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Theorem lcvfbr 34888
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 3118 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
4 fveq2 5872 . . . . . . . . 9  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
5 lcvfbr.s . . . . . . . . 9  |-  S  =  ( LSubSp `  W )
64, 5syl6eqr 2516 . . . . . . . 8  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  S )
76eleq2d 2527 . . . . . . 7  |-  ( w  =  W  ->  (
t  e.  ( LSubSp `  w )  <->  t  e.  S ) )
86eleq2d 2527 . . . . . . 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 3061 . . . . . . . 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 4520 . . . 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 34887 . . . 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 5882 . . . . . . 7  |-  ( LSubSp `  W )  e.  _V
175, 16eqeltri 2541 . . . . . 6  |-  S  e. 
_V
1817, 17xpex 6603 . . . . 5  |-  ( S  X.  S )  e. 
_V
19 opabssxp 5083 . . . . 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 4601 . . . 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 5956 . . 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 2510 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 1395    e. wcel 1819   E.wrex 2808   _Vcvv 3109    C. wpss 3472   {copab 4514    X. cxp 5006   ` cfv 5594   LSubSpclss 17705    <oLL clcv 34886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-lcv 34887
This theorem is referenced by:  lcvbr  34889
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