MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  kqnrmlem1 Structured version   Unicode version

Theorem kqnrmlem1 20744
Description: A Kolmogorov quotient of a normal space is normal. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
Assertion
Ref Expression
kqnrmlem1  |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  ->  (KQ `  J )  e.  Nrm )
Distinct variable groups:    x, y, J    x, X, y
Allowed substitution hints:    F( x, y)

Proof of Theorem kqnrmlem1
Dummy variables  m  w  z  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . 5  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
21kqtopon 20728 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
32adantr 466 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  ->  (KQ `  J )  e.  (TopOn `  ran  F ) )
4 topontop 19927 . . 3  |-  ( (KQ
`  J )  e.  (TopOn `  ran  F )  ->  (KQ `  J
)  e.  Top )
53, 4syl 17 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  ->  (KQ `  J )  e.  Top )
6 simplr 760 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  J  e.  Nrm )
71kqid 20729 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
87ad2antrr 730 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
9 simprl 762 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  z  e.  (KQ `  J ) )
10 cnima 20267 . . . . . 6  |-  ( ( F  e.  ( J  Cn  (KQ `  J
) )  /\  z  e.  (KQ `  J ) )  ->  ( `' F " z )  e.  J )
118, 9, 10syl2anc 665 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  ( `' F " z )  e.  J )
12 inss1 3682 . . . . . . 7  |-  ( (
Clsd `  (KQ `  J
) )  i^i  ~P z )  C_  ( Clsd `  (KQ `  J
) )
13 simprr 764 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) )
1412, 13sseldi 3462 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  w  e.  ( Clsd `  (KQ `  J
) ) )
15 cnclima 20270 . . . . . 6  |-  ( ( F  e.  ( J  Cn  (KQ `  J
) )  /\  w  e.  ( Clsd `  (KQ `  J ) ) )  ->  ( `' F " w )  e.  (
Clsd `  J )
)
168, 14, 15syl2anc 665 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  ( `' F " w )  e.  ( Clsd `  J
) )
17 inss2 3683 . . . . . . 7  |-  ( (
Clsd `  (KQ `  J
) )  i^i  ~P z )  C_  ~P z
1817, 13sseldi 3462 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  w  e.  ~P z )
19 elpwi 3988 . . . . . 6  |-  ( w  e.  ~P z  ->  w  C_  z )
20 imass2 5219 . . . . . 6  |-  ( w 
C_  z  ->  ( `' F " w ) 
C_  ( `' F " z ) )
2118, 19, 203syl 18 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  ( `' F " w )  C_  ( `' F " z ) )
22 nrmsep3 20357 . . . . 5  |-  ( ( J  e.  Nrm  /\  ( ( `' F " z )  e.  J  /\  ( `' F "
w )  e.  (
Clsd `  J )  /\  ( `' F "
w )  C_  ( `' F " z ) ) )  ->  E. u  e.  J  ( ( `' F " w ) 
C_  u  /\  (
( cls `  J
) `  u )  C_  ( `' F "
z ) ) )
236, 11, 16, 21, 22syl13anc 1266 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  E. u  e.  J  ( ( `' F " w ) 
C_  u  /\  (
( cls `  J
) `  u )  C_  ( `' F "
z ) ) )
24 simplll 766 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  J  e.  (TopOn `  X ) )
25 simprl 762 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  u  e.  J )
261kqopn 20735 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  u  e.  J )  ->  ( F " u )  e.  (KQ `  J ) )
2724, 25, 26syl2anc 665 . . . . 5  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( F " u )  e.  (KQ
`  J ) )
28 simprrl 772 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( `' F " w )  C_  u )
291kqffn 20726 . . . . . . . 8  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
30 fnfun 5687 . . . . . . . 8  |-  ( F  Fn  X  ->  Fun  F )
3124, 29, 303syl 18 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  Fun  F )
3214adantr 466 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  w  e.  ( Clsd `  (KQ `  J
) ) )
33 eqid 2422 . . . . . . . . . 10  |-  U. (KQ `  J )  =  U. (KQ `  J )
3433cldss 20030 . . . . . . . . 9  |-  ( w  e.  ( Clsd `  (KQ `  J ) )  ->  w  C_  U. (KQ `  J ) )
3532, 34syl 17 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  w  C_  U. (KQ `  J ) )
36 toponuni 19928 . . . . . . . . 9  |-  ( (KQ
`  J )  e.  (TopOn `  ran  F )  ->  ran  F  =  U. (KQ `  J ) )
3724, 2, 363syl 18 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ran  F  = 
U. (KQ `  J
) )
3835, 37sseqtr4d 3501 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  w  C_  ran  F )
39 funimass1 5670 . . . . . . 7  |-  ( ( Fun  F  /\  w  C_ 
ran  F )  -> 
( ( `' F " w )  C_  u  ->  w  C_  ( F " u ) ) )
4031, 38, 39syl2anc 665 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( ( `' F " w ) 
C_  u  ->  w  C_  ( F " u
) ) )
4128, 40mpd 15 . . . . 5  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  w  C_  ( F " u ) )
42 topontop 19927 . . . . . . . . . 10  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
4324, 42syl 17 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  J  e.  Top )
44 elssuni 4245 . . . . . . . . . 10  |-  ( u  e.  J  ->  u  C_ 
U. J )
4544ad2antrl 732 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  u  C_  U. J
)
46 eqid 2422 . . . . . . . . . 10  |-  U. J  =  U. J
4746clscld 20048 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  u  C_  U. J )  ->  ( ( cls `  J ) `  u
)  e.  ( Clsd `  J ) )
4843, 45, 47syl2anc 665 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( ( cls `  J ) `  u )  e.  (
Clsd `  J )
)
491kqcld 20736 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  (
( cls `  J
) `  u )  e.  ( Clsd `  J
) )  ->  ( F " ( ( cls `  J ) `  u
) )  e.  (
Clsd `  (KQ `  J
) ) )
5024, 48, 49syl2anc 665 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( F " ( ( cls `  J
) `  u )
)  e.  ( Clsd `  (KQ `  J ) ) )
5146sscls 20057 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  u  C_  U. J )  ->  u  C_  (
( cls `  J
) `  u )
)
5243, 45, 51syl2anc 665 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  u  C_  (
( cls `  J
) `  u )
)
53 imass2 5219 . . . . . . . 8  |-  ( u 
C_  ( ( cls `  J ) `  u
)  ->  ( F " u )  C_  ( F " ( ( cls `  J ) `  u
) ) )
5452, 53syl 17 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( F " u )  C_  ( F " ( ( cls `  J ) `  u
) ) )
5533clsss2 20074 . . . . . . 7  |-  ( ( ( F " (
( cls `  J
) `  u )
)  e.  ( Clsd `  (KQ `  J ) )  /\  ( F
" u )  C_  ( F " ( ( cls `  J ) `
 u ) ) )  ->  ( ( cls `  (KQ `  J
) ) `  ( F " u ) ) 
C_  ( F "
( ( cls `  J
) `  u )
) )
5650, 54, 55syl2anc 665 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( ( cls `  (KQ `  J
) ) `  ( F " u ) ) 
C_  ( F "
( ( cls `  J
) `  u )
) )
57 simprrr 773 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( ( cls `  J ) `  u )  C_  ( `' F " z ) )
5846clsss3 20060 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  u  C_  U. J )  ->  ( ( cls `  J ) `  u
)  C_  U. J )
5943, 45, 58syl2anc 665 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( ( cls `  J ) `  u )  C_  U. J
)
60 fndm 5689 . . . . . . . . . . 11  |-  ( F  Fn  X  ->  dom  F  =  X )
6124, 29, 603syl 18 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  dom  F  =  X )
62 toponuni 19928 . . . . . . . . . . 11  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
6324, 62syl 17 . . . . . . . . . 10  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  X  =  U. J )
6461, 63eqtrd 2463 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  dom  F  = 
U. J )
6559, 64sseqtr4d 3501 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( ( cls `  J ) `  u )  C_  dom  F )
66 funimass3 6009 . . . . . . . 8  |-  ( ( Fun  F  /\  (
( cls `  J
) `  u )  C_ 
dom  F )  -> 
( ( F "
( ( cls `  J
) `  u )
)  C_  z  <->  ( ( cls `  J ) `  u )  C_  ( `' F " z ) ) )
6731, 65, 66syl2anc 665 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( ( F " ( ( cls `  J ) `  u
) )  C_  z  <->  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) )
6857, 67mpbird 235 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( F " ( ( cls `  J
) `  u )
)  C_  z )
6956, 68sstrd 3474 . . . . 5  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  ( ( cls `  (KQ `  J
) ) `  ( F " u ) ) 
C_  z )
70 sseq2 3486 . . . . . . 7  |-  ( m  =  ( F "
u )  ->  (
w  C_  m  <->  w  C_  ( F " u ) ) )
71 fveq2 5877 . . . . . . . 8  |-  ( m  =  ( F "
u )  ->  (
( cls `  (KQ `  J ) ) `  m )  =  ( ( cls `  (KQ `  J ) ) `  ( F " u ) ) )
7271sseq1d 3491 . . . . . . 7  |-  ( m  =  ( F "
u )  ->  (
( ( cls `  (KQ `  J ) ) `  m )  C_  z  <->  ( ( cls `  (KQ `  J ) ) `  ( F " u ) )  C_  z )
)
7370, 72anbi12d 715 . . . . . 6  |-  ( m  =  ( F "
u )  ->  (
( w  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m )  C_  z
)  <->  ( w  C_  ( F " u )  /\  ( ( cls `  (KQ `  J ) ) `  ( F
" u ) ) 
C_  z ) ) )
7473rspcev 3182 . . . . 5  |-  ( ( ( F " u
)  e.  (KQ `  J )  /\  (
w  C_  ( F " u )  /\  (
( cls `  (KQ `  J ) ) `  ( F " u ) )  C_  z )
)  ->  E. m  e.  (KQ `  J ) ( w  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m )  C_  z
) )
7527, 41, 69, 74syl12anc 1262 . . . 4  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  ( z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  /\  ( u  e.  J  /\  (
( `' F "
w )  C_  u  /\  ( ( cls `  J
) `  u )  C_  ( `' F "
z ) ) ) )  ->  E. m  e.  (KQ `  J ) ( w  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m )  C_  z
) )
7623, 75rexlimddv 2921 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  /\  (
z  e.  (KQ `  J )  /\  w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) ) )  ->  E. m  e.  (KQ `  J ) ( w  C_  m  /\  ( ( cls `  (KQ `  J ) ) `  m )  C_  z
) )
7776ralrimivva 2846 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  ->  A. z  e.  (KQ `  J ) A. w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) E. m  e.  (KQ `  J ) ( w 
C_  m  /\  (
( cls `  (KQ `  J ) ) `  m )  C_  z
) )
78 isnrm 20337 . 2  |-  ( (KQ
`  J )  e. 
Nrm 
<->  ( (KQ `  J
)  e.  Top  /\  A. z  e.  (KQ `  J ) A. w  e.  ( ( Clsd `  (KQ `  J ) )  i^i 
~P z ) E. m  e.  (KQ `  J ) ( w 
C_  m  /\  (
( cls `  (KQ `  J ) ) `  m )  C_  z
) ) )
795, 77, 78sylanbrc 668 1  |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Nrm )  ->  (KQ `  J )  e.  Nrm )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1868   A.wral 2775   E.wrex 2776   {crab 2779    i^i cin 3435    C_ wss 3436   ~Pcpw 3979   U.cuni 4216    |-> cmpt 4479   `'ccnv 4848   dom cdm 4849   ran crn 4850   "cima 4852   Fun wfun 5591    Fn wfn 5592   ` cfv 5597  (class class class)co 6301   Topctop 19903  TopOnctopon 19904   Clsdccld 20017   clsccl 20019    Cn ccn 20226   Nrmcnrm 20312  KQckq 20694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-map 7478  df-qtop 15393  df-top 19907  df-topon 19909  df-cld 20020  df-cls 20022  df-cn 20229  df-nrm 20319  df-kq 20695
This theorem is referenced by:  kqnrm  20753
  Copyright terms: Public domain W3C validator