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Theorem iocmnfcld 18294
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 10472 . . . . . . 7  |-  -oo  e.  RR*
21a1i 10 . . . . . 6  |-  ( A  e.  RR  ->  -oo  e.  RR* )
3 rexr 8893 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  RR* )
4 pnfxr 10471 . . . . . . 7  |-  +oo  e.  RR*
54a1i 10 . . . . . 6  |-  ( A  e.  RR  ->  +oo  e.  RR* )
6 mnflt 10480 . . . . . 6  |-  ( A  e.  RR  ->  -oo  <  A )
7 ltpnf 10479 . . . . . 6  |-  ( A  e.  RR  ->  A  <  +oo )
8 df-ioc 10677 . . . . . . 7  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
9 df-ioo 10676 . . . . . . 7  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
10 xrltnle 8907 . . . . . . 7  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
11 xrlelttr 10503 . . . . . . 7  |-  ( ( w  e.  RR*  /\  A  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
w  <_  A  /\  A  <  +oo )  ->  w  <  +oo ) )
12 xrlttr 10490 . . . . . . 7  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR*  /\  w  e. 
RR* )  ->  (
(  -oo  <  A  /\  A  <  w )  ->  -oo  <  w ) )
138, 9, 10, 9, 11, 12ixxun 10688 . . . . . 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 1190 . . . . 5  |-  ( A  e.  RR  ->  (
(  -oo (,] A )  u.  ( A (,)  +oo ) )  =  ( 
-oo (,)  +oo ) )
15 ioomax 10740 . . . . 5  |-  (  -oo (,) 
+oo )  =  RR
1614, 15syl6eq 2344 . . . 4  |-  ( A  e.  RR  ->  (
(  -oo (,] A )  u.  ( A (,)  +oo ) )  =  RR )
17 iocssre 10745 . . . . . 6  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR )  ->  (  -oo (,] A )  C_  RR )
181, 17mpan 651 . . . . 5  |-  ( A  e.  RR  ->  (  -oo (,] A )  C_  RR )
198, 9, 10ixxdisj 10687 . . . . . 6  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR*  /\  +oo  e.  RR* )  ->  ( (  -oo (,] A )  i^i  ( A (,)  +oo ) )  =  (/) )
202, 3, 5, 19syl3anc 1182 . . . . 5  |-  ( A  e.  RR  ->  (
(  -oo (,] A )  i^i  ( A (,)  +oo ) )  =  (/) )
21 uneqdifeq 3555 . . . . 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 642 . . . 4  |-  ( A  e.  RR  ->  (
( (  -oo (,] A )  u.  ( A (,)  +oo ) )  =  RR  <->  ( RR  \ 
(  -oo (,] A ) )  =  ( A (,)  +oo ) ) )
2316, 22mpbid 201 . . 3  |-  ( A  e.  RR  ->  ( RR  \  (  -oo (,] A ) )  =  ( A (,)  +oo ) )
24 iooretop 18291 . . 3  |-  ( A (,)  +oo )  e.  (
topGen `  ran  (,) )
2523, 24syl6eqel 2384 . 2  |-  ( A  e.  RR  ->  ( RR  \  (  -oo (,] A ) )  e.  ( topGen `  ran  (,) )
)
26 retop 18286 . . 3  |-  ( topGen ` 
ran  (,) )  e.  Top
27 uniretop 18287 . . . 4  |-  RR  =  U. ( topGen `  ran  (,) )
2827iscld2 16781 . . 3  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  (  -oo (,] A )  C_  RR )  ->  ( (  -oo (,] A )  e.  (
Clsd `  ( topGen ` 
ran  (,) ) )  <->  ( RR  \  (  -oo (,] A
) )  e.  (
topGen `  ran  (,) )
) )
2926, 18, 28sylancr 644 . 2  |-  ( A  e.  RR  ->  (
(  -oo (,] A )  e.  ( Clsd `  ( topGen `
 ran  (,) )
)  <->  ( RR  \ 
(  -oo (,] A ) )  e.  ( topGen ` 
ran  (,) ) ) )
3025, 29mpbird 223 1  |-  ( A  e.  RR  ->  (  -oo (,] A )  e.  ( Clsd `  ( topGen `
 ran  (,) )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   class class class wbr 4039   ran crn 4706   ` cfv 5271  (class class class)co 5874   RRcr 8752    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882    < clt 8883    <_ cle 8884   (,)cioo 10672   (,]cioc 10673   topGenctg 13358   Topctop 16647   Clsdccld 16769
This theorem is referenced by:  logdmopn  20012  orvclteel  23688  rfcnpre4  27808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-ioo 10676  df-ioc 10677  df-topgen 13360  df-top 16652  df-bases 16654  df-cld 16772
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