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Theorem ustex3sym 19919
Description: In an uniform structure, for any entourage  V, there exists a symmetrical entourage smaller than a third of  V. (Contributed by Thierry Arnoux, 16-Jan-2018.)
Assertion
Ref Expression
ustex3sym  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  ( w  o.  (
w  o.  w ) )  C_  V )
)
Distinct variable groups:    w, U    w, V    w, X

Proof of Theorem ustex3sym
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 ustex2sym 19918 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  v  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  ( w  o.  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  o.  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  o.  w )  C_  v ) )  ->  `' w  =  w
)
6 simp-5l 767 . . . . . . . . 9  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  /\  ( `' w  =  w  /\  ( w  o.  w )  C_  v ) )  ->  U  e.  (UnifOn `  X
) )
7 simplr 754 . . . . . . . . 9  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  /\  ( `' w  =  w  /\  ( w  o.  w )  C_  v ) )  ->  w  e.  U )
8 ustssco 19916 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  w  e.  U )  ->  w  C_  ( w  o.  w
) )
96, 7, 8syl2anc 661 . . . . . . . 8  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  /\  ( `' w  =  w  /\  ( w  o.  w )  C_  v ) )  ->  w  C_  ( w  o.  w ) )
10 simprr 756 . . . . . . . 8  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  /\  ( `' w  =  w  /\  ( w  o.  w )  C_  v ) )  -> 
( w  o.  w
)  C_  v )
11 coss2 5099 . . . . . . . . . 10  |-  ( ( w  o.  w ) 
C_  v  ->  (
w  o.  ( w  o.  w ) ) 
C_  ( w  o.  v ) )
1211adantl 466 . . . . . . . . 9  |-  ( ( w  C_  ( w  o.  w )  /\  (
w  o.  w ) 
C_  v )  -> 
( w  o.  (
w  o.  w ) )  C_  ( w  o.  v ) )
13 sstr 3467 . . . . . . . . . 10  |-  ( ( w  C_  ( w  o.  w )  /\  (
w  o.  w ) 
C_  v )  ->  w  C_  v )
14 coss1 5098 . . . . . . . . . 10  |-  ( w 
C_  v  ->  (
w  o.  v ) 
C_  ( v  o.  v ) )
1513, 14syl 16 . . . . . . . . 9  |-  ( ( w  C_  ( w  o.  w )  /\  (
w  o.  w ) 
C_  v )  -> 
( w  o.  v
)  C_  ( v  o.  v ) )
1612, 15sstrd 3469 . . . . . . . 8  |-  ( ( w  C_  ( w  o.  w )  /\  (
w  o.  w ) 
C_  v )  -> 
( w  o.  (
w  o.  w ) )  C_  ( v  o.  v ) )
179, 10, 16syl2anc 661 . . . . . . 7  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  /\  ( `' w  =  w  /\  ( w  o.  w )  C_  v ) )  -> 
( w  o.  (
w  o.  w ) )  C_  ( v  o.  v ) )
18 simpllr 758 . . . . . . 7  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  /\  ( `' w  =  w  /\  ( w  o.  w )  C_  v ) )  -> 
( v  o.  v
)  C_  V )
1917, 18sstrd 3469 . . . . . 6  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  /\  ( `' w  =  w  /\  ( w  o.  w )  C_  v ) )  -> 
( w  o.  (
w  o.  w ) )  C_  V )
205, 19jca 532 . . . . 5  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  /\  ( `' w  =  w  /\  ( w  o.  w )  C_  v ) )  -> 
( `' w  =  w  /\  ( w  o.  ( w  o.  w ) )  C_  V ) )
2120ex 434 . . . 4  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  ->  ( ( `' w  =  w  /\  (
w  o.  w ) 
C_  v )  -> 
( `' w  =  w  /\  ( w  o.  ( w  o.  w ) )  C_  V ) ) )
2221reximdva 2928 . . 3  |-  ( ( ( ( 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 )  ->  E. w  e.  U  ( `' w  =  w  /\  ( w  o.  (
w  o.  w ) )  C_  V )
) )
234, 22mpd 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  o.  w ) )  C_  V )
)
24 ustexhalf 19912 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. v  e.  U  ( v  o.  v )  C_  V
)
2523, 24r19.29a 2962 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  ( w  o.  (
w  o.  w ) )  C_  V )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2797    C_ wss 3431   `'ccnv 4942    o. ccom 4947   ` cfv 5521  UnifOncust 19901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-iota 5484  df-fun 5523  df-fv 5529  df-ust 19902
This theorem is referenced by:  utopreg  19954
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