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Theorem iocpnfordt 16961
Description: An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
iocpnfordt  |-  ( A (,]  +oo )  e.  (ordTop `  <_  )

Proof of Theorem iocpnfordt
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  =  ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )
2 eqid 2296 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  (  -oo [,) x ) )  =  ran  ( x  e. 
RR*  |->  (  -oo [,) x ) )
3 eqid 2296 . . . . . . . . 9  |-  ran  (,)  =  ran  (,)
41, 2, 3leordtval 16959 . . . . . . . 8  |-  (ordTop `  <_  )  =  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)
5 letop 16952 . . . . . . . 8  |-  (ordTop `  <_  )  e.  Top
64, 5eqeltrri 2367 . . . . . . 7  |-  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top
7 tgclb 16724 . . . . . . 7  |-  ( ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  e. 
TopBases  <-> 
( topGen `  ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u.  ran  ( x  e.  RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top )
86, 7mpbir 200 . . . . . 6  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  e. 
TopBases
9 bastg 16720 . . . . . 6  |-  ( ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  e. 
TopBases  ->  ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u.  ran  ( x  e.  RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  C_  ( topGen `  ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u.  ran  ( x  e.  RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
) )
108, 9ax-mp 8 . . . . 5  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  C_  ( topGen `  ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u.  ran  ( x  e.  RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)
1110, 4sseqtr4i 3224 . . . 4  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  C_  (ordTop `  <_  )
12 ssun1 3351 . . . . 5  |-  ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u.  ran  ( x  e.  RR*  |->  (  -oo [,) x ) ) ) 
C_  ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u.  ran  ( x  e.  RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
13 ssun1 3351 . . . . . 6  |-  ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  C_  ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )
14 eqid 2296 . . . . . . . 8  |-  ( A (,]  +oo )  =  ( A (,]  +oo )
15 oveq1 5881 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x (,]  +oo )  =  ( A (,]  +oo ) )
1615eqeq2d 2307 . . . . . . . . 9  |-  ( x  =  A  ->  (
( A (,]  +oo )  =  ( x (,]  +oo )  <->  ( A (,]  +oo )  =  ( A (,]  +oo )
) )
1716rspcev 2897 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  ( A (,]  +oo )  =  ( A (,]  +oo )
)  ->  E. x  e.  RR*  ( A (,]  +oo )  =  ( x (,]  +oo ) )
1814, 17mpan2 652 . . . . . . 7  |-  ( A  e.  RR*  ->  E. x  e.  RR*  ( A (,]  +oo )  =  ( x (,]  +oo ) )
19 eqid 2296 . . . . . . . 8  |-  ( x  e.  RR*  |->  ( x (,]  +oo ) )  =  ( x  e.  RR*  |->  ( x (,]  +oo ) )
20 ovex 5899 . . . . . . . 8  |-  ( x (,]  +oo )  e.  _V
2119, 20elrnmpti 4946 . . . . . . 7  |-  ( ( A (,]  +oo )  e.  ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  <->  E. x  e.  RR*  ( A (,]  +oo )  =  ( x (,]  +oo ) )
2218, 21sylibr 203 . . . . . 6  |-  ( A  e.  RR*  ->  ( A (,]  +oo )  e.  ran  ( x  e.  RR*  |->  ( x (,]  +oo ) ) )
2313, 22sseldi 3191 . . . . 5  |-  ( A  e.  RR*  ->  ( A (,]  +oo )  e.  ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) ) )
2412, 23sseldi 3191 . . . 4  |-  ( A  e.  RR*  ->  ( A (,]  +oo )  e.  ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)
2511, 24sseldi 3191 . . 3  |-  ( A  e.  RR*  ->  ( A (,]  +oo )  e.  (ordTop `  <_  ) )
2625adantr 451 . 2  |-  ( ( A  e.  RR*  /\  +oo  e.  RR* )  ->  ( A (,]  +oo )  e.  (ordTop `  <_  ) )
27 df-ioc 10677 . . . . . 6  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
2827ixxf 10682 . . . . 5  |-  (,] :
( RR*  X.  RR* ) --> ~P RR*
2928fdmi 5410 . . . 4  |-  dom  (,]  =  ( RR*  X.  RR* )
3029ndmov 6020 . . 3  |-  ( -.  ( A  e.  RR*  /\ 
+oo  e.  RR* )  -> 
( A (,]  +oo )  =  (/) )
31 0opn 16666 . . . 4  |-  ( (ordTop `  <_  )  e.  Top  -> 
(/)  e.  (ordTop `  <_  ) )
325, 31ax-mp 8 . . 3  |-  (/)  e.  (ordTop `  <_  )
3330, 32syl6eqel 2384 . 2  |-  ( -.  ( A  e.  RR*  /\ 
+oo  e.  RR* )  -> 
( A (,]  +oo )  e.  (ordTop `  <_  ) )
3426, 33pm2.61i 156 1  |-  ( A (,]  +oo )  e.  (ordTop `  <_  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    u. cun 3163    C_ wss 3165   (/)c0 3468   ~Pcpw 3638    e. cmpt 4093    X. cxp 4703   ran crn 4706   ` cfv 5271  (class class class)co 5874    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882    < clt 8883    <_ cle 8884   (,)cioo 10672   (,]cioc 10673   [,)cico 10674   topGenctg 13358  ordTopcordt 13414   Topctop 16647   TopBasesctb 16651
This theorem is referenced by:  xrlimcnp  20279  pnfneige0  23389  lmxrge0  23390
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
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-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-int 3879  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-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-topgen 13360  df-ordt 13418  df-ps 14322  df-tsr 14323  df-top 16652  df-bases 16654  df-topon 16655
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