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Theorem icomnfordt 16946
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 2283 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  =  ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )
2 eqid 2283 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  (  -oo [,) x ) )  =  ran  ( x  e. 
RR*  |->  (  -oo [,) x ) )
3 eqid 2283 . . . . . . . . 9  |-  ran  (,)  =  ran  (,)
41, 2, 3leordtval 16943 . . . . . . . 8  |-  (ordTop `  <_  )  =  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)
5 letop 16936 . . . . . . . 8  |-  (ordTop `  <_  )  e.  Top
64, 5eqeltrri 2354 . . . . . . 7  |-  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top
7 tgclb 16708 . . . . . . 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 16704 . . . . . 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 3211 . . . 4  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  C_  (ordTop `  <_  )
12 ssun1 3338 . . . . 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 3339 . . . . . 6  |-  ran  (
x  e.  RR*  |->  (  -oo [,) x ) )  C_  ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )
14 eqid 2283 . . . . . . . 8  |-  (  -oo [,) A )  =  ( 
-oo [,) A )
15 oveq2 5866 . . . . . . . . . 10  |-  ( x  =  A  ->  (  -oo [,) x )  =  (  -oo [,) A
) )
1615eqeq2d 2294 . . . . . . . . 9  |-  ( x  =  A  ->  (
(  -oo [,) A )  =  (  -oo [,) x )  <->  (  -oo [,) A )  =  ( 
-oo [,) A ) ) )
1716rspcev 2884 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  (  -oo [,) A )  =  (  -oo [,) A
) )  ->  E. x  e.  RR*  (  -oo [,) A )  =  ( 
-oo [,) x ) )
1814, 17mpan2 652 . . . . . . 7  |-  ( A  e.  RR*  ->  E. x  e.  RR*  (  -oo [,) A )  =  ( 
-oo [,) x ) )
19 eqid 2283 . . . . . . . 8  |-  ( x  e.  RR*  |->  (  -oo [,) x ) )  =  ( x  e.  RR*  |->  (  -oo [,) x ) )
20 ovex 5883 . . . . . . . 8  |-  (  -oo [,) x )  e.  _V
2119, 20elrnmpti 4930 . . . . . . 7  |-  ( ( 
-oo [,) A )  e. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) )  <->  E. x  e.  RR*  (  -oo [,) A )  =  ( 
-oo [,) x ) )
2218, 21sylibr 203 . . . . . 6  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  ran  ( x  e.  RR*  |->  (  -oo [,) x ) ) )
2313, 22sseldi 3178 . . . . 5  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) ) )
2412, 23sseldi 3178 . . . 4  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)
2511, 24sseldi 3178 . . 3  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  (ordTop `  <_  ) )
2625adantl 452 . 2  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR* )  ->  (  -oo [,) A )  e.  (ordTop `  <_  ) )
27 df-ico 10662 . . . . . 6  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
2827ixxf 10666 . . . . 5  |-  [,) :
( RR*  X.  RR* ) --> ~P RR*
2928fdmi 5394 . . . 4  |-  dom  [,)  =  ( RR*  X.  RR* )
3029ndmov 6004 . . 3  |-  ( -.  (  -oo  e.  RR*  /\  A  e.  RR* )  ->  (  -oo [,) A
)  =  (/) )
31 0opn 16650 . . . 4  |-  ( (ordTop `  <_  )  e.  Top  -> 
(/)  e.  (ordTop `  <_  ) )
325, 31ax-mp 8 . . 3  |-  (/)  e.  (ordTop `  <_  )
3330, 32syl6eqel 2371 . 2  |-  ( -.  (  -oo  e.  RR*  /\  A  e.  RR* )  ->  (  -oo [,) A
)  e.  (ordTop `  <_  ) )
3426, 33pm2.61i 156 1  |-  (  -oo [,) A )  e.  (ordTop `  <_  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    u. cun 3150    C_ wss 3152   (/)c0 3455   ~Pcpw 3625    e. cmpt 4077    X. cxp 4687   ran crn 4690   ` cfv 5255  (class class class)co 5858    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867    <_ cle 8868   (,)cioo 10656   (,]cioc 10657   [,)cico 10658   topGenctg 13342  ordTopcordt 13398   Topctop 16631   TopBasesctb 16635
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
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-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-int 3863  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-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-topgen 13344  df-ordt 13402  df-ps 14306  df-tsr 14307  df-top 16636  df-bases 16638  df-topon 16639
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