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Theorem limsupgord 11962
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 8893 . . . . . 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 10678 . . . . . 6  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
5 xrletr 10505 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <_  B  /\  B  <_  w )  ->  A  <_  w
) )
64, 4, 5ixxss1 10690 . . . . 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 5065 . . . 4  |-  ( ( B [,)  +oo )  C_  ( A [,)  +oo )  ->  ( F "
( B [,)  +oo ) )  C_  ( F " ( A [,)  +oo ) ) )
9 ssrin 3407 . . . 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 3403 . . . . . 6  |-  ( ( F " ( A [,)  +oo ) )  i^i  RR* )  C_  RR*
12 supxrcl 10649 . . . . . 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 10500 . . . . 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 10661 . . . . 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 3250 . . 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 3403 . . 3  |-  ( ( F " ( B [,)  +oo ) )  i^i  RR* )  C_  RR*
22 supxrleub 10661 . . 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 1696   A.wral 2556    i^i cin 3164    C_ wss 3165   class class class wbr 4039   "cima 4708  (class class class)co 5874   supcsup 7209   RRcr 8752    +oocpnf 8880   RR*cxr 8882    < clt 8883    <_ cle 8884   [,)cico 10674
This theorem is referenced by:  limsupval2  11970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-ico 10678
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