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Theorem iocmnfcld 18278
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 10456 . . . . . . 7  |-  -oo  e.  RR*
21a1i 10 . . . . . 6  |-  ( A  e.  RR  ->  -oo  e.  RR* )
3 rexr 8877 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  RR* )
4 pnfxr 10455 . . . . . . 7  |-  +oo  e.  RR*
54a1i 10 . . . . . 6  |-  ( A  e.  RR  ->  +oo  e.  RR* )
6 mnflt 10464 . . . . . 6  |-  ( A  e.  RR  ->  -oo  <  A )
7 ltpnf 10463 . . . . . 6  |-  ( A  e.  RR  ->  A  <  +oo )
8 df-ioc 10661 . . . . . . 7  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
9 df-ioo 10660 . . . . . . 7  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
10 xrltnle 8891 . . . . . . 7  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
11 xrlelttr 10487 . . . . . . 7  |-  ( ( w  e.  RR*  /\  A  e.  RR*  /\  +oo  e.  RR* )  ->  ( (
w  <_  A  /\  A  <  +oo )  ->  w  <  +oo ) )
12 xrlttr 10474 . . . . . . 7  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR*  /\  w  e. 
RR* )  ->  (
(  -oo  <  A  /\  A  <  w )  ->  -oo  <  w ) )
138, 9, 10, 9, 11, 12ixxun 10672 . . . . . 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 10724 . . . . 5  |-  (  -oo (,) 
+oo )  =  RR
1614, 15syl6eq 2331 . . . 4  |-  ( A  e.  RR  ->  (
(  -oo (,] A )  u.  ( A (,)  +oo ) )  =  RR )
17 iocssre 10729 . . . . . 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 10671 . . . . . 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 3542 . . . . 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 18275 . . 3  |-  ( A (,)  +oo )  e.  (
topGen `  ran  (,) )
2523, 24syl6eqel 2371 . 2  |-  ( A  e.  RR  ->  ( RR  \  (  -oo (,] A ) )  e.  ( topGen `  ran  (,) )
)
26 retop 18270 . . 3  |-  ( topGen ` 
ran  (,) )  e.  Top
27 uniretop 18271 . . . 4  |-  RR  =  U. ( topGen `  ran  (,) )
2827iscld2 16765 . . 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 1623    e. wcel 1684    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   class class class wbr 4023   ran crn 4690   ` cfv 5255  (class class class)co 5858   RRcr 8736    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867    <_ cle 8868   (,)cioo 10656   (,]cioc 10657   topGenctg 13342   Topctop 16631   Clsdccld 16753
This theorem is referenced by:  logdmopn  19996  orvclteel  23673  rfcnpre4  27705
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-ioo 10660  df-ioc 10661  df-topgen 13344  df-top 16636  df-bases 16638  df-cld 16756
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