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Theorem icomnfordt 17242
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
icomnfordt  |-  (  -oo [,) A )  e.  (ordTop `  <_  )

Proof of Theorem icomnfordt
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2412 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  =  ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )
2 eqid 2412 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  (  -oo [,) x ) )  =  ran  ( x  e. 
RR*  |->  (  -oo [,) x ) )
3 eqid 2412 . . . . . . . . 9  |-  ran  (,)  =  ran  (,)
41, 2, 3leordtval 17239 . . . . . . . 8  |-  (ordTop `  <_  )  =  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)
5 letop 17232 . . . . . . . 8  |-  (ordTop `  <_  )  e.  Top
64, 5eqeltrri 2483 . . . . . . 7  |-  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top
7 tgclb 16998 . . . . . . 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 201 . . . . . 6  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  e. 
TopBases
9 bastg 16994 . . . . . 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 3349 . . . 4  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  C_  (ordTop `  <_  )
12 ssun1 3478 . . . . 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 ssun2 3479 . . . . . 6  |-  ran  (
x  e.  RR*  |->  (  -oo [,) x ) )  C_  ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )
14 eqid 2412 . . . . . . . 8  |-  (  -oo [,) A )  =  ( 
-oo [,) A )
15 oveq2 6056 . . . . . . . . . 10  |-  ( x  =  A  ->  (  -oo [,) x )  =  (  -oo [,) A
) )
1615eqeq2d 2423 . . . . . . . . 9  |-  ( x  =  A  ->  (
(  -oo [,) A )  =  (  -oo [,) x )  <->  (  -oo [,) A )  =  ( 
-oo [,) A ) ) )
1716rspcev 3020 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  (  -oo [,) A )  =  (  -oo [,) A
) )  ->  E. x  e.  RR*  (  -oo [,) A )  =  ( 
-oo [,) x ) )
1814, 17mpan2 653 . . . . . . 7  |-  ( A  e.  RR*  ->  E. x  e.  RR*  (  -oo [,) A )  =  ( 
-oo [,) x ) )
19 eqid 2412 . . . . . . . 8  |-  ( x  e.  RR*  |->  (  -oo [,) x ) )  =  ( x  e.  RR*  |->  (  -oo [,) x ) )
20 ovex 6073 . . . . . . . 8  |-  (  -oo [,) x )  e.  _V
2119, 20elrnmpti 5088 . . . . . . 7  |-  ( ( 
-oo [,) A )  e. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) )  <->  E. x  e.  RR*  (  -oo [,) A )  =  ( 
-oo [,) x ) )
2218, 21sylibr 204 . . . . . 6  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  ran  ( x  e.  RR*  |->  (  -oo [,) x ) ) )
2313, 22sseldi 3314 . . . . 5  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) ) )
2412, 23sseldi 3314 . . . 4  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)
2511, 24sseldi 3314 . . 3  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  (ordTop `  <_  ) )
2625adantl 453 . 2  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR* )  ->  (  -oo [,) A )  e.  (ordTop `  <_  ) )
27 df-ico 10886 . . . . . 6  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
2827ixxf 10890 . . . . 5  |-  [,) :
( RR*  X.  RR* ) --> ~P RR*
2928fdmi 5563 . . . 4  |-  dom  [,)  =  ( RR*  X.  RR* )
3029ndmov 6198 . . 3  |-  ( -.  (  -oo  e.  RR*  /\  A  e.  RR* )  ->  (  -oo [,) A
)  =  (/) )
31 0opn 16940 . . . 4  |-  ( (ordTop `  <_  )  e.  Top  -> 
(/)  e.  (ordTop `  <_  ) )
325, 31ax-mp 8 . . 3  |-  (/)  e.  (ordTop `  <_  )
3330, 32syl6eqel 2500 . 2  |-  ( -.  (  -oo  e.  RR*  /\  A  e.  RR* )  ->  (  -oo [,) A
)  e.  (ordTop `  <_  ) )
3426, 33pm2.61i 158 1  |-  (  -oo [,) A )  e.  (ordTop `  <_  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2675    u. cun 3286    C_ wss 3288   (/)c0 3596   ~Pcpw 3767    e. cmpt 4234    X. cxp 4843   ran crn 4846   ` cfv 5421  (class class class)co 6048    +oocpnf 9081    -oocmnf 9082   RR*cxr 9083    < clt 9084    <_ cle 9085   (,)cioo 10880   (,]cioc 10881   [,)cico 10882   topGenctg 13628  ordTopcordt 13684   Topctop 16921   TopBasesctb 16925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-fi 7382  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-ioo 10884  df-ioc 10885  df-ico 10886  df-icc 10887  df-topgen 13630  df-ordt 13688  df-ps 14592  df-tsr 14593  df-top 16926  df-bases 16928  df-topon 16929
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