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Theorem icomnfordt 17052
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 2358 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  =  ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )
2 eqid 2358 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  (  -oo [,) x ) )  =  ran  ( x  e. 
RR*  |->  (  -oo [,) x ) )
3 eqid 2358 . . . . . . . . 9  |-  ran  (,)  =  ran  (,)
41, 2, 3leordtval 17049 . . . . . . . 8  |-  (ordTop `  <_  )  =  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)
5 letop 17042 . . . . . . . 8  |-  (ordTop `  <_  )  e.  Top
64, 5eqeltrri 2429 . . . . . . 7  |-  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top
7 tgclb 16814 . . . . . . 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 16810 . . . . . 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 3287 . . . 4  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  C_  (ordTop `  <_  )
12 ssun1 3414 . . . . 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 3415 . . . . . 6  |-  ran  (
x  e.  RR*  |->  (  -oo [,) x ) )  C_  ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )
14 eqid 2358 . . . . . . . 8  |-  (  -oo [,) A )  =  ( 
-oo [,) A )
15 oveq2 5953 . . . . . . . . . 10  |-  ( x  =  A  ->  (  -oo [,) x )  =  (  -oo [,) A
) )
1615eqeq2d 2369 . . . . . . . . 9  |-  ( x  =  A  ->  (
(  -oo [,) A )  =  (  -oo [,) x )  <->  (  -oo [,) A )  =  ( 
-oo [,) A ) ) )
1716rspcev 2960 . . . . . . . 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 2358 . . . . . . . 8  |-  ( x  e.  RR*  |->  (  -oo [,) x ) )  =  ( x  e.  RR*  |->  (  -oo [,) x ) )
20 ovex 5970 . . . . . . . 8  |-  (  -oo [,) x )  e.  _V
2119, 20elrnmpti 5012 . . . . . . 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 3254 . . . . 5  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) ) )
2412, 23sseldi 3254 . . . 4  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)
2511, 24sseldi 3254 . . 3  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  (ordTop `  <_  ) )
2625adantl 452 . 2  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR* )  ->  (  -oo [,) A )  e.  (ordTop `  <_  ) )
27 df-ico 10754 . . . . . 6  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
2827ixxf 10758 . . . . 5  |-  [,) :
( RR*  X.  RR* ) --> ~P RR*
2928fdmi 5477 . . . 4  |-  dom  [,)  =  ( RR*  X.  RR* )
3029ndmov 6091 . . 3  |-  ( -.  (  -oo  e.  RR*  /\  A  e.  RR* )  ->  (  -oo [,) A
)  =  (/) )
31 0opn 16756 . . . 4  |-  ( (ordTop `  <_  )  e.  Top  -> 
(/)  e.  (ordTop `  <_  ) )
325, 31ax-mp 8 . . 3  |-  (/)  e.  (ordTop `  <_  )
3330, 32syl6eqel 2446 . 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 1642    e. wcel 1710   E.wrex 2620    u. cun 3226    C_ wss 3228   (/)c0 3531   ~Pcpw 3701    e. cmpt 4158    X. cxp 4769   ran crn 4772   ` cfv 5337  (class class class)co 5945    +oocpnf 8954    -oocmnf 8955   RR*cxr 8956    < clt 8957    <_ cle 8958   (,)cioo 10748   (,]cioc 10749   [,)cico 10750   topGenctg 13441  ordTopcordt 13497   Topctop 16737   TopBasesctb 16741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-fi 7255  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-ioo 10752  df-ioc 10753  df-ico 10754  df-icc 10755  df-topgen 13443  df-ordt 13501  df-ps 14405  df-tsr 14406  df-top 16742  df-bases 16744  df-topon 16745
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