MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iocpnfordt Structured version   Unicode version

Theorem iocpnfordt 19475
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 2460 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  =  ran  ( x  e.  RR*  |->  ( x (,] +oo ) )
2 eqid 2460 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( -oo [,) x ) )  =  ran  ( x  e. 
RR*  |->  ( -oo [,) x ) )
3 eqid 2460 . . . . . . . . 9  |-  ran  (,)  =  ran  (,)
41, 2, 3leordtval 19473 . . . . . . . 8  |-  (ordTop `  <_  )  =  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)
5 letop 19466 . . . . . . . 8  |-  (ordTop `  <_  )  e.  Top
64, 5eqeltrri 2545 . . . . . . 7  |-  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top
7 tgclb 19231 . . . . . . 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 19227 . . . . . 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 3530 . . . 4  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  C_  (ordTop `  <_  )
12 ssun1 3660 . . . . 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 3660 . . . . . 6  |-  ran  (
x  e.  RR*  |->  ( x (,] +oo ) ) 
C_  ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )
14 eqid 2460 . . . . . . . 8  |-  ( A (,] +oo )  =  ( A (,] +oo )
15 oveq1 6282 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x (,] +oo )  =  ( A (,] +oo ) )
1615eqeq2d 2474 . . . . . . . . 9  |-  ( x  =  A  ->  (
( A (,] +oo )  =  ( x (,] +oo )  <->  ( A (,] +oo )  =  ( A (,] +oo )
) )
1716rspcev 3207 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  ( A (,] +oo )  =  ( A (,] +oo ) )  ->  E. x  e.  RR*  ( A (,] +oo )  =  ( x (,] +oo ) )
1814, 17mpan2 671 . . . . . . 7  |-  ( A  e.  RR*  ->  E. x  e.  RR*  ( A (,] +oo )  =  ( x (,] +oo ) )
19 eqid 2460 . . . . . . . 8  |-  ( x  e.  RR*  |->  ( x (,] +oo ) )  =  ( x  e. 
RR*  |->  ( x (,] +oo ) )
20 ovex 6300 . . . . . . . 8  |-  ( x (,] +oo )  e. 
_V
2119, 20elrnmpti 5244 . . . . . . 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 3495 . . . . 5  |-  ( A  e.  RR*  ->  ( A (,] +oo )  e.  ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) ) )
2412, 23sseldi 3495 . . . 4  |-  ( A  e.  RR*  ->  ( A (,] +oo )  e.  ( ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)
2511, 24sseldi 3495 . . 3  |-  ( A  e.  RR*  ->  ( A (,] +oo )  e.  (ordTop `  <_  ) )
2625adantr 465 . 2  |-  ( ( A  e.  RR*  /\ +oo  e.  RR* )  ->  ( A (,] +oo )  e.  (ordTop `  <_  ) )
27 df-ioc 11523 . . . . . 6  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
2827ixxf 11528 . . . . 5  |-  (,] :
( RR*  X.  RR* ) --> ~P RR*
2928fdmi 5727 . . . 4  |-  dom  (,]  =  ( RR*  X.  RR* )
3029ndmov 6434 . . 3  |-  ( -.  ( A  e.  RR*  /\ +oo  e.  RR* )  ->  ( A (,] +oo )  =  (/) )
31 0opn 19173 . . . 4  |-  ( (ordTop `  <_  )  e.  Top  -> 
(/)  e.  (ordTop `  <_  ) )
325, 31ax-mp 5 . . 3  |-  (/)  e.  (ordTop `  <_  )
3330, 32syl6eqel 2556 . 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 1374    e. wcel 1762   E.wrex 2808    u. cun 3467    C_ wss 3469   (/)c0 3778   ~Pcpw 4003    |-> cmpt 4498    X. cxp 4990   ran crn 4993   ` cfv 5579  (class class class)co 6275   +oocpnf 9614   -oocmnf 9615   RR*cxr 9616    < clt 9617    <_ cle 9618   (,)cioo 11518   (,]cioc 11519   [,)cico 11520   topGenctg 14682  ordTopcordt 14743   Topctop 19154   TopBasesctb 19158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fi 7860  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-ioo 11522  df-ioc 11523  df-ico 11524  df-icc 11525  df-topgen 14688  df-ordt 14745  df-ps 15676  df-tsr 15677  df-top 19159  df-bases 19161  df-topon 19162
This theorem is referenced by:  xrlimcnp  23019  pnfneige0  27555  lmxrge0  27556
  Copyright terms: Public domain W3C validator