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Theorem icccmp 18856
Description: A closed interval in  RR is compact. (Contributed by Mario Carneiro, 13-Jun-2014.)
Hypotheses
Ref Expression
icccmp.1  |-  J  =  ( topGen `  ran  (,) )
icccmp.2  |-  T  =  ( Jt  ( A [,] B ) )
Assertion
Ref Expression
icccmp  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  T  e.  Comp )

Proof of Theorem icccmp
Dummy variables  u  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 icccmp.2 . 2  |-  T  =  ( Jt  ( A [,] B ) )
2 icccmp.1 . . . . . . . 8  |-  J  =  ( topGen `  ran  (,) )
3 eqid 2436 . . . . . . . 8  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
4 eqid 2436 . . . . . . . 8  |-  { x  e.  ( A [,] B
)  |  E. z  e.  ( ~P u  i^i 
Fin ) ( A [,] x )  C_  U. z }  =  {
x  e.  ( A [,] B )  |  E. z  e.  ( ~P u  i^i  Fin ) ( A [,] x )  C_  U. z }
5 simplll 735 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
u  e.  ~P J  /\  ( A [,] B
)  C_  U. u
) )  ->  A  e.  RR )
6 simpllr 736 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
u  e.  ~P J  /\  ( A [,] B
)  C_  U. u
) )  ->  B  e.  RR )
7 simplr 732 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
u  e.  ~P J  /\  ( A [,] B
)  C_  U. u
) )  ->  A  <_  B )
8 elpwi 3807 . . . . . . . . 9  |-  ( u  e.  ~P J  ->  u  C_  J )
98ad2antrl 709 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
u  e.  ~P J  /\  ( A [,] B
)  C_  U. u
) )  ->  u  C_  J )
10 simprr 734 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
u  e.  ~P J  /\  ( A [,] B
)  C_  U. u
) )  ->  ( A [,] B )  C_  U. u )
112, 1, 3, 4, 5, 6, 7, 9, 10icccmplem3 18855 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
u  e.  ~P J  /\  ( A [,] B
)  C_  U. u
) )  ->  B  e.  { x  e.  ( A [,] B )  |  E. z  e.  ( ~P u  i^i 
Fin ) ( A [,] x )  C_  U. z } )
12 oveq2 6089 . . . . . . . . . . 11  |-  ( x  =  B  ->  ( A [,] x )  =  ( A [,] B
) )
1312sseq1d 3375 . . . . . . . . . 10  |-  ( x  =  B  ->  (
( A [,] x
)  C_  U. z  <->  ( A [,] B ) 
C_  U. z ) )
1413rexbidv 2726 . . . . . . . . 9  |-  ( x  =  B  ->  ( E. z  e.  ( ~P u  i^i  Fin )
( A [,] x
)  C_  U. z  <->  E. z  e.  ( ~P u  i^i  Fin )
( A [,] B
)  C_  U. z
) )
1514elrab 3092 . . . . . . . 8  |-  ( B  e.  { x  e.  ( A [,] B
)  |  E. z  e.  ( ~P u  i^i 
Fin ) ( A [,] x )  C_  U. z }  <->  ( B  e.  ( A [,] B
)  /\  E. z  e.  ( ~P u  i^i 
Fin ) ( A [,] B )  C_  U. z ) )
1615simprbi 451 . . . . . . 7  |-  ( B  e.  { x  e.  ( A [,] B
)  |  E. z  e.  ( ~P u  i^i 
Fin ) ( A [,] x )  C_  U. z }  ->  E. z  e.  ( ~P u  i^i 
Fin ) ( A [,] B )  C_  U. z )
1711, 16syl 16 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
u  e.  ~P J  /\  ( A [,] B
)  C_  U. u
) )  ->  E. z  e.  ( ~P u  i^i 
Fin ) ( A [,] B )  C_  U. z )
1817expr 599 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  u  e.  ~P J )  -> 
( ( A [,] B )  C_  U. u  ->  E. z  e.  ( ~P u  i^i  Fin ) ( A [,] B )  C_  U. z
) )
1918ralrimiva 2789 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  A. u  e.  ~P  J ( ( A [,] B ) 
C_  U. u  ->  E. z  e.  ( ~P u  i^i 
Fin ) ( A [,] B )  C_  U. z ) )
20 retop 18795 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  Top
212, 20eqeltri 2506 . . . . 5  |-  J  e. 
Top
22 iccssre 10992 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
2322adantr 452 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( A [,] B )  C_  RR )
24 uniretop 18796 . . . . . . 7  |-  RR  =  U. ( topGen `  ran  (,) )
252unieqi 4025 . . . . . . 7  |-  U. J  =  U. ( topGen `  ran  (,) )
2624, 25eqtr4i 2459 . . . . . 6  |-  RR  =  U. J
2726cmpsub 17463 . . . . 5  |-  ( ( J  e.  Top  /\  ( A [,] B ) 
C_  RR )  -> 
( ( Jt  ( A [,] B ) )  e.  Comp  <->  A. u  e.  ~P  J ( ( A [,] B )  C_  U. u  ->  E. z  e.  ( ~P u  i^i 
Fin ) ( A [,] B )  C_  U. z ) ) )
2821, 23, 27sylancr 645 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( ( Jt  ( A [,] B ) )  e.  Comp  <->  A. u  e.  ~P  J ( ( A [,] B ) 
C_  U. u  ->  E. z  e.  ( ~P u  i^i 
Fin ) ( A [,] B )  C_  U. z ) ) )
2919, 28mpbird 224 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( Jt  ( A [,] B ) )  e.  Comp )
30 rexr 9130 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
31 rexr 9130 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
32 icc0 10964 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,] B
)  =  (/)  <->  B  <  A ) )
3330, 31, 32syl2an 464 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A [,] B )  =  (/)  <->  B  <  A ) )
3433biimpar 472 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( A [,] B )  =  (/) )
3534oveq2d 6097 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( Jt  ( A [,] B ) )  =  ( Jt  (/) ) )
36 rest0 17233 . . . . . 6  |-  ( J  e.  Top  ->  ( Jt  (/) )  =  { (/) } )
3721, 36ax-mp 8 . . . . 5  |-  ( Jt  (/) )  =  { (/) }
3835, 37syl6eq 2484 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( Jt  ( A [,] B ) )  =  { (/) } )
39 0cmp 17457 . . . 4  |-  { (/) }  e.  Comp
4038, 39syl6eqel 2524 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( Jt  ( A [,] B ) )  e.  Comp )
41 lelttric 9180 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  \/  B  <  A ) )
4229, 40, 41mpjaodan 762 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Jt  ( A [,] B ) )  e. 
Comp )
431, 42syl5eqel 2520 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  T  e.  Comp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   {crab 2709    i^i cin 3319    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   {csn 3814   U.cuni 4015   class class class wbr 4212    X. cxp 4876   ran crn 4879    |` cres 4880    o. ccom 4882   ` cfv 5454  (class class class)co 6081   Fincfn 7109   RRcr 8989   RR*cxr 9119    < clt 9120    <_ cle 9121    - cmin 9291   (,)cioo 10916   [,]cicc 10919   abscabs 12039   ↾t crest 13648   topGenctg 13665   Topctop 16958   Compccmp 17449
This theorem is referenced by:  iicmp  18916  cnheiborlem  18979  evthicc  19356  ovolicc2  19418  dvcnvrelem2  19902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-icc 10923  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-rest 13650  df-topgen 13667  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-top 16963  df-bases 16965  df-topon 16966  df-cmp 17450
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