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Theorem iocpnfordt 20009
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 2402 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  =  ran  ( x  e.  RR*  |->  ( x (,] +oo ) )
2 eqid 2402 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( -oo [,) x ) )  =  ran  ( x  e. 
RR*  |->  ( -oo [,) x ) )
3 eqid 2402 . . . . . . . . 9  |-  ran  (,)  =  ran  (,)
41, 2, 3leordtval 20007 . . . . . . . 8  |-  (ordTop `  <_  )  =  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)
5 letop 20000 . . . . . . . 8  |-  (ordTop `  <_  )  e.  Top
64, 5eqeltrri 2487 . . . . . . 7  |-  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top
7 tgclb 19764 . . . . . . 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 209 . . . . . 6  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  e. 
TopBases
9 bastg 19759 . . . . . 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 5 . . . . 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 3475 . . . 4  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  C_  (ordTop `  <_  )
12 ssun1 3606 . . . . 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 3606 . . . . . 6  |-  ran  (
x  e.  RR*  |->  ( x (,] +oo ) ) 
C_  ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )
14 eqid 2402 . . . . . . . 8  |-  ( A (,] +oo )  =  ( A (,] +oo )
15 oveq1 6285 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x (,] +oo )  =  ( A (,] +oo ) )
1615eqeq2d 2416 . . . . . . . . 9  |-  ( x  =  A  ->  (
( A (,] +oo )  =  ( x (,] +oo )  <->  ( A (,] +oo )  =  ( A (,] +oo )
) )
1716rspcev 3160 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  ( A (,] +oo )  =  ( A (,] +oo ) )  ->  E. x  e.  RR*  ( A (,] +oo )  =  ( x (,] +oo ) )
1814, 17mpan2 669 . . . . . . 7  |-  ( A  e.  RR*  ->  E. x  e.  RR*  ( A (,] +oo )  =  ( x (,] +oo ) )
19 eqid 2402 . . . . . . . 8  |-  ( x  e.  RR*  |->  ( x (,] +oo ) )  =  ( x  e. 
RR*  |->  ( x (,] +oo ) )
20 ovex 6306 . . . . . . . 8  |-  ( x (,] +oo )  e. 
_V
2119, 20elrnmpti 5074 . . . . . . 7  |-  ( ( A (,] +oo )  e.  ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  <->  E. x  e.  RR*  ( A (,] +oo )  =  ( x (,] +oo ) )
2218, 21sylibr 212 . . . . . 6  |-  ( A  e.  RR*  ->  ( A (,] +oo )  e. 
ran  ( x  e. 
RR*  |->  ( x (,] +oo ) ) )
2313, 22sseldi 3440 . . . . 5  |-  ( A  e.  RR*  ->  ( A (,] +oo )  e.  ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) ) )
2412, 23sseldi 3440 . . . 4  |-  ( A  e.  RR*  ->  ( A (,] +oo )  e.  ( ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)
2511, 24sseldi 3440 . . 3  |-  ( A  e.  RR*  ->  ( A (,] +oo )  e.  (ordTop `  <_  ) )
2625adantr 463 . 2  |-  ( ( A  e.  RR*  /\ +oo  e.  RR* )  ->  ( A (,] +oo )  e.  (ordTop `  <_  ) )
27 df-ioc 11587 . . . . . 6  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
2827ixxf 11592 . . . . 5  |-  (,] :
( RR*  X.  RR* ) --> ~P RR*
2928fdmi 5719 . . . 4  |-  dom  (,]  =  ( RR*  X.  RR* )
3029ndmov 6440 . . 3  |-  ( -.  ( A  e.  RR*  /\ +oo  e.  RR* )  ->  ( A (,] +oo )  =  (/) )
31 0opn 19705 . . . 4  |-  ( (ordTop `  <_  )  e.  Top  -> 
(/)  e.  (ordTop `  <_  ) )
325, 31ax-mp 5 . . 3  |-  (/)  e.  (ordTop `  <_  )
3330, 32syl6eqel 2498 . 2  |-  ( -.  ( A  e.  RR*  /\ +oo  e.  RR* )  ->  ( A (,] +oo )  e.  (ordTop `  <_  ) )
3426, 33pm2.61i 164 1  |-  ( A (,] +oo )  e.  (ordTop `  <_  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 367    = wceq 1405    e. wcel 1842   E.wrex 2755    u. cun 3412    C_ wss 3414   (/)c0 3738   ~Pcpw 3955    |-> cmpt 4453    X. cxp 4821   ran crn 4824   ` cfv 5569  (class class class)co 6278   +oocpnf 9655   -oocmnf 9656   RR*cxr 9657    < clt 9658    <_ cle 9659   (,)cioo 11582   (,]cioc 11583   [,)cico 11584   topGenctg 15052  ordTopcordt 15113   Topctop 19686   TopBasesctb 19690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fi 7905  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-ioo 11586  df-ioc 11587  df-ico 11588  df-icc 11589  df-topgen 15058  df-ordt 15115  df-ps 16154  df-tsr 16155  df-top 19691  df-bases 19693  df-topon 19694
This theorem is referenced by:  xrlimcnp  23624  pnfneige0  28386  lmxrge0  28387
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