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Theorem iocmnfcld 21008
Description: Left-unbounded closed intervals are closed sets of the standard topology on  RR. (Contributed by Mario Carneiro, 17-Feb-2015.)
Assertion
Ref Expression
iocmnfcld  |-  ( A  e.  RR  ->  ( -oo (,] A )  e.  ( Clsd `  ( topGen `
 ran  (,) )
) )

Proof of Theorem iocmnfcld
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mnfxr 11319 . . . . . . 7  |- -oo  e.  RR*
21a1i 11 . . . . . 6  |-  ( A  e.  RR  -> -oo  e.  RR* )
3 rexr 9635 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  RR* )
4 pnfxr 11317 . . . . . . 7  |- +oo  e.  RR*
54a1i 11 . . . . . 6  |-  ( A  e.  RR  -> +oo  e.  RR* )
6 mnflt 11329 . . . . . 6  |-  ( A  e.  RR  -> -oo  <  A )
7 ltpnf 11327 . . . . . 6  |-  ( A  e.  RR  ->  A  < +oo )
8 df-ioc 11530 . . . . . . 7  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
9 df-ioo 11529 . . . . . . 7  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
10 xrltnle 9649 . . . . . . 7  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
11 xrlelttr 11355 . . . . . . 7  |-  ( ( w  e.  RR*  /\  A  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
w  <_  A  /\  A  < +oo )  ->  w  < +oo ) )
12 xrlttr 11342 . . . . . . 7  |-  ( ( -oo  e.  RR*  /\  A  e.  RR*  /\  w  e. 
RR* )  ->  (
( -oo  <  A  /\  A  <  w )  -> -oo  <  w ) )
138, 9, 10, 9, 11, 12ixxun 11541 . . . . . 6  |-  ( ( ( -oo  e.  RR*  /\  A  e.  RR*  /\ +oo  e.  RR* )  /\  ( -oo  <  A  /\  A  < +oo ) )  -> 
( ( -oo (,] A )  u.  ( A (,) +oo ) )  =  ( -oo (,) +oo ) )
142, 3, 5, 6, 7, 13syl32anc 1236 . . . . 5  |-  ( A  e.  RR  ->  (
( -oo (,] A )  u.  ( A (,) +oo ) )  =  ( -oo (,) +oo )
)
15 ioomax 11595 . . . . 5  |-  ( -oo (,) +oo )  =  RR
1614, 15syl6eq 2524 . . . 4  |-  ( A  e.  RR  ->  (
( -oo (,] A )  u.  ( A (,) +oo ) )  =  RR )
17 iocssre 11600 . . . . . 6  |-  ( ( -oo  e.  RR*  /\  A  e.  RR )  ->  ( -oo (,] A )  C_  RR )
181, 17mpan 670 . . . . 5  |-  ( A  e.  RR  ->  ( -oo (,] A )  C_  RR )
198, 9, 10ixxdisj 11540 . . . . . 6  |-  ( ( -oo  e.  RR*  /\  A  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( -oo (,] A )  i^i  ( A (,) +oo ) )  =  (/) )
202, 3, 5, 19syl3anc 1228 . . . . 5  |-  ( A  e.  RR  ->  (
( -oo (,] A )  i^i  ( A (,) +oo ) )  =  (/) )
21 uneqdifeq 3915 . . . . 5  |-  ( ( ( -oo (,] A
)  C_  RR  /\  (
( -oo (,] A )  i^i  ( A (,) +oo ) )  =  (/) )  ->  ( ( ( -oo (,] A )  u.  ( A (,) +oo ) )  =  RR  <->  ( RR  \  ( -oo (,] A ) )  =  ( A (,) +oo ) ) )
2218, 20, 21syl2anc 661 . . . 4  |-  ( A  e.  RR  ->  (
( ( -oo (,] A )  u.  ( A (,) +oo ) )  =  RR  <->  ( RR  \  ( -oo (,] A
) )  =  ( A (,) +oo )
) )
2316, 22mpbid 210 . . 3  |-  ( A  e.  RR  ->  ( RR  \  ( -oo (,] A ) )  =  ( A (,) +oo ) )
24 iooretop 21005 . . 3  |-  ( A (,) +oo )  e.  ( topGen `  ran  (,) )
2523, 24syl6eqel 2563 . 2  |-  ( A  e.  RR  ->  ( RR  \  ( -oo (,] A ) )  e.  ( topGen `  ran  (,) )
)
26 retop 21000 . . 3  |-  ( topGen ` 
ran  (,) )  e.  Top
27 uniretop 21001 . . . 4  |-  RR  =  U. ( topGen `  ran  (,) )
2827iscld2 19292 . . 3  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( -oo (,] A )  C_  RR )  ->  ( ( -oo (,] A )  e.  (
Clsd `  ( topGen ` 
ran  (,) ) )  <->  ( RR  \  ( -oo (,] A
) )  e.  (
topGen `  ran  (,) )
) )
2926, 18, 28sylancr 663 . 2  |-  ( A  e.  RR  ->  (
( -oo (,] A )  e.  ( Clsd `  ( topGen `
 ran  (,) )
)  <->  ( RR  \ 
( -oo (,] A ) )  e.  ( topGen ` 
ran  (,) ) ) )
3025, 29mpbird 232 1  |-  ( A  e.  RR  ->  ( -oo (,] A )  e.  ( Clsd `  ( topGen `
 ran  (,) )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767    \ cdif 3473    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   class class class wbr 4447   ran crn 5000   ` cfv 5586  (class class class)co 6282   RRcr 9487   +oocpnf 9621   -oocmnf 9622   RR*cxr 9623    < clt 9624    <_ cle 9625   (,)cioo 11525   (,]cioc 11526   topGenctg 14686   Topctop 19158   Clsdccld 19280
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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-ioo 11529  df-ioc 11530  df-topgen 14692  df-top 19163  df-bases 19165  df-cld 19283
This theorem is referenced by:  logdmopn  22755  orvclteel  28048  dvasin  29678  dvacos  29679  dvreasin  29680  dvreacos  29681  rfcnpre4  30987
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