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

Theorem llynlly 17532
Description: A locally  A space is n-locally  A: the "n-locally" predicate is the weaker notion. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llynlly  |-  ( J  e. Locally  A  ->  J  e. 𝑛Locally  A )

Proof of Theorem llynlly
Dummy variables  u  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 17527 . 2  |-  ( J  e. Locally  A  ->  J  e. 
Top )
2 llyi 17529 . . . . 5  |-  ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  ->  E. u  e.  J  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) )
3 simpl1 960 . . . . . . . . . . 11  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  ->  J  e. Locally  A )
43, 1syl 16 . . . . . . . . . 10  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  ->  J  e.  Top )
5 simprl 733 . . . . . . . . . 10  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  ->  u  e.  J )
6 simprr2 1006 . . . . . . . . . 10  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  -> 
y  e.  u )
7 opnneip 17175 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  u  e.  J  /\  y  e.  u )  ->  u  e.  ( ( nei `  J ) `
 { y } ) )
84, 5, 6, 7syl3anc 1184 . . . . . . . . 9  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  ->  u  e.  ( ( nei `  J ) `  { y } ) )
9 simprr1 1005 . . . . . . . . . 10  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  ->  u  C_  x )
10 vex 2951 . . . . . . . . . . 11  |-  u  e. 
_V
1110elpw 3797 . . . . . . . . . 10  |-  ( u  e.  ~P x  <->  u  C_  x
)
129, 11sylibr 204 . . . . . . . . 9  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  ->  u  e.  ~P x
)
13 elin 3522 . . . . . . . . 9  |-  ( u  e.  ( ( ( nei `  J ) `
 { y } )  i^i  ~P x
)  <->  ( u  e.  ( ( nei `  J
) `  { y } )  /\  u  e.  ~P x ) )
148, 12, 13sylanbrc 646 . . . . . . . 8  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  ->  u  e.  ( (
( nei `  J
) `  { y } )  i^i  ~P x ) )
15 simprr3 1007 . . . . . . . 8  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  -> 
( Jt  u )  e.  A
)
1614, 15jca 519 . . . . . . 7  |-  ( ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  /\  ( u  e.  J  /\  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) ) )  -> 
( u  e.  ( ( ( nei `  J
) `  { y } )  i^i  ~P x )  /\  ( Jt  u )  e.  A
) )
1716ex 424 . . . . . 6  |-  ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  ->  ( ( u  e.  J  /\  ( u 
C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
) )  ->  (
u  e.  ( ( ( nei `  J
) `  { y } )  i^i  ~P x )  /\  ( Jt  u )  e.  A
) ) )
1817reximdv2 2807 . . . . 5  |-  ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  ->  ( E. u  e.  J  ( u  C_  x  /\  y  e.  u  /\  ( Jt  u )  e.  A
)  ->  E. u  e.  ( ( ( nei `  J ) `  {
y } )  i^i 
~P x ) ( Jt  u )  e.  A
) )
192, 18mpd 15 . . . 4  |-  ( ( J  e. Locally  A  /\  x  e.  J  /\  y  e.  x )  ->  E. u  e.  ( ( ( nei `  J
) `  { y } )  i^i  ~P x ) ( Jt  u )  e.  A )
20193expb 1154 . . 3  |-  ( ( J  e. Locally  A  /\  ( x  e.  J  /\  y  e.  x
) )  ->  E. u  e.  ( ( ( nei `  J ) `  {
y } )  i^i 
~P x ) ( Jt  u )  e.  A
)
2120ralrimivva 2790 . 2  |-  ( J  e. Locally  A  ->  A. x  e.  J  A. y  e.  x  E. u  e.  ( ( ( nei `  J ) `  {
y } )  i^i 
~P x ) ( Jt  u )  e.  A
)
22 isnlly 17524 . 2  |-  ( J  e. 𝑛Locally  A  <->  ( J  e. 
Top  /\  A. x  e.  J  A. y  e.  x  E. u  e.  ( ( ( nei `  J ) `  {
y } )  i^i 
~P x ) ( Jt  u )  e.  A
) )
231, 21, 22sylanbrc 646 1  |-  ( J  e. Locally  A  ->  J  e. 𝑛Locally  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725   A.wral 2697   E.wrex 2698    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   {csn 3806   ` cfv 5446  (class class class)co 6073   ↾t crest 13640   Topctop 16950   neicnei 17153  Locally clly 17519  𝑛Locally cnlly 17520
This theorem is referenced by:  llyssnlly  17533  symgtgp  18123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-top 16955  df-nei 17154  df-lly 17521  df-nlly 17522
  Copyright terms: Public domain W3C validator