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Theorem lcvfbr 29832
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 2809 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
4 fveq2 5541 . . . . . . . . 9  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
5 lcvfbr.s . . . . . . . . 9  |-  S  =  ( LSubSp `  W )
64, 5syl6eqr 2346 . . . . . . . 8  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  S )
76eleq2d 2363 . . . . . . 7  |-  ( w  =  W  ->  (
t  e.  ( LSubSp `  w )  <->  t  e.  S ) )
86eleq2d 2363 . . . . . . 7  |-  ( w  =  W  ->  (
u  e.  ( LSubSp `  w )  <->  u  e.  S ) )
97, 8anbi12d 691 . . . . . 6  |-  ( w  =  W  ->  (
( t  e.  (
LSubSp `  w )  /\  u  e.  ( LSubSp `  w ) )  <->  ( t  e.  S  /\  u  e.  S ) ) )
106rexeqdv 2756 . . . . . . . 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 285 . . . . . . 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 684 . . . . . 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 691 . . . . 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 4098 . . . 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 29831 . . . 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 5555 . . . . . . 7  |-  ( LSubSp `  W )  e.  _V
175, 16eqeltri 2366 . . . . . 6  |-  S  e. 
_V
1817, 17xpex 4817 . . . . 5  |-  ( S  X.  S )  e. 
_V
19 opabssxp 4778 . . . . 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 4175 . . . 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 5618 . . 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 18 . 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 2340 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    C. wpss 3166   {copab 4092    X. cxp 4703   ` cfv 5271   LSubSpclss 15705    <oLL clcv 29830
This theorem is referenced by:  lcvbr  29833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-lcv 29831
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