Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pnfneige0 Structured version   Unicode version

Theorem pnfneige0 28622
Description: A neighborhood of +oo contains an unbounded interval based at a real number. See pnfnei 20160 (Contributed by Thierry Arnoux, 31-Jul-2017.)
Hypothesis
Ref Expression
pnfneige0.j  |-  J  =  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )
Assertion
Ref Expression
pnfneige0  |-  ( ( A  e.  J  /\ +oo  e.  A )  ->  E. x  e.  RR  ( x (,] +oo )  C_  A )
Distinct variable group:    x, A
Allowed substitution hint:    J( x)

Proof of Theorem pnfneige0
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 0red 9633 . . . 4  |-  ( ( ( ( ( A  e.  J  /\ +oo  e.  A )  /\  y  e.  RR )  /\  (
y (,] +oo )  C_  ( A  i^i  (
0 (,] +oo )
) )  /\  y  <  0 )  ->  0  e.  RR )
2 simpllr 767 . . . 4  |-  ( ( ( ( ( A  e.  J  /\ +oo  e.  A )  /\  y  e.  RR )  /\  (
y (,] +oo )  C_  ( A  i^i  (
0 (,] +oo )
) )  /\  -.  y  <  0 )  -> 
y  e.  RR )
31, 2ifclda 3938 . . 3  |-  ( ( ( ( A  e.  J  /\ +oo  e.  A )  /\  y  e.  RR )  /\  (
y (,] +oo )  C_  ( A  i^i  (
0 (,] +oo )
) )  ->  if ( y  <  0 ,  0 ,  y )  e.  RR )
4 rexr 9675 . . . . . . 7  |-  ( y  e.  RR  ->  y  e.  RR* )
5 0xr 9676 . . . . . . . 8  |-  0  e.  RR*
65a1i 11 . . . . . . 7  |-  ( y  e.  RR  ->  0  e.  RR* )
7 pnfxr 11401 . . . . . . . 8  |- +oo  e.  RR*
87a1i 11 . . . . . . 7  |-  ( y  e.  RR  -> +oo  e.  RR* )
9 iocinif 28225 . . . . . . 7  |-  ( ( y  e.  RR*  /\  0  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
y (,] +oo )  i^i  ( 0 (,] +oo ) )  =  if ( y  <  0 ,  ( 0 (,] +oo ) ,  ( y (,] +oo ) ) )
104, 6, 8, 9syl3anc 1264 . . . . . 6  |-  ( y  e.  RR  ->  (
( y (,] +oo )  i^i  ( 0 (,] +oo ) )  =  if ( y  <  0 ,  ( 0 (,] +oo ) ,  ( y (,] +oo ) ) )
11 ovif 6378 . . . . . 6  |-  ( if ( y  <  0 ,  0 ,  y ) (,] +oo )  =  if ( y  <  0 ,  ( 0 (,] +oo ) ,  ( y (,] +oo ) )
1210, 11syl6reqr 2480 . . . . 5  |-  ( y  e.  RR  ->  ( if ( y  <  0 ,  0 ,  y ) (,] +oo )  =  ( ( y (,] +oo )  i^i  ( 0 (,] +oo ) ) )
1312ad2antlr 731 . . . 4  |-  ( ( ( ( A  e.  J  /\ +oo  e.  A )  /\  y  e.  RR )  /\  (
y (,] +oo )  C_  ( A  i^i  (
0 (,] +oo )
) )  ->  ( if ( y  <  0 ,  0 ,  y ) (,] +oo )  =  ( ( y (,] +oo )  i^i  ( 0 (,] +oo ) ) )
14 iocssicc 11711 . . . . . 6  |-  ( 0 (,] +oo )  C_  ( 0 [,] +oo )
15 sslin 3685 . . . . . 6  |-  ( ( 0 (,] +oo )  C_  ( 0 [,] +oo )  ->  ( ( y (,] +oo )  i^i  ( 0 (,] +oo ) )  C_  (
( y (,] +oo )  i^i  ( 0 [,] +oo ) ) )
1614, 15mp1i 13 . . . . 5  |-  ( ( ( ( A  e.  J  /\ +oo  e.  A )  /\  y  e.  RR )  /\  (
y (,] +oo )  C_  ( A  i^i  (
0 (,] +oo )
) )  ->  (
( y (,] +oo )  i^i  ( 0 (,] +oo ) )  C_  (
( y (,] +oo )  i^i  ( 0 [,] +oo ) ) )
17 simpr 462 . . . . . 6  |-  ( ( ( ( A  e.  J  /\ +oo  e.  A )  /\  y  e.  RR )  /\  (
y (,] +oo )  C_  ( A  i^i  (
0 (,] +oo )
) )  ->  (
y (,] +oo )  C_  ( A  i^i  (
0 (,] +oo )
) )
18 ssin 3681 . . . . . . . 8  |-  ( ( ( y (,] +oo )  C_  A  /\  (
y (,] +oo )  C_  ( 0 (,] +oo ) )  <->  ( y (,] +oo )  C_  ( A  i^i  ( 0 (,] +oo ) ) )
1918biimpri 209 . . . . . . 7  |-  ( ( y (,] +oo )  C_  ( A  i^i  (
0 (,] +oo )
)  ->  ( (
y (,] +oo )  C_  A  /\  ( y (,] +oo )  C_  ( 0 (,] +oo ) ) )
2019simpld 460 . . . . . 6  |-  ( ( y (,] +oo )  C_  ( A  i^i  (
0 (,] +oo )
)  ->  ( y (,] +oo )  C_  A
)
21 ssinss1 3687 . . . . . 6  |-  ( ( y (,] +oo )  C_  A  ->  ( (
y (,] +oo )  i^i  ( 0 [,] +oo ) )  C_  A
)
2217, 20, 213syl 18 . . . . 5  |-  ( ( ( ( A  e.  J  /\ +oo  e.  A )  /\  y  e.  RR )  /\  (
y (,] +oo )  C_  ( A  i^i  (
0 (,] +oo )
) )  ->  (
( y (,] +oo )  i^i  ( 0 [,] +oo ) )  C_  A
)
2316, 22sstrd 3471 . . . 4  |-  ( ( ( ( A  e.  J  /\ +oo  e.  A )  /\  y  e.  RR )  /\  (
y (,] +oo )  C_  ( A  i^i  (
0 (,] +oo )
) )  ->  (
( y (,] +oo )  i^i  ( 0 (,] +oo ) )  C_  A
)
2413, 23eqsstrd 3495 . . 3  |-  ( ( ( ( A  e.  J  /\ +oo  e.  A )  /\  y  e.  RR )  /\  (
y (,] +oo )  C_  ( A  i^i  (
0 (,] +oo )
) )  ->  ( if ( y  <  0 ,  0 ,  y ) (,] +oo )  C_  A )
25 oveq1 6303 . . . . 5  |-  ( x  =  if ( y  <  0 ,  0 ,  y )  -> 
( x (,] +oo )  =  ( if ( y  <  0 ,  0 ,  y ) (,] +oo )
)
2625sseq1d 3488 . . . 4  |-  ( x  =  if ( y  <  0 ,  0 ,  y )  -> 
( ( x (,] +oo )  C_  A  <->  ( if ( y  <  0 ,  0 ,  y ) (,] +oo )  C_  A ) )
2726rspcev 3179 . . 3  |-  ( ( if ( y  <  0 ,  0 ,  y )  e.  RR  /\  ( if ( y  <  0 ,  0 ,  y ) (,] +oo )  C_  A )  ->  E. x  e.  RR  ( x (,] +oo )  C_  A )
283, 24, 27syl2anc 665 . 2  |-  ( ( ( ( A  e.  J  /\ +oo  e.  A )  /\  y  e.  RR )  /\  (
y (,] +oo )  C_  ( A  i^i  (
0 (,] +oo )
) )  ->  E. x  e.  RR  ( x (,] +oo )  C_  A )
29 letopon 20145 . . . . . . . . . 10  |-  (ordTop `  <_  )  e.  (TopOn `  RR* )
30 iccssxr 11706 . . . . . . . . . 10  |-  ( 0 [,] +oo )  C_  RR*
31 resttopon 20101 . . . . . . . . . 10  |-  ( ( (ordTop `  <_  )  e.  (TopOn `  RR* )  /\  ( 0 [,] +oo )  C_  RR* )  ->  (
(ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  (TopOn `  (
0 [,] +oo )
) )
3229, 30, 31mp2an 676 . . . . . . . . 9  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  (TopOn `  ( 0 [,] +oo ) )
3332topontopi 19870 . . . . . . . 8  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  Top
3433a1i 11 . . . . . . 7  |-  ( A  e.  J  ->  (
(ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  Top )
35 ovex 6324 . . . . . . . 8  |-  ( 0 (,] +oo )  e. 
_V
3635a1i 11 . . . . . . 7  |-  ( A  e.  J  ->  (
0 (,] +oo )  e.  _V )
37 pnfneige0.j . . . . . . . . . 10  |-  J  =  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )
38 xrge0topn 28614 . . . . . . . . . 10  |-  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
3937, 38eqtri 2449 . . . . . . . . 9  |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
4039eleq2i 2498 . . . . . . . 8  |-  ( A  e.  J  <->  A  e.  ( (ordTop `  <_  )t  ( 0 [,] +oo ) ) )
4140biimpi 197 . . . . . . 7  |-  ( A  e.  J  ->  A  e.  ( (ordTop `  <_  )t  ( 0 [,] +oo )
) )
42 elrestr 15279 . . . . . . 7  |-  ( ( ( (ordTop `  <_  )t  ( 0 [,] +oo )
)  e.  Top  /\  ( 0 (,] +oo )  e.  _V  /\  A  e.  ( (ordTop `  <_  )t  ( 0 [,] +oo )
) )  ->  ( A  i^i  ( 0 (,] +oo ) )  e.  ( ( (ordTop `  <_  )t  ( 0 [,] +oo )
)t  ( 0 (,] +oo ) ) )
4334, 36, 41, 42syl3anc 1264 . . . . . 6  |-  ( A  e.  J  ->  ( A  i^i  ( 0 (,] +oo ) )  e.  ( ( (ordTop `  <_  )t  ( 0 [,] +oo )
)t  ( 0 (,] +oo ) ) )
44 letop 20146 . . . . . . 7  |-  (ordTop `  <_  )  e.  Top
45 ovex 6324 . . . . . . 7  |-  ( 0 [,] +oo )  e. 
_V
46 restabs 20105 . . . . . . 7  |-  ( ( (ordTop `  <_  )  e. 
Top  /\  ( 0 (,] +oo )  C_  ( 0 [,] +oo )  /\  ( 0 [,] +oo )  e.  _V )  ->  ( ( (ordTop `  <_  )t  ( 0 [,] +oo ) )t  ( 0 (,] +oo ) )  =  ( (ordTop `  <_  )t  ( 0 (,] +oo ) ) )
4744, 14, 45, 46mp3an 1360 . . . . . 6  |-  ( ( (ordTop `  <_  )t  ( 0 [,] +oo ) )t  ( 0 (,] +oo )
)  =  ( (ordTop `  <_  )t  ( 0 (,] +oo ) )
4843, 47syl6eleq 2518 . . . . 5  |-  ( A  e.  J  ->  ( A  i^i  ( 0 (,] +oo ) )  e.  ( (ordTop `  <_  )t  ( 0 (,] +oo ) ) )
4944a1i 11 . . . . . 6  |-  ( A  e.  J  ->  (ordTop ` 
<_  )  e.  Top )
50 iocpnfordt 20155 . . . . . . 7  |-  ( 0 (,] +oo )  e.  (ordTop `  <_  )
5150a1i 11 . . . . . 6  |-  ( A  e.  J  ->  (
0 (,] +oo )  e.  (ordTop `  <_  ) )
52 ssid 3480 . . . . . . 7  |-  ( 0 (,] +oo )  C_  ( 0 (,] +oo )
5352a1i 11 . . . . . 6  |-  ( A  e.  J  ->  (
0 (,] +oo )  C_  ( 0 (,] +oo ) )
54 inss2 3680 . . . . . . 7  |-  ( A  i^i  ( 0 (,] +oo ) )  C_  (
0 (,] +oo )
5554a1i 11 . . . . . 6  |-  ( A  e.  J  ->  ( A  i^i  ( 0 (,] +oo ) )  C_  (
0 (,] +oo )
)
56 restopnb 20115 . . . . . 6  |-  ( ( ( (ordTop `  <_  )  e.  Top  /\  (
0 (,] +oo )  e.  _V )  /\  (
( 0 (,] +oo )  e.  (ordTop `  <_  )  /\  ( 0 (,] +oo )  C_  ( 0 (,] +oo )  /\  ( A  i^i  (
0 (,] +oo )
)  C_  ( 0 (,] +oo ) ) )  ->  ( ( A  i^i  ( 0 (,] +oo ) )  e.  (ordTop `  <_  )  <->  ( A  i^i  ( 0 (,] +oo ) )  e.  ( (ordTop `  <_  )t  ( 0 (,] +oo ) ) ) )
5749, 36, 51, 53, 55, 56syl23anc 1271 . . . . 5  |-  ( A  e.  J  ->  (
( A  i^i  (
0 (,] +oo )
)  e.  (ordTop `  <_  )  <->  ( A  i^i  ( 0 (,] +oo ) )  e.  ( (ordTop `  <_  )t  ( 0 (,] +oo ) ) ) )
5848, 57mpbird 235 . . . 4  |-  ( A  e.  J  ->  ( A  i^i  ( 0 (,] +oo ) )  e.  (ordTop `  <_  ) )
5958adantr 466 . . 3  |-  ( ( A  e.  J  /\ +oo  e.  A )  -> 
( A  i^i  (
0 (,] +oo )
)  e.  (ordTop `  <_  ) )
60 simpr 462 . . . 4  |-  ( ( A  e.  J  /\ +oo  e.  A )  -> +oo  e.  A )
61 0ltpnf 11413 . . . . . 6  |-  0  < +oo
62 ubioc1 11677 . . . . . 6  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  0  < +oo )  -> +oo  e.  ( 0 (,] +oo ) )
635, 7, 61, 62mp3an 1360 . . . . 5  |- +oo  e.  ( 0 (,] +oo )
6463a1i 11 . . . 4  |-  ( ( A  e.  J  /\ +oo  e.  A )  -> +oo  e.  ( 0 (,] +oo ) )
6560, 64elind 3647 . . 3  |-  ( ( A  e.  J  /\ +oo  e.  A )  -> +oo  e.  ( A  i^i  ( 0 (,] +oo ) ) )
66 pnfnei 20160 . . 3  |-  ( ( ( A  i^i  (
0 (,] +oo )
)  e.  (ordTop `  <_  )  /\ +oo  e.  ( A  i^i  (
0 (,] +oo )
) )  ->  E. y  e.  RR  ( y (,] +oo )  C_  ( A  i^i  ( 0 (,] +oo ) ) )
6759, 65, 66syl2anc 665 . 2  |-  ( ( A  e.  J  /\ +oo  e.  A )  ->  E. y  e.  RR  ( y (,] +oo )  C_  ( A  i^i  ( 0 (,] +oo ) ) )
6828, 67r19.29a 2968 1  |-  ( ( A  e.  J  /\ +oo  e.  A )  ->  E. x  e.  RR  ( x (,] +oo )  C_  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867   E.wrex 2774   _Vcvv 3078    i^i cin 3432    C_ wss 3433   ifcif 3906   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   RRcr 9527   0cc0 9528   +oocpnf 9661   RR*cxr 9663    < clt 9664    <_ cle 9665   (,]cioc 11625   [,]cicc 11627   ↾s cress 15074   ↾t crest 15271   TopOpenctopn 15272  ordTopcordt 15349   RR*scxrs 15350   Topctop 19841  TopOnctopon 19842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-fi 7922  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-ioo 11628  df-ioc 11629  df-ico 11630  df-icc 11631  df-fz 11772  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-ress 15080  df-plusg 15155  df-mulr 15156  df-tset 15161  df-ple 15162  df-ds 15164  df-rest 15273  df-topn 15274  df-topgen 15294  df-ordt 15351  df-xrs 15352  df-ps 16390  df-tsr 16391  df-top 19845  df-bases 19846  df-topon 19847
This theorem is referenced by:  lmxrge0  28623
  Copyright terms: Public domain W3C validator