MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustex2sym Structured version   Unicode version

Theorem ustex2sym 20482
Description: In an uniform structure, for any entourage  V, there exists a symmetrical entourage smaller than half  V. (Contributed by Thierry Arnoux, 16-Jan-2018.)
Assertion
Ref Expression
ustex2sym  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  ( w  o.  w
)  C_  V )
)
Distinct variable groups:    w, U    w, V    w, X

Proof of Theorem ustex2sym
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 simplll 757 . . . 4  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  ->  U  e.  (UnifOn `  X
) )
2 simplr 754 . . . 4  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  -> 
v  e.  U )
3 ustexsym 20481 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  v  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  w  C_  v ) )
41, 2, 3syl2anc 661 . . 3  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  ->  E. w  e.  U  ( `' w  =  w  /\  w  C_  v ) )
5 simprl 755 . . . . . 6  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  v ) )  ->  `' w  =  w
)
6 coss1 5158 . . . . . . . . 9  |-  ( w 
C_  v  ->  (
w  o.  w ) 
C_  ( v  o.  w ) )
7 coss2 5159 . . . . . . . . 9  |-  ( w 
C_  v  ->  (
v  o.  w ) 
C_  ( v  o.  v ) )
86, 7sstrd 3514 . . . . . . . 8  |-  ( w 
C_  v  ->  (
w  o.  w ) 
C_  ( v  o.  v ) )
98ad2antll 728 . . . . . . 7  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  v ) )  -> 
( w  o.  w
)  C_  ( v  o.  v ) )
10 simpllr 758 . . . . . . 7  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  v ) )  -> 
( v  o.  v
)  C_  V )
119, 10sstrd 3514 . . . . . 6  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  v ) )  -> 
( w  o.  w
)  C_  V )
125, 11jca 532 . . . . 5  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  v ) )  -> 
( `' w  =  w  /\  ( w  o.  w )  C_  V ) )
1312ex 434 . . . 4  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  ->  ( ( `' w  =  w  /\  w  C_  v )  ->  ( `' w  =  w  /\  ( w  o.  w
)  C_  V )
) )
1413reximdva 2938 . . 3  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  -> 
( E. w  e.  U  ( `' w  =  w  /\  w  C_  v )  ->  E. w  e.  U  ( `' w  =  w  /\  ( w  o.  w
)  C_  V )
) )
154, 14mpd 15 . 2  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  ->  E. w  e.  U  ( `' w  =  w  /\  ( w  o.  w
)  C_  V )
)
16 ustexhalf 20476 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. v  e.  U  ( v  o.  v )  C_  V
)
1715, 16r19.29a 3003 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  ( w  o.  w
)  C_  V )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815    C_ wss 3476   `'ccnv 4998    o. ccom 5003   ` cfv 5588  UnifOncust 20465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-iota 5551  df-fun 5590  df-fv 5596  df-ust 20466
This theorem is referenced by:  ustex3sym  20483
  Copyright terms: Public domain W3C validator