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Theorem icomnfordt 17285
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 2438 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  =  ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )
2 eqid 2438 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  (  -oo [,) x ) )  =  ran  ( x  e. 
RR*  |->  (  -oo [,) x ) )
3 eqid 2438 . . . . . . . . 9  |-  ran  (,)  =  ran  (,)
41, 2, 3leordtval 17282 . . . . . . . 8  |-  (ordTop `  <_  )  =  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)
5 letop 17275 . . . . . . . 8  |-  (ordTop `  <_  )  e.  Top
64, 5eqeltrri 2509 . . . . . . 7  |-  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top
7 tgclb 17040 . . . . . . 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 17036 . . . . . 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 3383 . . . 4  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )  C_  (ordTop `  <_  )
12 ssun1 3512 . . . . 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 3513 . . . . . 6  |-  ran  (
x  e.  RR*  |->  (  -oo [,) x ) )  C_  ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )
14 eqid 2438 . . . . . . . 8  |-  (  -oo [,) A )  =  ( 
-oo [,) A )
15 oveq2 6092 . . . . . . . . . 10  |-  ( x  =  A  ->  (  -oo [,) x )  =  (  -oo [,) A
) )
1615eqeq2d 2449 . . . . . . . . 9  |-  ( x  =  A  ->  (
(  -oo [,) A )  =  (  -oo [,) x )  <->  (  -oo [,) A )  =  ( 
-oo [,) A ) ) )
1716rspcev 3054 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  (  -oo [,) A )  =  (  -oo [,) A
) )  ->  E. x  e.  RR*  (  -oo [,) A )  =  ( 
-oo [,) x ) )
1814, 17mpan2 654 . . . . . . 7  |-  ( A  e.  RR*  ->  E. x  e.  RR*  (  -oo [,) A )  =  ( 
-oo [,) x ) )
19 eqid 2438 . . . . . . . 8  |-  ( x  e.  RR*  |->  (  -oo [,) x ) )  =  ( x  e.  RR*  |->  (  -oo [,) x ) )
20 ovex 6109 . . . . . . . 8  |-  (  -oo [,) x )  e.  _V
2119, 20elrnmpti 5124 . . . . . . 7  |-  ( ( 
-oo [,) A )  e. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) )  <->  E. x  e.  RR*  (  -oo [,) A )  =  ( 
-oo [,) x ) )
2218, 21sylibr 205 . . . . . 6  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  ran  ( x  e.  RR*  |->  (  -oo [,) x ) ) )
2313, 22sseldi 3348 . . . . 5  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  ( ran  ( x  e. 
RR*  |->  ( x (,] 
+oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) ) )
2412, 23sseldi 3348 . . . 4  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  ( ( ran  ( x  e.  RR*  |->  ( x (,]  +oo ) )  u. 
ran  ( x  e. 
RR*  |->  (  -oo [,) x ) ) )  u.  ran  (,) )
)
2511, 24sseldi 3348 . . 3  |-  ( A  e.  RR*  ->  (  -oo [,) A )  e.  (ordTop `  <_  ) )
2625adantl 454 . 2  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR* )  ->  (  -oo [,) A )  e.  (ordTop `  <_  ) )
27 df-ico 10927 . . . . . 6  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
2827ixxf 10931 . . . . 5  |-  [,) :
( RR*  X.  RR* ) --> ~P RR*
2928fdmi 5599 . . . 4  |-  dom  [,)  =  ( RR*  X.  RR* )
3029ndmov 6234 . . 3  |-  ( -.  (  -oo  e.  RR*  /\  A  e.  RR* )  ->  (  -oo [,) A
)  =  (/) )
31 0opn 16982 . . . 4  |-  ( (ordTop `  <_  )  e.  Top  -> 
(/)  e.  (ordTop `  <_  ) )
325, 31ax-mp 5 . . 3  |-  (/)  e.  (ordTop `  <_  )
3330, 32syl6eqel 2526 . 2  |-  ( -.  (  -oo  e.  RR*  /\  A  e.  RR* )  ->  (  -oo [,) A
)  e.  (ordTop `  <_  ) )
3426, 33pm2.61i 159 1  |-  (  -oo [,) A )  e.  (ordTop `  <_  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708    u. cun 3320    C_ wss 3322   (/)c0 3630   ~Pcpw 3801    e. cmpt 4269    X. cxp 4879   ran crn 4882   ` cfv 5457  (class class class)co 6084    +oocpnf 9122    -oocmnf 9123   RR*cxr 9124    < clt 9125    <_ cle 9126   (,)cioo 10921   (,]cioc 10922   [,)cico 10923   topGenctg 13670  ordTopcordt 13726   Topctop 16963   TopBasesctb 16967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-ioo 10925  df-ioc 10926  df-ico 10927  df-icc 10928  df-topgen 13672  df-ordt 13730  df-ps 14634  df-tsr 14635  df-top 16968  df-bases 16970  df-topon 16971
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