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Theorem iocmnfcld 18795
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 10706 . . . . . . 7  |-  -oo  e.  RR*
21a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  -oo  e.  RR* )
3 rexr 9122 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  RR* )
4 pnfxr 10705 . . . . . . 7  |-  +oo  e.  RR*
54a1i 11 . . . . . 6  |-  ( A  e.  RR  ->  +oo  e.  RR* )
6 mnflt 10714 . . . . . 6  |-  ( A  e.  RR  ->  -oo  <  A )
7 ltpnf 10713 . . . . . 6  |-  ( A  e.  RR  ->  A  <  +oo )
8 df-ioc 10913 . . . . . . 7  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
9 df-ioo 10912 . . . . . . 7  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
10 xrltnle 9136 . . . . . . 7  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
11 xrlelttr 10738 . . . . . . 7  |-  ( ( w  e.  RR*  /\  A  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
w  <_  A  /\  A  <  +oo )  ->  w  <  +oo ) )
12 xrlttr 10725 . . . . . . 7  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR*  /\  w  e. 
RR* )  ->  (
(  -oo  <  A  /\  A  <  w )  ->  -oo  <  w ) )
138, 9, 10, 9, 11, 12ixxun 10924 . . . . . 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 1192 . . . . 5  |-  ( A  e.  RR  ->  (
(  -oo (,] A )  u.  ( A (,)  +oo ) )  =  ( 
-oo (,)  +oo ) )
15 ioomax 10977 . . . . 5  |-  (  -oo (,) 
+oo )  =  RR
1614, 15syl6eq 2483 . . . 4  |-  ( A  e.  RR  ->  (
(  -oo (,] A )  u.  ( A (,)  +oo ) )  =  RR )
17 iocssre 10982 . . . . . 6  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR )  ->  (  -oo (,] A )  C_  RR )
181, 17mpan 652 . . . . 5  |-  ( A  e.  RR  ->  (  -oo (,] A )  C_  RR )
198, 9, 10ixxdisj 10923 . . . . . 6  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR*  /\  +oo  e.  RR* )  ->  ( (  -oo (,] A )  i^i  ( A (,)  +oo ) )  =  (/) )
202, 3, 5, 19syl3anc 1184 . . . . 5  |-  ( A  e.  RR  ->  (
(  -oo (,] A )  i^i  ( A (,)  +oo ) )  =  (/) )
21 uneqdifeq 3708 . . . . 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 643 . . . 4  |-  ( A  e.  RR  ->  (
( (  -oo (,] A )  u.  ( A (,)  +oo ) )  =  RR  <->  ( RR  \ 
(  -oo (,] A ) )  =  ( A (,)  +oo ) ) )
2316, 22mpbid 202 . . 3  |-  ( A  e.  RR  ->  ( RR  \  (  -oo (,] A ) )  =  ( A (,)  +oo ) )
24 iooretop 18792 . . 3  |-  ( A (,)  +oo )  e.  (
topGen `  ran  (,) )
2523, 24syl6eqel 2523 . 2  |-  ( A  e.  RR  ->  ( RR  \  (  -oo (,] A ) )  e.  ( topGen `  ran  (,) )
)
26 retop 18787 . . 3  |-  ( topGen ` 
ran  (,) )  e.  Top
27 uniretop 18788 . . . 4  |-  RR  =  U. ( topGen `  ran  (,) )
2827iscld2 17084 . . 3  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  (  -oo (,] A )  C_  RR )  ->  ( (  -oo (,] A )  e.  (
Clsd `  ( topGen ` 
ran  (,) ) )  <->  ( RR  \  (  -oo (,] A
) )  e.  (
topGen `  ran  (,) )
) )
2926, 18, 28sylancr 645 . 2  |-  ( A  e.  RR  ->  (
(  -oo (,] A )  e.  ( Clsd `  ( topGen `
 ran  (,) )
)  <->  ( RR  \ 
(  -oo (,] A ) )  e.  ( topGen ` 
ran  (,) ) ) )
3025, 29mpbird 224 1  |-  ( A  e.  RR  ->  (  -oo (,] A )  e.  ( Clsd `  ( topGen `
 ran  (,) )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725    \ cdif 3309    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   class class class wbr 4204   ran crn 4871   ` cfv 5446  (class class class)co 6073   RRcr 8981    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111    < clt 9112    <_ cle 9113   (,)cioo 10908   (,]cioc 10909   topGenctg 13657   Topctop 16950   Clsdccld 17072
This theorem is referenced by:  logdmopn  20532  orvclteel  24722  rfcnpre4  27662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-ioo 10912  df-ioc 10913  df-topgen 13659  df-top 16955  df-bases 16957  df-cld 17075
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