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Theorem iocpnfordt 17280
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 2437 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  =  ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )
2 eqid 2437 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  (  -oo [,) x ) )  =  ran  ( x  e. 
RR*  |->  (  -oo [,) x ) )
3 eqid 2437 . . . . . . . . 9  |-  ran  (,)  =  ran  (,)
41, 2, 3leordtval 17278 . . . . . . . 8  |-  (ordTop `  <_  )  =  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)
5 letop 17271 . . . . . . . 8  |-  (ordTop `  <_  )  e.  Top
64, 5eqeltrri 2508 . . . . . . 7  |-  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top
7 tgclb 17036 . . . . . . 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 202 . . . . . 6  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  e. 
TopBases
9 bastg 17032 . . . . . 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 3382 . . . 4  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  C_  (ordTop `  <_  )
12 ssun1 3511 . . . . 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 3511 . . . . . 6  |-  ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  C_  ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )
14 eqid 2437 . . . . . . . 8  |-  ( A (,]  +oo )  =  ( A (,]  +oo )
15 oveq1 6089 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x (,]  +oo )  =  ( A (,]  +oo ) )
1615eqeq2d 2448 . . . . . . . . 9  |-  ( x  =  A  ->  (
( A (,]  +oo )  =  ( x (,]  +oo )  <->  ( A (,]  +oo )  =  ( A (,]  +oo )
) )
1716rspcev 3053 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  ( A (,]  +oo )  =  ( A (,]  +oo )
)  ->  E. x  e.  RR*  ( A (,]  +oo )  =  ( x (,]  +oo ) )
1814, 17mpan2 654 . . . . . . 7  |-  ( A  e.  RR*  ->  E. x  e.  RR*  ( A (,]  +oo )  =  ( x (,]  +oo ) )
19 eqid 2437 . . . . . . . 8  |-  ( x  e.  RR*  |->  ( x (,]  +oo ) )  =  ( x  e.  RR*  |->  ( x (,]  +oo ) )
20 ovex 6107 . . . . . . . 8  |-  ( x (,]  +oo )  e.  _V
2119, 20elrnmpti 5122 . . . . . . 7  |-  ( ( A (,]  +oo )  e.  ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  <->  E. x  e.  RR*  ( A (,]  +oo )  =  ( x (,]  +oo ) )
2218, 21sylibr 205 . . . . . 6  |-  ( A  e.  RR*  ->  ( A (,]  +oo )  e.  ran  ( x  e.  RR*  |->  ( x (,]  +oo ) ) )
2313, 22sseldi 3347 . . . . 5  |-  ( A  e.  RR*  ->  ( A (,]  +oo )  e.  ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) ) )
2412, 23sseldi 3347 . . . 4  |-  ( A  e.  RR*  ->  ( A (,]  +oo )  e.  ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)
2511, 24sseldi 3347 . . 3  |-  ( A  e.  RR*  ->  ( A (,]  +oo )  e.  (ordTop `  <_  ) )
2625adantr 453 . 2  |-  ( ( A  e.  RR*  /\  +oo  e.  RR* )  ->  ( A (,]  +oo )  e.  (ordTop `  <_  ) )
27 df-ioc 10922 . . . . . 6  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
2827ixxf 10927 . . . . 5  |-  (,] :
( RR*  X.  RR* ) --> ~P RR*
2928fdmi 5597 . . . 4  |-  dom  (,]  =  ( RR*  X.  RR* )
3029ndmov 6232 . . 3  |-  ( -.  ( A  e.  RR*  /\ 
+oo  e.  RR* )  -> 
( A (,]  +oo )  =  (/) )
31 0opn 16978 . . . 4  |-  ( (ordTop `  <_  )  e.  Top  -> 
(/)  e.  (ordTop `  <_  ) )
325, 31ax-mp 8 . . 3  |-  (/)  e.  (ordTop `  <_  )
3330, 32syl6eqel 2525 . 2  |-  ( -.  ( A  e.  RR*  /\ 
+oo  e.  RR* )  -> 
( A (,]  +oo )  e.  (ordTop `  <_  ) )
3426, 33pm2.61i 159 1  |-  ( A (,]  +oo )  e.  (ordTop `  <_  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2707    u. cun 3319    C_ wss 3321   (/)c0 3629   ~Pcpw 3800    e. cmpt 4267    X. cxp 4877   ran crn 4880   ` cfv 5455  (class class class)co 6082    +oocpnf 9118    -oocmnf 9119   RR*cxr 9120    < clt 9121    <_ cle 9122   (,)cioo 10917   (,]cioc 10918   [,)cico 10919   topGenctg 13666  ordTopcordt 13722   Topctop 16959   TopBasesctb 16963
This theorem is referenced by:  xrlimcnp  20808  pnfneige0  24337  lmxrge0  24338
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-fi 7417  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-ioo 10921  df-ioc 10922  df-ico 10923  df-icc 10924  df-topgen 13668  df-ordt 13726  df-ps 14630  df-tsr 14631  df-top 16964  df-bases 16966  df-topon 16967
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