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Theorem limsupgord 11946
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 8877 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  RR* )
213ad2ant1 976 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  A  e.  RR* )
3 simp3 957 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  A  <_  B )
4 df-ico 10662 . . . . . 6  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
5 xrletr 10489 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <_  B  /\  B  <_  w )  ->  A  <_  w
) )
64, 4, 5ixxss1 10674 . . . . 5  |-  ( ( A  e.  RR*  /\  A  <_  B )  ->  ( B [,)  +oo )  C_  ( A [,)  +oo ) )
72, 3, 6syl2anc 642 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( B [,)  +oo )  C_  ( A [,)  +oo ) )
8 imass2 5049 . . . 4  |-  ( ( B [,)  +oo )  C_  ( A [,)  +oo )  ->  ( F "
( B [,)  +oo ) )  C_  ( F " ( A [,)  +oo ) ) )
9 ssrin 3394 . . . 4  |-  ( ( F " ( B [,)  +oo ) )  C_  ( F " ( A [,)  +oo ) )  -> 
( ( F "
( B [,)  +oo ) )  i^i  RR* )  C_  ( ( F
" ( A [,)  +oo ) )  i^i  RR* ) )
107, 8, 93syl 18 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  (
( F " ( B [,)  +oo ) )  i^i  RR* )  C_  ( ( F " ( A [,)  +oo ) )  i^i  RR* ) )
11 inss2 3390 . . . . . 6  |-  ( ( F " ( A [,)  +oo ) )  i^i  RR* )  C_  RR*
12 supxrcl 10633 . . . . . 6  |-  ( ( ( F " ( A [,)  +oo ) )  i^i  RR* )  C_  RR*  ->  sup ( ( ( F
" ( A [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )  e.  RR* )
1311, 12ax-mp 8 . . . . 5  |-  sup (
( ( F "
( A [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )  e.  RR*
14 xrleid 10484 . . . . 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 8 . . . 4  |-  sup (
( ( F "
( A [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )  <_  sup ( ( ( F " ( A [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
16 supxrleub 10645 . . . . 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 653 . . . 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 199 . . 3  |-  A. x  e.  ( ( F "
( A [,)  +oo ) )  i^i  RR* ) x  <_  sup (
( ( F "
( A [,)  +oo ) )  i^i  RR* ) ,  RR* ,  <  )
19 ssralv 3237 . . 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, 19ee10 1366 . 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 3390 . . 3  |-  ( ( F " ( B [,)  +oo ) )  i^i  RR* )  C_  RR*
22 supxrleub 10645 . . 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 653 . 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 203 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152   class class class wbr 4023   "cima 4692  (class class class)co 5858   supcsup 7193   RRcr 8736    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868   [,)cico 10658
This theorem is referenced by:  limsupval2  11954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-ico 10662
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