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Theorem icccmp 21496
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 2454 . . . . . . . 8  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
4 eqid 2454 . . . . . . . 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 753 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
u  e.  ~P J  /\  ( A [,] B
)  C_  U. u
) )  ->  A  <_  B )
8 elpwi 4008 . . . . . . . . 9  |-  ( u  e.  ~P J  ->  u  C_  J )
98ad2antrl 725 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B )  /\  (
u  e.  ~P J  /\  ( A [,] B
)  C_  U. u
) )  ->  u  C_  J )
10 simprr 755 . . . . . . . 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 21495 . . . . . . 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 6278 . . . . . . . . . . 11  |-  ( x  =  B  ->  ( A [,] x )  =  ( A [,] B
) )
1312sseq1d 3516 . . . . . . . . . 10  |-  ( x  =  B  ->  (
( A [,] x
)  C_  U. z  <->  ( A [,] B ) 
C_  U. z ) )
1413rexbidv 2965 . . . . . . . . 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 3254 . . . . . . . 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 462 . . . . . . 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 613 . . . . 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 2868 . . . 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 21434 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  Top
212, 20eqeltri 2538 . . . . 5  |-  J  e. 
Top
22 iccssre 11609 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
2322adantr 463 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <_  B
)  ->  ( A [,] B )  C_  RR )
24 uniretop 21435 . . . . . . 7  |-  RR  =  U. ( topGen `  ran  (,) )
252unieqi 4244 . . . . . . 7  |-  U. J  =  U. ( topGen `  ran  (,) )
2624, 25eqtr4i 2486 . . . . . 6  |-  RR  =  U. J
2726cmpsub 20067 . . . . 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 661 . . . 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 9628 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  RR* )
31 rexr 9628 . . . . . . . 8  |-  ( B  e.  RR  ->  B  e.  RR* )
32 icc0 11580 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A [,] B
)  =  (/)  <->  B  <  A ) )
3330, 31, 32syl2an 475 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A [,] B )  =  (/)  <->  B  <  A ) )
3433biimpar 483 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( A [,] B )  =  (/) )
3534oveq2d 6286 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( Jt  ( A [,] B ) )  =  ( Jt  (/) ) )
36 rest0 19837 . . . . . 6  |-  ( J  e.  Top  ->  ( Jt  (/) )  =  { (/) } )
3721, 36ax-mp 5 . . . . 5  |-  ( Jt  (/) )  =  { (/) }
3835, 37syl6eq 2511 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( Jt  ( A [,] B ) )  =  { (/) } )
39 0cmp 20061 . . . 4  |-  { (/) }  e.  Comp
4038, 39syl6eqel 2550 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  <  A
)  ->  ( Jt  ( A [,] B ) )  e.  Comp )
41 lelttric 9680 . . 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 2546 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  T  e.  Comp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   {crab 2808    i^i cin 3460    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016   U.cuni 4235   class class class wbr 4439    X. cxp 4986   ran crn 4989    |` cres 4990    o. ccom 4992   ` cfv 5570  (class class class)co 6270   Fincfn 7509   RRcr 9480   RR*cxr 9616    < clt 9617    <_ cle 9618    - cmin 9796   (,)cioo 11532   [,]cicc 11535   abscabs 13149   ↾t crest 14910   topGenctg 14927   Topctop 19561   Compccmp 20053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fi 7863  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-icc 11539  df-seq 12090  df-exp 12149  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-rest 14912  df-topgen 14933  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-top 19566  df-bases 19568  df-topon 19569  df-cmp 20054
This theorem is referenced by:  iicmp  21556  cnheiborlem  21620  evthicc  22037  ovolicc2  22099  dvcnvrelem2  22585  fourierdlem42  32170
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