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

Theorem pnfneige0 26379
Description: A neighborhood of +oo contains an unbounded interval based at a real number. See pnfnei 18822 (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 0re 9384 . . . . 5  |-  0  e.  RR
21a1i 11 . . . 4  |-  ( ( ( ( ( A  e.  J  /\ +oo  e.  A )  /\  y  e.  RR )  /\  (
y (,] +oo )  C_  ( A  i^i  (
0 (,] +oo )
) )  /\  y  <  0 )  ->  0  e.  RR )
3 simpllr 758 . . . 4  |-  ( ( ( ( ( A  e.  J  /\ +oo  e.  A )  /\  y  e.  RR )  /\  (
y (,] +oo )  C_  ( A  i^i  (
0 (,] +oo )
) )  /\  -.  y  <  0 )  -> 
y  e.  RR )
42, 3ifclda 3819 . . 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 )
5 rexr 9427 . . . . . . 7  |-  ( y  e.  RR  ->  y  e.  RR* )
6 0xr 9428 . . . . . . . 8  |-  0  e.  RR*
76a1i 11 . . . . . . 7  |-  ( y  e.  RR  ->  0  e.  RR* )
8 pnfxr 11090 . . . . . . . 8  |- +oo  e.  RR*
98a1i 11 . . . . . . 7  |-  ( y  e.  RR  -> +oo  e.  RR* )
10 iocinif 26069 . . . . . . 7  |-  ( ( y  e.  RR*  /\  0  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
y (,] +oo )  i^i  ( 0 (,] +oo ) )  =  if ( y  <  0 ,  ( 0 (,] +oo ) ,  ( y (,] +oo ) ) )
115, 7, 9, 10syl3anc 1218 . . . . . 6  |-  ( y  e.  RR  ->  (
( y (,] +oo )  i^i  ( 0 (,] +oo ) )  =  if ( y  <  0 ,  ( 0 (,] +oo ) ,  ( y (,] +oo ) ) )
12 ovif 6166 . . . . . 6  |-  ( if ( y  <  0 ,  0 ,  y ) (,] +oo )  =  if ( y  <  0 ,  ( 0 (,] +oo ) ,  ( y (,] +oo ) )
1311, 12syl6reqr 2492 . . . . 5  |-  ( y  e.  RR  ->  ( if ( y  <  0 ,  0 ,  y ) (,] +oo )  =  ( ( y (,] +oo )  i^i  ( 0 (,] +oo ) ) )
1413ad2antlr 726 . . . 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 ) ) )
15 iocssicc 26057 . . . . . 6  |-  ( 0 (,] +oo )  C_  ( 0 [,] +oo )
16 sslin 3574 . . . . . 6  |-  ( ( 0 (,] +oo )  C_  ( 0 [,] +oo )  ->  ( ( y (,] +oo )  i^i  ( 0 (,] +oo ) )  C_  (
( y (,] +oo )  i^i  ( 0 [,] +oo ) ) )
1715, 16mp1i 12 . . . . 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 ) ) )
18 simpr 461 . . . . . 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 )
) )
19 ssin 3570 . . . . . . . 8  |-  ( ( ( y (,] +oo )  C_  A  /\  (
y (,] +oo )  C_  ( 0 (,] +oo ) )  <->  ( y (,] +oo )  C_  ( A  i^i  ( 0 (,] +oo ) ) )
2019biimpri 206 . . . . . . 7  |-  ( ( y (,] +oo )  C_  ( A  i^i  (
0 (,] +oo )
)  ->  ( (
y (,] +oo )  C_  A  /\  ( y (,] +oo )  C_  ( 0 (,] +oo ) ) )
2120simpld 459 . . . . . 6  |-  ( ( y (,] +oo )  C_  ( A  i^i  (
0 (,] +oo )
)  ->  ( y (,] +oo )  C_  A
)
22 ssinss1 3576 . . . . . 6  |-  ( ( y (,] +oo )  C_  A  ->  ( (
y (,] +oo )  i^i  ( 0 [,] +oo ) )  C_  A
)
2318, 21, 223syl 20 . . . . 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
)
2417, 23sstrd 3364 . . . 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
)
2514, 24eqsstrd 3388 . . 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 )
26 oveq1 6096 . . . . 5  |-  ( x  =  if ( y  <  0 ,  0 ,  y )  -> 
( x (,] +oo )  =  ( if ( y  <  0 ,  0 ,  y ) (,] +oo )
)
2726sseq1d 3381 . . . 4  |-  ( x  =  if ( y  <  0 ,  0 ,  y )  -> 
( ( x (,] +oo )  C_  A  <->  ( if ( y  <  0 ,  0 ,  y ) (,] +oo )  C_  A ) )
2827rspcev 3071 . . 3  |-  ( ( if ( y  <  0 ,  0 ,  y )  e.  RR  /\  ( if ( y  <  0 ,  0 ,  y ) (,] +oo )  C_  A )  ->  E. x  e.  RR  ( x (,] +oo )  C_  A )
294, 25, 28syl2anc 661 . 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 )
30 letopon 18807 . . . . . . . . . 10  |-  (ordTop `  <_  )  e.  (TopOn `  RR* )
31 iccssxr 11376 . . . . . . . . . 10  |-  ( 0 [,] +oo )  C_  RR*
32 resttopon 18763 . . . . . . . . . 10  |-  ( ( (ordTop `  <_  )  e.  (TopOn `  RR* )  /\  ( 0 [,] +oo )  C_  RR* )  ->  (
(ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  (TopOn `  (
0 [,] +oo )
) )
3330, 31, 32mp2an 672 . . . . . . . . 9  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  (TopOn `  ( 0 [,] +oo ) )
3433topontopi 18534 . . . . . . . 8  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  Top
3534a1i 11 . . . . . . 7  |-  ( A  e.  J  ->  (
(ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  Top )
36 ovex 6114 . . . . . . . 8  |-  ( 0 (,] +oo )  e. 
_V
3736a1i 11 . . . . . . 7  |-  ( A  e.  J  ->  (
0 (,] +oo )  e.  _V )
38 pnfneige0.j . . . . . . . . . 10  |-  J  =  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )
39 xrge0topn 26371 . . . . . . . . . 10  |-  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
4038, 39eqtri 2461 . . . . . . . . 9  |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
4140eleq2i 2505 . . . . . . . 8  |-  ( A  e.  J  <->  A  e.  ( (ordTop `  <_  )t  ( 0 [,] +oo ) ) )
4241biimpi 194 . . . . . . 7  |-  ( A  e.  J  ->  A  e.  ( (ordTop `  <_  )t  ( 0 [,] +oo )
) )
43 elrestr 14365 . . . . . . 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 ) ) )
4435, 37, 42, 43syl3anc 1218 . . . . . 6  |-  ( A  e.  J  ->  ( A  i^i  ( 0 (,] +oo ) )  e.  ( ( (ordTop `  <_  )t  ( 0 [,] +oo )
)t  ( 0 (,] +oo ) ) )
45 letop 18808 . . . . . . 7  |-  (ordTop `  <_  )  e.  Top
46 ovex 6114 . . . . . . 7  |-  ( 0 [,] +oo )  e. 
_V
47 restabs 18767 . . . . . . 7  |-  ( ( (ordTop `  <_  )  e. 
Top  /\  ( 0 (,] +oo )  C_  ( 0 [,] +oo )  /\  ( 0 [,] +oo )  e.  _V )  ->  ( ( (ordTop `  <_  )t  ( 0 [,] +oo ) )t  ( 0 (,] +oo ) )  =  ( (ordTop `  <_  )t  ( 0 (,] +oo ) ) )
4845, 15, 46, 47mp3an 1314 . . . . . 6  |-  ( ( (ordTop `  <_  )t  ( 0 [,] +oo ) )t  ( 0 (,] +oo )
)  =  ( (ordTop `  <_  )t  ( 0 (,] +oo ) )
4944, 48syl6eleq 2531 . . . . 5  |-  ( A  e.  J  ->  ( A  i^i  ( 0 (,] +oo ) )  e.  ( (ordTop `  <_  )t  ( 0 (,] +oo ) ) )
5045a1i 11 . . . . . 6  |-  ( A  e.  J  ->  (ordTop ` 
<_  )  e.  Top )
51 iocpnfordt 18817 . . . . . . 7  |-  ( 0 (,] +oo )  e.  (ordTop `  <_  )
5251a1i 11 . . . . . 6  |-  ( A  e.  J  ->  (
0 (,] +oo )  e.  (ordTop `  <_  ) )
53 ssid 3373 . . . . . . 7  |-  ( 0 (,] +oo )  C_  ( 0 (,] +oo )
5453a1i 11 . . . . . 6  |-  ( A  e.  J  ->  (
0 (,] +oo )  C_  ( 0 (,] +oo ) )
55 inss2 3569 . . . . . . 7  |-  ( A  i^i  ( 0 (,] +oo ) )  C_  (
0 (,] +oo )
5655a1i 11 . . . . . 6  |-  ( A  e.  J  ->  ( A  i^i  ( 0 (,] +oo ) )  C_  (
0 (,] +oo )
)
57 restopnb 18777 . . . . . 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 ) ) ) )
5850, 37, 52, 54, 56, 57syl23anc 1225 . . . . 5  |-  ( A  e.  J  ->  (
( A  i^i  (
0 (,] +oo )
)  e.  (ordTop `  <_  )  <->  ( A  i^i  ( 0 (,] +oo ) )  e.  ( (ordTop `  <_  )t  ( 0 (,] +oo ) ) ) )
5949, 58mpbird 232 . . . 4  |-  ( A  e.  J  ->  ( A  i^i  ( 0 (,] +oo ) )  e.  (ordTop `  <_  ) )
6059adantr 465 . . 3  |-  ( ( A  e.  J  /\ +oo  e.  A )  -> 
( A  i^i  (
0 (,] +oo )
)  e.  (ordTop `  <_  ) )
61 simpr 461 . . . 4  |-  ( ( A  e.  J  /\ +oo  e.  A )  -> +oo  e.  A )
62 0ltpnf 11101 . . . . . 6  |-  0  < +oo
63 ubioc1 11347 . . . . . 6  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  0  < +oo )  -> +oo  e.  ( 0 (,] +oo ) )
646, 8, 62, 63mp3an 1314 . . . . 5  |- +oo  e.  ( 0 (,] +oo )
6564a1i 11 . . . 4  |-  ( ( A  e.  J  /\ +oo  e.  A )  -> +oo  e.  ( 0 (,] +oo ) )
6661, 65elind 3538 . . 3  |-  ( ( A  e.  J  /\ +oo  e.  A )  -> +oo  e.  ( A  i^i  ( 0 (,] +oo ) ) )
67 pnfnei 18822 . . 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 ) ) )
6860, 66, 67syl2anc 661 . 2  |-  ( ( A  e.  J  /\ +oo  e.  A )  ->  E. y  e.  RR  ( y (,] +oo )  C_  ( A  i^i  ( 0 (,] +oo ) ) )
6929, 68r19.29a 2860 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 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2714   _Vcvv 2970    i^i cin 3325    C_ wss 3326   ifcif 3789   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   RRcr 9279   0cc0 9280   +oocpnf 9413   RR*cxr 9415    < clt 9416    <_ cle 9417   (,]cioc 11299   [,]cicc 11301   ↾s cress 14173   ↾t crest 14357   TopOpenctopn 14358  ordTopcordt 14435   RR*scxrs 14436   Topctop 18496  TopOnctopon 18497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-fi 7659  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-7 10383  df-8 10384  df-9 10385  df-10 10386  df-n0 10578  df-z 10645  df-dec 10754  df-uz 10860  df-ioo 11302  df-ioc 11303  df-ico 11304  df-icc 11305  df-fz 11436  df-struct 14174  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-tset 14255  df-ple 14256  df-ds 14258  df-rest 14359  df-topn 14360  df-topgen 14380  df-ordt 14437  df-xrs 14438  df-ps 15368  df-tsr 15369  df-top 18501  df-bases 18503  df-topon 18504
This theorem is referenced by:  lmxrge0  26380
  Copyright terms: Public domain W3C validator