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Theorem iocpnfordt 18661
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 2433 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  =  ran  ( x  e.  RR*  |->  ( x (,] +oo ) )
2 eqid 2433 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( -oo [,) x ) )  =  ran  ( x  e. 
RR*  |->  ( -oo [,) x ) )
3 eqid 2433 . . . . . . . . 9  |-  ran  (,)  =  ran  (,)
41, 2, 3leordtval 18659 . . . . . . . 8  |-  (ordTop `  <_  )  =  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)
5 letop 18652 . . . . . . . 8  |-  (ordTop `  <_  )  e.  Top
64, 5eqeltrri 2504 . . . . . . 7  |-  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top
7 tgclb 18417 . . . . . . 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 18413 . . . . . 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 3377 . . . 4  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  C_  (ordTop `  <_  )
12 ssun1 3507 . . . . 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 3507 . . . . . 6  |-  ran  (
x  e.  RR*  |->  ( x (,] +oo ) ) 
C_  ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )
14 eqid 2433 . . . . . . . 8  |-  ( A (,] +oo )  =  ( A (,] +oo )
15 oveq1 6087 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x (,] +oo )  =  ( A (,] +oo ) )
1615eqeq2d 2444 . . . . . . . . 9  |-  ( x  =  A  ->  (
( A (,] +oo )  =  ( x (,] +oo )  <->  ( A (,] +oo )  =  ( A (,] +oo )
) )
1716rspcev 3062 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  ( A (,] +oo )  =  ( A (,] +oo ) )  ->  E. x  e.  RR*  ( A (,] +oo )  =  ( x (,] +oo ) )
1814, 17mpan2 664 . . . . . . 7  |-  ( A  e.  RR*  ->  E. x  e.  RR*  ( A (,] +oo )  =  ( x (,] +oo ) )
19 eqid 2433 . . . . . . . 8  |-  ( x  e.  RR*  |->  ( x (,] +oo ) )  =  ( x  e. 
RR*  |->  ( x (,] +oo ) )
20 ovex 6105 . . . . . . . 8  |-  ( x (,] +oo )  e. 
_V
2119, 20elrnmpti 5077 . . . . . . 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 3342 . . . . 5  |-  ( A  e.  RR*  ->  ( A (,] +oo )  e.  ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) ) )
2412, 23sseldi 3342 . . . 4  |-  ( A  e.  RR*  ->  ( A (,] +oo )  e.  ( ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)
2511, 24sseldi 3342 . . 3  |-  ( A  e.  RR*  ->  ( A (,] +oo )  e.  (ordTop `  <_  ) )
2625adantr 462 . 2  |-  ( ( A  e.  RR*  /\ +oo  e.  RR* )  ->  ( A (,] +oo )  e.  (ordTop `  <_  ) )
27 df-ioc 11293 . . . . . 6  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
2827ixxf 11298 . . . . 5  |-  (,] :
( RR*  X.  RR* ) --> ~P RR*
2928fdmi 5552 . . . 4  |-  dom  (,]  =  ( RR*  X.  RR* )
3029ndmov 6236 . . 3  |-  ( -.  ( A  e.  RR*  /\ +oo  e.  RR* )  ->  ( A (,] +oo )  =  (/) )
31 0opn 18359 . . . 4  |-  ( (ordTop `  <_  )  e.  Top  -> 
(/)  e.  (ordTop `  <_  ) )
325, 31ax-mp 5 . . 3  |-  (/)  e.  (ordTop `  <_  )
3330, 32syl6eqel 2521 . 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 369    = wceq 1362    e. wcel 1755   E.wrex 2706    u. cun 3314    C_ wss 3316   (/)c0 3625   ~Pcpw 3848    e. cmpt 4338    X. cxp 4825   ran crn 4828   ` cfv 5406  (class class class)co 6080   +oocpnf 9403   -oocmnf 9404   RR*cxr 9405    < clt 9406    <_ cle 9407   (,)cioo 11288   (,]cioc 11289   [,)cico 11290   topGenctg 14359  ordTopcordt 14420   Topctop 18340   TopBasesctb 18344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fi 7649  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-ioo 11292  df-ioc 11293  df-ico 11294  df-icc 11295  df-topgen 14365  df-ordt 14422  df-ps 15353  df-tsr 15354  df-top 18345  df-bases 18347  df-topon 18348
This theorem is referenced by:  xrlimcnp  22247  pnfneige0  26235  lmxrge0  26236
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