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Theorem ustex3sym 18200
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 735 . . . 4  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  ->  U  e.  (UnifOn `  X
) )
2 simplr 732 . . . 4  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  -> 
v  e.  U )
3 ustex2sym 18199 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  v  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  ( w  o.  w
)  C_  v )
)
41, 2, 3syl2anc 643 . . 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 733 . . . . . 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 745 . . . . . . . . 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 732 . . . . . . . . 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 18197 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  w  e.  U )  ->  w  C_  ( w  o.  w
) )
96, 7, 8syl2anc 643 . . . . . . . 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 734 . . . . . . . 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 4988 . . . . . . . . . 10  |-  ( ( w  o.  w ) 
C_  v  ->  (
w  o.  ( w  o.  w ) ) 
C_  ( w  o.  v ) )
1211adantl 453 . . . . . . . . 9  |-  ( ( w  C_  ( w  o.  w )  /\  (
w  o.  w ) 
C_  v )  -> 
( w  o.  (
w  o.  w ) )  C_  ( w  o.  v ) )
13 sstr 3316 . . . . . . . . . 10  |-  ( ( w  C_  ( w  o.  w )  /\  (
w  o.  w ) 
C_  v )  ->  w  C_  v )
14 coss1 4987 . . . . . . . . . 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 3318 . . . . . . . 8  |-  ( ( w  C_  ( w  o.  w )  /\  (
w  o.  w ) 
C_  v )  -> 
( w  o.  (
w  o.  w ) )  C_  ( v  o.  v ) )
179, 10, 16syl2anc 643 . . . . . . 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 736 . . . . . . 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 3318 . . . . . 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 519 . . . . 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 424 . . . 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 2778 . . 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 18193 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. v  e.  U  ( v  o.  v )  C_  V
)
2523, 24r19.29a 2810 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667    C_ wss 3280   `'ccnv 4836    o. ccom 4841   ` cfv 5413  UnifOncust 18182
This theorem is referenced by:  utopreg  18235
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-iota 5377  df-fun 5415  df-fv 5421  df-ust 18183
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