Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lcvfbr Structured version   Unicode version

Theorem lcvfbr 32670
Description: The covers relation for a left vector space (or a left module). (Contributed by NM, 7-Jan-2015.)
Hypotheses
Ref Expression
lcvfbr.s  |-  S  =  ( LSubSp `  W )
lcvfbr.c  |-  C  =  (  <oLL  `  W )
lcvfbr.w  |-  ( ph  ->  W  e.  X )
Assertion
Ref Expression
lcvfbr  |-  ( ph  ->  C  =  { <. t ,  u >.  |  ( ( t  e.  S  /\  u  e.  S
)  /\  ( t  C.  u  /\  -.  E. s  e.  S  (
t  C.  s  /\  s  C.  u ) ) ) } )
Distinct variable groups:    t, s, u, S    W, s, t, u
Allowed substitution hints:    ph( u, t, s)    C( u, t, s)    X( u, t, s)

Proof of Theorem lcvfbr
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 lcvfbr.c . 2  |-  C  =  (  <oLL  `  W )
2 lcvfbr.w . . 3  |-  ( ph  ->  W  e.  X )
3 elex 2986 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
4 fveq2 5696 . . . . . . . . 9  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
5 lcvfbr.s . . . . . . . . 9  |-  S  =  ( LSubSp `  W )
64, 5syl6eqr 2493 . . . . . . . 8  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  S )
76eleq2d 2510 . . . . . . 7  |-  ( w  =  W  ->  (
t  e.  ( LSubSp `  w )  <->  t  e.  S ) )
86eleq2d 2510 . . . . . . 7  |-  ( w  =  W  ->  (
u  e.  ( LSubSp `  w )  <->  u  e.  S ) )
97, 8anbi12d 710 . . . . . 6  |-  ( w  =  W  ->  (
( t  e.  (
LSubSp `  w )  /\  u  e.  ( LSubSp `  w ) )  <->  ( t  e.  S  /\  u  e.  S ) ) )
106rexeqdv 2929 . . . . . . . 8  |-  ( w  =  W  ->  ( E. s  e.  ( LSubSp `
 w ) ( t  C.  s  /\  s  C.  u )  <->  E. s  e.  S  ( t  C.  s  /\  s  C.  u ) ) )
1110notbid 294 . . . . . . 7  |-  ( w  =  W  ->  ( -.  E. s  e.  (
LSubSp `  w ) ( t  C.  s  /\  s  C.  u )  <->  -.  E. s  e.  S  ( t  C.  s  /\  s  C.  u ) ) )
1211anbi2d 703 . . . . . 6  |-  ( w  =  W  ->  (
( t  C.  u  /\  -.  E. s  e.  ( LSubSp `  w )
( t  C.  s  /\  s  C.  u ) )  <->  ( t  C.  u  /\  -.  E. s  e.  S  ( t  C.  s  /\  s  C.  u ) ) ) )
139, 12anbi12d 710 . . . . 5  |-  ( w  =  W  ->  (
( ( t  e.  ( LSubSp `  w )  /\  u  e.  ( LSubSp `
 w ) )  /\  ( t  C.  u  /\  -.  E. s  e.  ( LSubSp `  w )
( t  C.  s  /\  s  C.  u ) ) )  <->  ( (
t  e.  S  /\  u  e.  S )  /\  ( t  C.  u  /\  -.  E. s  e.  S  ( t  C.  s  /\  s  C.  u
) ) ) ) )
1413opabbidv 4360 . . . 4  |-  ( w  =  W  ->  { <. t ,  u >.  |  ( ( t  e.  (
LSubSp `  w )  /\  u  e.  ( LSubSp `  w ) )  /\  ( t  C.  u  /\  -.  E. s  e.  ( LSubSp `  w )
( t  C.  s  /\  s  C.  u ) ) ) }  =  { <. t ,  u >.  |  ( ( t  e.  S  /\  u  e.  S )  /\  (
t  C.  u  /\  -.  E. s  e.  S  ( t  C.  s  /\  s  C.  u ) ) ) } )
15 df-lcv 32669 . . . 4  |-  <oLL  =  (
w  e.  _V  |->  {
<. t ,  u >.  |  ( ( t  e.  ( LSubSp `  w )  /\  u  e.  ( LSubSp `
 w ) )  /\  ( t  C.  u  /\  -.  E. s  e.  ( LSubSp `  w )
( t  C.  s  /\  s  C.  u ) ) ) } )
16 fvex 5706 . . . . . . 7  |-  ( LSubSp `  W )  e.  _V
175, 16eqeltri 2513 . . . . . 6  |-  S  e. 
_V
1817, 17xpex 6513 . . . . 5  |-  ( S  X.  S )  e. 
_V
19 opabssxp 4916 . . . . 5  |-  { <. t ,  u >.  |  ( ( t  e.  S  /\  u  e.  S
)  /\  ( t  C.  u  /\  -.  E. s  e.  S  (
t  C.  s  /\  s  C.  u ) ) ) }  C_  ( S  X.  S )
2018, 19ssexi 4442 . . . 4  |-  { <. t ,  u >.  |  ( ( t  e.  S  /\  u  e.  S
)  /\  ( t  C.  u  /\  -.  E. s  e.  S  (
t  C.  s  /\  s  C.  u ) ) ) }  e.  _V
2114, 15, 20fvmpt 5779 . . 3  |-  ( W  e.  _V  ->  (  <oLL  `  W )  =  { <. t ,  u >.  |  ( ( t  e.  S  /\  u  e.  S )  /\  (
t  C.  u  /\  -.  E. s  e.  S  ( t  C.  s  /\  s  C.  u ) ) ) } )
222, 3, 213syl 20 . 2  |-  ( ph  ->  (  <oLL  `  W )  =  { <. t ,  u >.  |  ( ( t  e.  S  /\  u  e.  S )  /\  (
t  C.  u  /\  -.  E. s  e.  S  ( t  C.  s  /\  s  C.  u ) ) ) } )
231, 22syl5eq 2487 1  |-  ( ph  ->  C  =  { <. t ,  u >.  |  ( ( t  e.  S  /\  u  e.  S
)  /\  ( t  C.  u  /\  -.  E. s  e.  S  (
t  C.  s  /\  s  C.  u ) ) ) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2721   _Vcvv 2977    C. wpss 3334   {copab 4354    X. cxp 4843   ` cfv 5423   LSubSpclss 17018    <oLL clcv 32668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-lcv 32669
This theorem is referenced by:  lcvbr  32671
  Copyright terms: Public domain W3C validator