Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  inttop4 Unicode version

Theorem inttop4 25517
Description: The intersection of two topologies is a topology. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
inttop4  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  i^i  K
)  e.  Top )

Proof of Theorem inttop4
StepHypRef Expression
1 prssg 3770 . . . 4  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( ( J  e. 
Top  /\  K  e.  Top )  <->  { J ,  K }  C_  Top ) )
21biimpd 198 . . 3  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( ( J  e. 
Top  /\  K  e.  Top )  ->  { J ,  K }  C_  Top ) )
3 intprg 3896 . . . . . . 7  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  |^| { J ,  K }  =  ( J  i^i  K ) )
43eqcomd 2288 . . . . . 6  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  i^i  K
)  =  |^| { J ,  K } )
54adantl 452 . . . . 5  |-  ( ( { J ,  K }  C_  Top  /\  ( J  e.  Top  /\  K  e.  Top ) )  -> 
( J  i^i  K
)  =  |^| { J ,  K } )
6 prnzg 3746 . . . . . . . 8  |-  ( J  e.  Top  ->  { J ,  K }  =/=  (/) )
7 inttop3 25516 . . . . . . . . 9  |-  ( ( { J ,  K }  =/=  (/)  /\  { J ,  K }  C_  Top )  ->  |^| { J ,  K }  e.  Top )
87ex 423 . . . . . . . 8  |-  ( { J ,  K }  =/=  (/)  ->  ( { J ,  K }  C_ 
Top  ->  |^| { J ,  K }  e.  Top ) )
96, 8syl 15 . . . . . . 7  |-  ( J  e.  Top  ->  ( { J ,  K }  C_ 
Top  ->  |^| { J ,  K }  e.  Top ) )
109adantr 451 . . . . . 6  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( { J ,  K }  C_  Top  ->  |^|
{ J ,  K }  e.  Top )
)
1110impcom 419 . . . . 5  |-  ( ( { J ,  K }  C_  Top  /\  ( J  e.  Top  /\  K  e.  Top ) )  ->  |^| { J ,  K }  e.  Top )
125, 11eqeltrd 2357 . . . 4  |-  ( ( { J ,  K }  C_  Top  /\  ( J  e.  Top  /\  K  e.  Top ) )  -> 
( J  i^i  K
)  e.  Top )
1312ex 423 . . 3  |-  ( { J ,  K }  C_ 
Top  ->  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  i^i  K )  e. 
Top ) )
142, 13syli 33 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( ( J  e. 
Top  /\  K  e.  Top )  ->  ( J  i^i  K )  e. 
Top ) )
1514pm2.43i 43 1  |-  ( ( J  e.  Top  /\  K  e.  Top )  ->  ( J  i^i  K
)  e.  Top )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    i^i cin 3151    C_ wss 3152   (/)c0 3455   {cpr 3641   |^|cint 3862   Topctop 16631
This theorem is referenced by:  intcont  25543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-pw 3627  df-sn 3646  df-pr 3647  df-uni 3828  df-int 3863  df-iin 3908  df-top 16636
  Copyright terms: Public domain W3C validator