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

Theorem iocpnfordt 18822
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 2443 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  =  ran  ( x  e.  RR*  |->  ( x (,] +oo ) )
2 eqid 2443 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( -oo [,) x ) )  =  ran  ( x  e. 
RR*  |->  ( -oo [,) x ) )
3 eqid 2443 . . . . . . . . 9  |-  ran  (,)  =  ran  (,)
41, 2, 3leordtval 18820 . . . . . . . 8  |-  (ordTop `  <_  )  =  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)
5 letop 18813 . . . . . . . 8  |-  (ordTop `  <_  )  e.  Top
64, 5eqeltrri 2514 . . . . . . 7  |-  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top
7 tgclb 18578 . . . . . . 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 18574 . . . . . 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 3392 . . . 4  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  C_  (ordTop `  <_  )
12 ssun1 3522 . . . . 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 3522 . . . . . 6  |-  ran  (
x  e.  RR*  |->  ( x (,] +oo ) ) 
C_  ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )
14 eqid 2443 . . . . . . . 8  |-  ( A (,] +oo )  =  ( A (,] +oo )
15 oveq1 6101 . . . . . . . . . 10  |-  ( x  =  A  ->  (
x (,] +oo )  =  ( A (,] +oo ) )
1615eqeq2d 2454 . . . . . . . . 9  |-  ( x  =  A  ->  (
( A (,] +oo )  =  ( x (,] +oo )  <->  ( A (,] +oo )  =  ( A (,] +oo )
) )
1716rspcev 3076 . . . . . . . 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 2443 . . . . . . . 8  |-  ( x  e.  RR*  |->  ( x (,] +oo ) )  =  ( x  e. 
RR*  |->  ( x (,] +oo ) )
20 ovex 6119 . . . . . . . 8  |-  ( x (,] +oo )  e. 
_V
2119, 20elrnmpti 5093 . . . . . . 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 3357 . . . . 5  |-  ( A  e.  RR*  ->  ( A (,] +oo )  e.  ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) ) )
2412, 23sseldi 3357 . . . 4  |-  ( A  e.  RR*  ->  ( A (,] +oo )  e.  ( ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)
2511, 24sseldi 3357 . . 3  |-  ( A  e.  RR*  ->  ( A (,] +oo )  e.  (ordTop `  <_  ) )
2625adantr 465 . 2  |-  ( ( A  e.  RR*  /\ +oo  e.  RR* )  ->  ( A (,] +oo )  e.  (ordTop `  <_  ) )
27 df-ioc 11308 . . . . . 6  |-  (,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <  z  /\  z  <_  y ) } )
2827ixxf 11313 . . . . 5  |-  (,] :
( RR*  X.  RR* ) --> ~P RR*
2928fdmi 5567 . . . 4  |-  dom  (,]  =  ( RR*  X.  RR* )
3029ndmov 6250 . . 3  |-  ( -.  ( A  e.  RR*  /\ +oo  e.  RR* )  ->  ( A (,] +oo )  =  (/) )
31 0opn 18520 . . . 4  |-  ( (ordTop `  <_  )  e.  Top  -> 
(/)  e.  (ordTop `  <_  ) )
325, 31ax-mp 5 . . 3  |-  (/)  e.  (ordTop `  <_  )
3330, 32syl6eqel 2531 . 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 1369    e. wcel 1756   E.wrex 2719    u. cun 3329    C_ wss 3331   (/)c0 3640   ~Pcpw 3863    e. cmpt 4353    X. cxp 4841   ran crn 4844   ` cfv 5421  (class class class)co 6094   +oocpnf 9418   -oocmnf 9419   RR*cxr 9420    < clt 9421    <_ cle 9422   (,)cioo 11303   (,]cioc 11304   [,)cico 11305   topGenctg 14379  ordTopcordt 14440   Topctop 18501   TopBasesctb 18505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-oadd 6927  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-fi 7664  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-ioo 11307  df-ioc 11308  df-ico 11309  df-icc 11310  df-topgen 14385  df-ordt 14442  df-ps 15373  df-tsr 15374  df-top 18506  df-bases 18508  df-topon 18509
This theorem is referenced by:  xrlimcnp  22365  pnfneige0  26384  lmxrge0  26385
  Copyright terms: Public domain W3C validator