <_  A
) )
3029ralima 6135 . . . 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 2474 . . . . . . . 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 3628 . . . . . . . 8  |-  ( j  e.  ( B  i^i  ( C [,) +oo )
)  <->  ( j  e.  B  /\  j  e.  ( C [,) +oo ) ) )
3432, 33syl6bb 263 . . . . . . 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 1003 . . . . . . . . 9  |-  ( ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e. 
RR* )  /\  j  e.  B )  ->  C  e.  RR )
36 simp1l 1023 . . . . . . . . . 10  |-  ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e.  RR* )  ->  B  C_  RR )
3736sselda 3444 . . . . . . . . 9  |-  ( ( ( ( B  C_  RR  /\  F : B --> RR* )  /\  C  e.  RR  /\  A  e. 
RR* )  /\  j  e.  B )  ->  j  e.  RR )
38 elicopnf 11676 . . . . . . . . . 10  |-  ( C  e.  RR  ->  (
j  e.  ( C [,) +oo )  <->  ( j  e.  RR  /\  C  <_ 
j ) ) )
3938baibd 912 . . . . . . . . 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 255 . . . . . 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 263 . . . 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 2841 . . 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 281 . 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 281 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 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844   A.wral 2756    i^i cin 3415    C_ wss 3416   class class class wbr 4397    |-> cmpt 4455   dom cdm 4825   ran crn 4826    |` cres 4827   "cima 4828    Fn wfn 5566   -->wf 5567   ` cfv 5571  (class class class)co 6280   supcsup 7936   RRcr 9523   +oocpnf 9657   RR*cxr 9659    < clt 9660    <_ cle 9661   [,)cico 11586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-po 4746  df-so 4747  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-sup 7937  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-ico 11590
This theorem is referenced by:  limsupgre  13455  limsupbnd1  13456  limsupbnd2  13457  mbflimsup  22367
  Copyright terms: Public domain W3C validator