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Theorem icccmp 21198
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 2467 . . . . . . . 8  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
4 eqid 2467 . . . . . . . 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 757 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
u  e.  ~P J  /\  ( A [,] B
)  C_  U. u
) )  ->  A  e.  RR )
6 simpllr 758 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
u  e.  ~P J  /\  ( A [,] B
)  C_  U. u
) )  ->  B  e.  RR )
7 simplr 754 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
u  e.  ~P J  /\  ( A [,] B
)  C_  U. u
) )  ->  A  <_  B )
8 elpwi 4025 . . . . . . . . 9  |-  ( u  e.  ~P J  ->  u  C_  J )
98ad2antrl 727 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
u  e.  ~P J  /\  ( A [,] B
)  C_  U. u
) )  ->  u  C_  J )
10 simprr 756 . . . . . . . 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 21197 . . . . . . 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 6303 . . . . . . . . . . 11  |-  ( x  =  B  ->  ( A [,] x )  =  ( A [,] B
) )
1312sseq1d 3536 . . . . . . . . . 10  |-  ( x  =  B  ->  (
( A [,] x
)  C_  U. z  <->  ( A [,] B ) 
C_  U. z ) )
1413rexbidv 2978 . . . . . . . . 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 3266 . . . . . . . 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 464 . . . . . . 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 615 . . . . 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 2881 . . . 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 21136 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  Top
212, 20eqeltri 2551 . . . . 5  |-  J  e. 
Top
22 iccssre 11618 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
2322adantr 465 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( A [,] B )  C_  RR )
24 uniretop 21137 . . . . . . 7  |-  RR  =  U. ( topGen `  ran  (,) )
252unieqi 4260 . . . . . . 7  |-  U. J  =  U. ( topGen `  ran  (,) )
2624, 25eqtr4i 2499 . . . . . 6  |-  RR  =  U. J
2726cmpsub 19768 . . . . 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 663 . . . 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 232 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( Jt  ( A [,] B ) )  e.  Comp )
30 rexr 9651 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
31 rexr 9651 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
32 icc0 11589 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,] B
)  =  (/)  <->  B  <  A ) )
3330, 31, 32syl2an 477 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A [,] B )  =  (/)  <->  B  <  A ) )
3433biimpar 485 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( A [,] B )  =  (/) )
3534oveq2d 6311 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( Jt  ( A [,] B ) )  =  ( Jt  (/) ) )
36 rest0 19538 . . . . . 6  |-  ( J  e.  Top  ->  ( Jt  (/) )  =  { (/) } )
3721, 36ax-mp 5 . . . . 5  |-  ( Jt  (/) )  =  { (/) }
3835, 37syl6eq 2524 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( Jt  ( A [,] B ) )  =  { (/) } )
39 0cmp 19762 . . . 4  |-  { (/) }  e.  Comp
4038, 39syl6eqel 2563 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( Jt  ( A [,] B ) )  e.  Comp )
41 lelttric 9703 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  \/  B  <  A ) )
4229, 40, 41mpjaodan 784 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Jt  ( A [,] B ) )  e. 
Comp )
431, 42syl5eqel 2559 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  T  e.  Comp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   {crab 2821    i^i cin 3480    C_ wss 3481   (/)c0 3790   ~Pcpw 4016   {csn 4033   U.cuni 4251   class class class wbr 4453    X. cxp 5003   ran crn 5006    |` cres 5007    o. ccom 5009   ` cfv 5594  (class class class)co 6295   Fincfn 7528   RRcr 9503   RR*cxr 9639    < clt 9640    <_ cle 9641    - cmin 9817   (,)cioo 11541   [,]cicc 11544   abscabs 13047   ↾t crest 14693   topGenctg 14710   Topctop 19263   Compccmp 19754
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fi 7883  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-icc 11548  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-rest 14695  df-topgen 14716  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-top 19268  df-bases 19270  df-topon 19271  df-cmp 19755
This theorem is referenced by:  iicmp  21258  cnheiborlem  21322  evthicc  21739  ovolicc2  21801  dvcnvrelem2  22287  fourierdlem42  31772
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