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Theorem icccmp 18809
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 2404 . . . . . . . 8  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
4 eqid 2404 . . . . . . . 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 3767 . . . . . . . . 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 18808 . . . . . . 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 6048 . . . . . . . . . . 11  |-  ( x  =  B  ->  ( A [,] x )  =  ( A [,] B
) )
1312sseq1d 3335 . . . . . . . . . 10  |-  ( x  =  B  ->  (
( A [,] x
)  C_  U. z  <->  ( A [,] B ) 
C_  U. z ) )
1413rexbidv 2687 . . . . . . . . 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 3052 . . . . . . . 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 2749 . . . 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 18748 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  Top
212, 20eqeltri 2474 . . . . 5  |-  J  e. 
Top
22 iccssre 10948 . . . . . 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 18749 . . . . . . 7  |-  RR  =  U. ( topGen `  ran  (,) )
252unieqi 3985 . . . . . . 7  |-  U. J  =  U. ( topGen `  ran  (,) )
2624, 25eqtr4i 2427 . . . . . 6  |-  RR  =  U. J
2726cmpsub 17417 . . . . 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 9086 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
31 rexr 9086 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
32 icc0 10920 . . . . . . . 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 6056 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( Jt  ( A [,] B ) )  =  ( Jt  (/) ) )
36 rest0 17187 . . . . . 6  |-  ( J  e.  Top  ->  ( Jt  (/) )  =  { (/) } )
3721, 36ax-mp 8 . . . . 5  |-  ( Jt  (/) )  =  { (/) }
3835, 37syl6eq 2452 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( Jt  ( A [,] B ) )  =  { (/) } )
39 0cmp 17411 . . . 4  |-  { (/) }  e.  Comp
4038, 39syl6eqel 2492 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( Jt  ( A [,] B ) )  e.  Comp )
41 lelttric 9136 . . 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 2488 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 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   {crab 2670    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   {csn 3774   U.cuni 3975   class class class wbr 4172    X. cxp 4835   ran crn 4838    |` cres 4839    o. ccom 4841   ` cfv 5413  (class class class)co 6040   Fincfn 7068   RRcr 8945   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247   (,)cioo 10872   [,]cicc 10875   abscabs 11994   ↾t crest 13603   topGenctg 13620   Topctop 16913   Compccmp 17403
This theorem is referenced by:  iicmp  18869  cnheiborlem  18932  evthicc  19309  ovolicc2  19371  dvcnvrelem2  19855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-icc 10879  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-rest 13605  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-top 16918  df-bases 16920  df-topon 16921  df-cmp 17404
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