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Theorem iocmnfcld 21442
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 11326 . . . . . . 7  |- -oo  e.  RR*
21a1i 11 . . . . . 6  |-  ( A  e.  RR  -> -oo  e.  RR* )
3 rexr 9628 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  RR* )
4 pnfxr 11324 . . . . . . 7  |- +oo  e.  RR*
54a1i 11 . . . . . 6  |-  ( A  e.  RR  -> +oo  e.  RR* )
6 mnflt 11336 . . . . . 6  |-  ( A  e.  RR  -> -oo  <  A )
7 ltpnf 11334 . . . . . 6  |-  ( A  e.  RR  ->  A  < +oo )
8 df-ioc 11537 . . . . . . 7  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
9 df-ioo 11536 . . . . . . 7  |-  (,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <  y ) } )
10 xrltnle 9642 . . . . . . 7  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  <->  -.  w  <_  A ) )
11 xrlelttr 11362 . . . . . . 7  |-  ( ( w  e.  RR*  /\  A  e.  RR*  /\ +oo  e.  RR* )  ->  ( (
w  <_  A  /\  A  < +oo )  ->  w  < +oo ) )
12 xrlttr 11349 . . . . . . 7  |-  ( ( -oo  e.  RR*  /\  A  e.  RR*  /\  w  e. 
RR* )  ->  (
( -oo  <  A  /\  A  <  w )  -> -oo  <  w ) )
138, 9, 10, 9, 11, 12ixxun 11548 . . . . . 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 1234 . . . . 5  |-  ( A  e.  RR  ->  (
( -oo (,] A )  u.  ( A (,) +oo ) )  =  ( -oo (,) +oo )
)
15 ioomax 11602 . . . . 5  |-  ( -oo (,) +oo )  =  RR
1614, 15syl6eq 2511 . . . 4  |-  ( A  e.  RR  ->  (
( -oo (,] A )  u.  ( A (,) +oo ) )  =  RR )
17 iocssre 11607 . . . . . 6  |-  ( ( -oo  e.  RR*  /\  A  e.  RR )  ->  ( -oo (,] A )  C_  RR )
181, 17mpan 668 . . . . 5  |-  ( A  e.  RR  ->  ( -oo (,] A )  C_  RR )
198, 9, 10ixxdisj 11547 . . . . . 6  |-  ( ( -oo  e.  RR*  /\  A  e.  RR*  /\ +oo  e.  RR* )  ->  ( ( -oo (,] A )  i^i  ( A (,) +oo ) )  =  (/) )
202, 3, 5, 19syl3anc 1226 . . . . 5  |-  ( A  e.  RR  ->  (
( -oo (,] A )  i^i  ( A (,) +oo ) )  =  (/) )
21 uneqdifeq 3904 . . . . 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 659 . . . 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 21439 . . 3  |-  ( A (,) +oo )  e.  ( topGen `  ran  (,) )
2523, 24syl6eqel 2550 . 2  |-  ( A  e.  RR  ->  ( RR  \  ( -oo (,] A ) )  e.  ( topGen `  ran  (,) )
)
26 retop 21434 . . 3  |-  ( topGen ` 
ran  (,) )  e.  Top
27 uniretop 21435 . . . 4  |-  RR  =  U. ( topGen `  ran  (,) )
2827iscld2 19696 . . 3  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( -oo (,] A )  C_  RR )  ->  ( ( -oo (,] A )  e.  (
Clsd `  ( topGen ` 
ran  (,) ) )  <->  ( RR  \  ( -oo (,] A
) )  e.  (
topGen `  ran  (,) )
) )
2926, 18, 28sylancr 661 . 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 1398    e. wcel 1823    \ cdif 3458    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3783   class class class wbr 4439   ran crn 4989   ` cfv 5570  (class class class)co 6270   RRcr 9480   +oocpnf 9614   -oocmnf 9615   RR*cxr 9616    < clt 9617    <_ cle 9618   (,)cioo 11532   (,]cioc 11533   topGenctg 14927   Topctop 19561   Clsdccld 19684
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-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-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-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  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-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-ioo 11536  df-ioc 11537  df-topgen 14933  df-top 19566  df-bases 19568  df-cld 19687
This theorem is referenced by:  logdmopn  23198  orvclteel  28675  dvasin  30343  dvacos  30344  dvreasin  30345  dvreacos  30346  rfcnpre4  31649
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