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Theorem icccmp 20361
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 2441 . . . . . . . 8  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
4 eqid 2441 . . . . . . . 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 752 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
u  e.  ~P J  /\  ( A [,] B
)  C_  U. u
) )  ->  A  e.  RR )
6 simpllr 753 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
u  e.  ~P J  /\  ( A [,] B
)  C_  U. u
) )  ->  B  e.  RR )
7 simplr 749 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
u  e.  ~P J  /\  ( A [,] B
)  C_  U. u
) )  ->  A  <_  B )
8 elpwi 3866 . . . . . . . . 9  |-  ( u  e.  ~P J  ->  u  C_  J )
98ad2antrl 722 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
u  e.  ~P J  /\  ( A [,] B
)  C_  U. u
) )  ->  u  C_  J )
10 simprr 751 . . . . . . . 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 20360 . . . . . . 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 6098 . . . . . . . . . . 11  |-  ( x  =  B  ->  ( A [,] x )  =  ( A [,] B
) )
1312sseq1d 3380 . . . . . . . . . 10  |-  ( x  =  B  ->  (
( A [,] x
)  C_  U. z  <->  ( A [,] B ) 
C_  U. z ) )
1413rexbidv 2734 . . . . . . . . 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 3114 . . . . . . . 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 461 . . . . . . 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 612 . . . . 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 2797 . . . 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 20299 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  Top
212, 20eqeltri 2511 . . . . 5  |-  J  e. 
Top
22 iccssre 11373 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
2322adantr 462 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( A [,] B )  C_  RR )
24 uniretop 20300 . . . . . . 7  |-  RR  =  U. ( topGen `  ran  (,) )
252unieqi 4097 . . . . . . 7  |-  U. J  =  U. ( topGen `  ran  (,) )
2624, 25eqtr4i 2464 . . . . . 6  |-  RR  =  U. J
2726cmpsub 18962 . . . . 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 658 . . . 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 9425 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
31 rexr 9425 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
32 icc0 11344 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,] B
)  =  (/)  <->  B  <  A ) )
3330, 31, 32syl2an 474 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A [,] B )  =  (/)  <->  B  <  A ) )
3433biimpar 482 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( A [,] B )  =  (/) )
3534oveq2d 6106 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( Jt  ( A [,] B ) )  =  ( Jt  (/) ) )
36 rest0 18732 . . . . . 6  |-  ( J  e.  Top  ->  ( Jt  (/) )  =  { (/) } )
3721, 36ax-mp 5 . . . . 5  |-  ( Jt  (/) )  =  { (/) }
3835, 37syl6eq 2489 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( Jt  ( A [,] B ) )  =  { (/) } )
39 0cmp 18956 . . . 4  |-  { (/) }  e.  Comp
4038, 39syl6eqel 2529 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( Jt  ( A [,] B ) )  e.  Comp )
41 lelttric 9477 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  \/  B  <  A ) )
4229, 40, 41mpjaodan 779 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Jt  ( A [,] B ) )  e. 
Comp )
431, 42syl5eqel 2525 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 1364    e. wcel 1761   A.wral 2713   E.wrex 2714   {crab 2717    i^i cin 3324    C_ wss 3325   (/)c0 3634   ~Pcpw 3857   {csn 3874   U.cuni 4088   class class class wbr 4289    X. cxp 4834   ran crn 4837    |` cres 4838    o. ccom 4840   ` cfv 5415  (class class class)co 6090   Fincfn 7306   RRcr 9277   RR*cxr 9413    < clt 9414    <_ cle 9415    - cmin 9591   (,)cioo 11296   [,]cicc 11299   abscabs 12719   ↾t crest 14355   topGenctg 14372   Topctop 18457   Compccmp 18948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  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-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fi 7657  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-icc 11303  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-rest 14357  df-topgen 14378  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-top 18462  df-bases 18464  df-topon 18465  df-cmp 18949
This theorem is referenced by:  iicmp  20421  cnheiborlem  20485  evthicc  20902  ovolicc2  20964  dvcnvrelem2  21449
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