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Theorem pnfneige0 27597
Description: A neighborhood of +oo contains an unbounded interval based at a real number. See pnfnei 19515 (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 9596 . . . . 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 3971 . . 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 9639 . . . . . . 7  |-  ( y  e.  RR  ->  y  e.  RR* )
6 0xr 9640 . . . . . . . 8  |-  0  e.  RR*
76a1i 11 . . . . . . 7  |-  ( y  e.  RR  ->  0  e.  RR* )
8 pnfxr 11321 . . . . . . . 8  |- +oo  e.  RR*
98a1i 11 . . . . . . 7  |-  ( y  e.  RR  -> +oo  e.  RR* )
10 iocinif 27288 . . . . . . 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 1228 . . . . . 6  |-  ( y  e.  RR  ->  (
( y (,] +oo )  i^i  ( 0 (,] +oo ) )  =  if ( y  <  0 ,  ( 0 (,] +oo ) ,  ( y (,] +oo ) ) )
12 ovif 6363 . . . . . 6  |-  ( if ( y  <  0 ,  0 ,  y ) (,] +oo )  =  if ( y  <  0 ,  ( 0 (,] +oo ) ,  ( y (,] +oo ) )
1311, 12syl6reqr 2527 . . . . 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 11612 . . . . . 6  |-  ( 0 (,] +oo )  C_  ( 0 [,] +oo )
16 sslin 3724 . . . . . 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 3720 . . . . . . . 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 3726 . . . . . 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 3514 . . . 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 3538 . . 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 6291 . . . . 5  |-  ( x  =  if ( y  <  0 ,  0 ,  y )  -> 
( x (,] +oo )  =  ( if ( y  <  0 ,  0 ,  y ) (,] +oo )
)
2726sseq1d 3531 . . . 4  |-  ( x  =  if ( y  <  0 ,  0 ,  y )  -> 
( ( x (,] +oo )  C_  A  <->  ( if ( y  <  0 ,  0 ,  y ) (,] +oo )  C_  A ) )
2827rspcev 3214 . . 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 19500 . . . . . . . . . 10  |-  (ordTop `  <_  )  e.  (TopOn `  RR* )
31 iccssxr 11607 . . . . . . . . . 10  |-  ( 0 [,] +oo )  C_  RR*
32 resttopon 19456 . . . . . . . . . 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 19227 . . . . . . . 8  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  Top
3534a1i 11 . . . . . . 7  |-  ( A  e.  J  ->  (
(ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  Top )
36 ovex 6309 . . . . . . . 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 27589 . . . . . . . . . 10  |-  ( TopOpen `  ( RR*ss  ( 0 [,] +oo ) ) )  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
4038, 39eqtri 2496 . . . . . . . . 9  |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
4140eleq2i 2545 . . . . . . . 8  |-  ( A  e.  J  <->  A  e.  ( (ordTop `  <_  )t  ( 0 [,] +oo ) ) )
4241biimpi 194 . . . . . . 7  |-  ( A  e.  J  ->  A  e.  ( (ordTop `  <_  )t  ( 0 [,] +oo )
) )
43 elrestr 14684 . . . . . . 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 1228 . . . . . 6  |-  ( A  e.  J  ->  ( A  i^i  ( 0 (,] +oo ) )  e.  ( ( (ordTop `  <_  )t  ( 0 [,] +oo )
)t  ( 0 (,] +oo ) ) )
45 letop 19501 . . . . . . 7  |-  (ordTop `  <_  )  e.  Top
46 ovex 6309 . . . . . . 7  |-  ( 0 [,] +oo )  e. 
_V
47 restabs 19460 . . . . . . 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 1324 . . . . . 6  |-  ( ( (ordTop `  <_  )t  ( 0 [,] +oo ) )t  ( 0 (,] +oo )
)  =  ( (ordTop `  <_  )t  ( 0 (,] +oo ) )
4944, 48syl6eleq 2565 . . . . 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 19510 . . . . . . 7  |-  ( 0 (,] +oo )  e.  (ordTop `  <_  )
5251a1i 11 . . . . . 6  |-  ( A  e.  J  ->  (
0 (,] +oo )  e.  (ordTop `  <_  ) )
53 ssid 3523 . . . . . . 7  |-  ( 0 (,] +oo )  C_  ( 0 (,] +oo )
5453a1i 11 . . . . . 6  |-  ( A  e.  J  ->  (
0 (,] +oo )  C_  ( 0 (,] +oo ) )
55 inss2 3719 . . . . . . 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 19470 . . . . . 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 1235 . . . . 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 11332 . . . . . 6  |-  0  < +oo
63 ubioc1 11578 . . . . . 6  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  0  < +oo )  -> +oo  e.  ( 0 (,] +oo ) )
646, 8, 62, 63mp3an 1324 . . . . 5  |- +oo  e.  ( 0 (,] +oo )
6564a1i 11 . . . 4  |-  ( ( A  e.  J  /\ +oo  e.  A )  -> +oo  e.  ( 0 (,] +oo ) )
6661, 65elind 3688 . . 3  |-  ( ( A  e.  J  /\ +oo  e.  A )  -> +oo  e.  ( A  i^i  ( 0 (,] +oo ) ) )
67 pnfnei 19515 . . 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 3003 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 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113    i^i cin 3475    C_ wss 3476   ifcif 3939   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   RRcr 9491   0cc0 9492   +oocpnf 9625   RR*cxr 9627    < clt 9628    <_ cle 9629   (,]cioc 11530   [,]cicc 11532   ↾s cress 14491   ↾t crest 14676   TopOpenctopn 14677  ordTopcordt 14754   RR*scxrs 14755   Topctop 19189  TopOnctopon 19190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fi 7871  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-ioo 11533  df-ioc 11534  df-ico 11535  df-icc 11536  df-fz 11673  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-tset 14574  df-ple 14575  df-ds 14577  df-rest 14678  df-topn 14679  df-topgen 14699  df-ordt 14756  df-xrs 14757  df-ps 15687  df-tsr 15688  df-top 19194  df-bases 19196  df-topon 19197
This theorem is referenced by:  lmxrge0  27598
  Copyright terms: Public domain W3C validator