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

Theorem limsupgord 13306
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 9656 . . . . . 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 11560 . . . . . 6  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
5 xrletr 11386 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A  <_  B  /\  B  <_  w )  ->  A  <_  w
) )
64, 4, 5ixxss1 11572 . . . . 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 5382 . . . 4  |-  ( ( B [,) +oo )  C_  ( A [,) +oo )  ->  ( F "
( B [,) +oo ) )  C_  ( F " ( A [,) +oo ) ) )
9 ssrin 3719 . . . 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 3715 . . . . . 6  |-  ( ( F " ( A [,) +oo ) )  i^i  RR* )  C_  RR*
12 supxrcl 11531 . . . . . 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 11381 . . . . 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 11543 . . . . 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 3560 . . 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 3715 . . 3  |-  ( ( F " ( B [,) +oo ) )  i^i  RR* )  C_  RR*
22 supxrleub 11543 . . 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 1819   A.wral 2807    i^i cin 3470    C_ wss 3471   class class class wbr 4456   "cima 5011  (class class class)co 6296   supcsup 7918   RRcr 9508   +oocpnf 9642   RR*cxr 9644    < clt 9645    <_ cle 9646   [,)cico 11556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-ico 11560
This theorem is referenced by:  limsupval2  13314
  Copyright terms: Public domain W3C validator