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Theorem limsupgord 13261
Description: Ordering property of the superior limit function. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.)
Assertion
Ref Expression
limsupgord  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  sup ( ( ( F
" ( B [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  <_  sup ( ( ( F " ( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )

Proof of Theorem limsupgord
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexr 9640 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  RR* )
213ad2ant1 1017 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  A  e.  RR* )
3 simp3 998 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  A  <_  B )
4 df-ico 11536 . . . . . 6  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
5 xrletr 11362 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <_  B  /\  B  <_  w )  ->  A  <_  w
) )
64, 4, 5ixxss1 11548 . . . . 5  |-  ( ( A  e.  RR*  /\  A  <_  B )  ->  ( B [,) +oo )  C_  ( A [,) +oo )
)
72, 3, 6syl2anc 661 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( B [,) +oo )  C_  ( A [,) +oo )
)
8 imass2 5372 . . . 4  |-  ( ( B [,) +oo )  C_  ( A [,) +oo )  ->  ( F "
( B [,) +oo ) )  C_  ( F " ( A [,) +oo ) ) )
9 ssrin 3723 . . . 4  |-  ( ( F " ( B [,) +oo ) ) 
C_  ( F "
( A [,) +oo ) )  ->  (
( F " ( B [,) +oo ) )  i^i  RR* )  C_  (
( F " ( A [,) +oo ) )  i^i  RR* ) )
107, 8, 93syl 20 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( F " ( B [,) +oo ) )  i^i  RR* )  C_  (
( F " ( A [,) +oo ) )  i^i  RR* ) )
11 inss2 3719 . . . . . 6  |-  ( ( F " ( A [,) +oo ) )  i^i  RR* )  C_  RR*
12 supxrcl 11507 . . . . . 6  |-  ( ( ( F " ( A [,) +oo ) )  i^i  RR* )  C_  RR*  ->  sup ( ( ( F
" ( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  e.  RR* )
1311, 12ax-mp 5 . . . . 5  |-  sup (
( ( F "
( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  e.  RR*
14 xrleid 11357 . . . . 5  |-  ( sup ( ( ( F
" ( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  e.  RR*  ->  sup (
( ( F "
( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  <_  sup ( ( ( F " ( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
1513, 14ax-mp 5 . . . 4  |-  sup (
( ( F "
( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  <_  sup ( ( ( F " ( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )
16 supxrleub 11519 . . . . 5  |-  ( ( ( ( F "
( A [,) +oo ) )  i^i  RR* )  C_  RR*  /\  sup (
( ( F "
( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  e.  RR* )  ->  ( sup ( ( ( F
" ( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  <_  sup ( ( ( F " ( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  <->  A. x  e.  ( ( F "
( A [,) +oo ) )  i^i  RR* ) x  <_  sup (
( ( F "
( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) )
1711, 13, 16mp2an 672 . . . 4  |-  ( sup ( ( ( F
" ( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  <_  sup ( ( ( F " ( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  <->  A. x  e.  ( ( F "
( A [,) +oo ) )  i^i  RR* ) x  <_  sup (
( ( F "
( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
1815, 17mpbi 208 . . 3  |-  A. x  e.  ( ( F "
( A [,) +oo ) )  i^i  RR* ) x  <_  sup (
( ( F "
( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )
19 ssralv 3564 . . 3  |-  ( ( ( F " ( B [,) +oo ) )  i^i  RR* )  C_  (
( F " ( A [,) +oo ) )  i^i  RR* )  ->  ( A. x  e.  (
( F " ( A [,) +oo ) )  i^i  RR* ) x  <_  sup ( ( ( F
" ( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  ->  A. x  e.  ( ( F " ( B [,) +oo ) )  i^i  RR* ) x  <_  sup ( ( ( F
" ( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) )
2010, 18, 19mpisyl 18 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  A. x  e.  ( ( F "
( B [,) +oo ) )  i^i  RR* ) x  <_  sup (
( ( F "
( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
21 inss2 3719 . . 3  |-  ( ( F " ( B [,) +oo ) )  i^i  RR* )  C_  RR*
22 supxrleub 11519 . . 3  |-  ( ( ( ( F "
( B [,) +oo ) )  i^i  RR* )  C_  RR*  /\  sup (
( ( F "
( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  e.  RR* )  ->  ( sup ( ( ( F
" ( B [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  <_  sup ( ( ( F " ( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  <->  A. x  e.  ( ( F "
( B [,) +oo ) )  i^i  RR* ) x  <_  sup (
( ( F "
( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) ) )
2321, 13, 22mp2an 672 . 2  |-  ( sup ( ( ( F
" ( B [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  <_  sup ( ( ( F " ( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  <->  A. x  e.  ( ( F "
( B [,) +oo ) )  i^i  RR* ) x  <_  sup (
( ( F "
( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
2420, 23sylibr 212 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  sup ( ( ( F
" ( B [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  )  <_  sup ( ( ( F " ( A [,) +oo ) )  i^i  RR* ) ,  RR* ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    e. wcel 1767   A.wral 2814    i^i cin 3475    C_ wss 3476   class class class wbr 4447   "cima 5002  (class class class)co 6285   supcsup 7901   RRcr 9492   +oocpnf 9626   RR*cxr 9628    < clt 9629    <_ cle 9630   [,)cico 11532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-ico 11536
This theorem is referenced by:  limsupval2  13269
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