MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nrmsep3 Structured version   Unicode version

Theorem nrmsep3 17424
Description: In a normal space, given a closed set  B inside an open set  A, there is an open set  x such that  B  C_  x  C_  cls ( x )  C_  A. (Contributed by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
nrmsep3  |-  ( ( J  e.  Nrm  /\  ( A  e.  J  /\  B  e.  ( Clsd `  J )  /\  B  C_  A ) )  ->  E. x  e.  J  ( B  C_  x  /\  ( ( cls `  J
) `  x )  C_  A ) )
Distinct variable groups:    x, A    x, B    x, J

Proof of Theorem nrmsep3
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnrm 17404 . . . . . 6  |-  ( J  e.  Nrm  <->  ( J  e.  Top  /\  A. y  e.  J  A. z  e.  ( ( Clsd `  J
)  i^i  ~P y
) E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J ) `  x
)  C_  y )
) )
21simprbi 452 . . . . 5  |-  ( J  e.  Nrm  ->  A. y  e.  J  A. z  e.  ( ( Clsd `  J
)  i^i  ~P y
) E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J ) `  x
)  C_  y )
)
3 pweq 3804 . . . . . . . 8  |-  ( y  =  A  ->  ~P y  =  ~P A
)
43ineq2d 3544 . . . . . . 7  |-  ( y  =  A  ->  (
( Clsd `  J )  i^i  ~P y )  =  ( ( Clsd `  J
)  i^i  ~P A
) )
5 sseq2 3372 . . . . . . . . 9  |-  ( y  =  A  ->  (
( ( cls `  J
) `  x )  C_  y  <->  ( ( cls `  J ) `  x
)  C_  A )
)
65anbi2d 686 . . . . . . . 8  |-  ( y  =  A  ->  (
( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  y )  <->  ( z  C_  x  /\  ( ( cls `  J ) `
 x )  C_  A ) ) )
76rexbidv 2728 . . . . . . 7  |-  ( y  =  A  ->  ( E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  y )  <->  E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J ) `
 x )  C_  A ) ) )
84, 7raleqbidv 2918 . . . . . 6  |-  ( y  =  A  ->  ( A. z  e.  (
( Clsd `  J )  i^i  ~P y ) E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  y )  <->  A. z  e.  ( ( Clsd `  J
)  i^i  ~P A
) E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J ) `  x
)  C_  A )
) )
98rspccv 3051 . . . . 5  |-  ( A. y  e.  J  A. z  e.  ( ( Clsd `  J )  i^i 
~P y ) E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  y )  ->  ( A  e.  J  ->  A. z  e.  ( (
Clsd `  J )  i^i  ~P A ) E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  A ) ) )
102, 9syl 16 . . . 4  |-  ( J  e.  Nrm  ->  ( A  e.  J  ->  A. z  e.  ( (
Clsd `  J )  i^i  ~P A ) E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  A ) ) )
11 elin 3532 . . . . . 6  |-  ( B  e.  ( ( Clsd `  J )  i^i  ~P A )  <->  ( B  e.  ( Clsd `  J
)  /\  B  e.  ~P A ) )
12 elpwg 3808 . . . . . . 7  |-  ( B  e.  ( Clsd `  J
)  ->  ( B  e.  ~P A  <->  B  C_  A
) )
1312pm5.32i 620 . . . . . 6  |-  ( ( B  e.  ( Clsd `  J )  /\  B  e.  ~P A )  <->  ( B  e.  ( Clsd `  J
)  /\  B  C_  A
) )
1411, 13bitri 242 . . . . 5  |-  ( B  e.  ( ( Clsd `  J )  i^i  ~P A )  <->  ( B  e.  ( Clsd `  J
)  /\  B  C_  A
) )
15 sseq1 3371 . . . . . . . 8  |-  ( z  =  B  ->  (
z  C_  x  <->  B  C_  x
) )
1615anbi1d 687 . . . . . . 7  |-  ( z  =  B  ->  (
( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  A )  <->  ( B  C_  x  /\  ( ( cls `  J ) `
 x )  C_  A ) ) )
1716rexbidv 2728 . . . . . 6  |-  ( z  =  B  ->  ( E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  A )  <->  E. x  e.  J  ( B  C_  x  /\  ( ( cls `  J ) `
 x )  C_  A ) ) )
1817rspccv 3051 . . . . 5  |-  ( A. z  e.  ( ( Clsd `  J )  i^i 
~P A ) E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  A )  ->  ( B  e.  ( ( Clsd `  J )  i^i 
~P A )  ->  E. x  e.  J  ( B  C_  x  /\  ( ( cls `  J
) `  x )  C_  A ) ) )
1914, 18syl5bir 211 . . . 4  |-  ( A. z  e.  ( ( Clsd `  J )  i^i 
~P A ) E. x  e.  J  ( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  A )  ->  (
( B  e.  (
Clsd `  J )  /\  B  C_  A )  ->  E. x  e.  J  ( B  C_  x  /\  ( ( cls `  J
) `  x )  C_  A ) ) )
2010, 19syl6 32 . . 3  |-  ( J  e.  Nrm  ->  ( A  e.  J  ->  ( ( B  e.  (
Clsd `  J )  /\  B  C_  A )  ->  E. x  e.  J  ( B  C_  x  /\  ( ( cls `  J
) `  x )  C_  A ) ) ) )
2120exp4a 591 . 2  |-  ( J  e.  Nrm  ->  ( A  e.  J  ->  ( B  e.  ( Clsd `  J )  ->  ( B  C_  A  ->  E. x  e.  J  ( B  C_  x  /\  ( ( cls `  J ) `
 x )  C_  A ) ) ) ) )
22213imp2 1169 1  |-  ( ( J  e.  Nrm  /\  ( A  e.  J  /\  B  e.  ( Clsd `  J )  /\  B  C_  A ) )  ->  E. x  e.  J  ( B  C_  x  /\  ( ( cls `  J
) `  x )  C_  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   ` cfv 5457   Topctop 16963   Clsdccld 17085   clsccl 17087   Nrmcnrm 17379
This theorem is referenced by:  nrmsep2  17425  kqnrmlem1  17780  kqnrmlem2  17781  nrmr0reg  17786  nrmhmph  17831
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-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-nrm 17386
  Copyright terms: Public domain W3C validator