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Theorem nrmsep3 20448
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 20428 . . . . . 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 471 . . . . 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 3945 . . . . . . . 8  |-  ( y  =  A  ->  ~P y  =  ~P A
)
43ineq2d 3625 . . . . . . 7  |-  ( y  =  A  ->  (
( Clsd `  J )  i^i  ~P y )  =  ( ( Clsd `  J
)  i^i  ~P A
) )
5 sseq2 3440 . . . . . . . . 9  |-  ( y  =  A  ->  (
( ( cls `  J
) `  x )  C_  y  <->  ( ( cls `  J ) `  x
)  C_  A )
)
65anbi2d 718 . . . . . . . 8  |-  ( y  =  A  ->  (
( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  y )  <->  ( z  C_  x  /\  ( ( cls `  J ) `
 x )  C_  A ) ) )
76rexbidv 2892 . . . . . . 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 2987 . . . . . 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 3133 . . . . 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 17 . . . 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 3608 . . . . . 6  |-  ( B  e.  ( ( Clsd `  J )  i^i  ~P A )  <->  ( B  e.  ( Clsd `  J
)  /\  B  e.  ~P A ) )
12 elpwg 3950 . . . . . . 7  |-  ( B  e.  ( Clsd `  J
)  ->  ( B  e.  ~P A  <->  B  C_  A
) )
1312pm5.32i 649 . . . . . 6  |-  ( ( B  e.  ( Clsd `  J )  /\  B  e.  ~P A )  <->  ( B  e.  ( Clsd `  J
)  /\  B  C_  A
) )
1411, 13bitri 257 . . . . 5  |-  ( B  e.  ( ( Clsd `  J )  i^i  ~P A )  <->  ( B  e.  ( Clsd `  J
)  /\  B  C_  A
) )
15 sseq1 3439 . . . . . . . 8  |-  ( z  =  B  ->  (
z  C_  x  <->  B  C_  x
) )
1615anbi1d 719 . . . . . . 7  |-  ( z  =  B  ->  (
( z  C_  x  /\  ( ( cls `  J
) `  x )  C_  A )  <->  ( B  C_  x  /\  ( ( cls `  J ) `
 x )  C_  A ) ) )
1716rexbidv 2892 . . . . . 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 3133 . . . . 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 226 . . . 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 33 . . 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 617 . 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 1248 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 setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757    i^i cin 3389    C_ wss 3390   ~Pcpw 3942   ` cfv 5589   Topctop 19994   Clsdccld 20108   clsccl 20110   Nrmcnrm 20403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-iota 5553  df-fv 5597  df-nrm 20410
This theorem is referenced by:  nrmsep2  20449  kqnrmlem1  20835  kqnrmlem2  20836  nrmr0reg  20841  nrmhmph  20886
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