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Theorem lcvbr 33029
Description: The covers relation for a left vector space (or a left module). (cvbr 25865 analog.) (Contributed by NM, 9-Jan-2015.)
Hypotheses
Ref Expression
lcvfbr.s  |-  S  =  ( LSubSp `  W )
lcvfbr.c  |-  C  =  (  <oLL  `  W )
lcvfbr.w  |-  ( ph  ->  W  e.  X )
lcvfbr.t  |-  ( ph  ->  T  e.  S )
lcvfbr.u  |-  ( ph  ->  U  e.  S )
Assertion
Ref Expression
lcvbr  |-  ( ph  ->  ( T C U  <-> 
( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U
) ) ) )
Distinct variable groups:    S, s    W, s    T, s    U, s
Allowed substitution hints:    ph( s)    C( s)    X( s)

Proof of Theorem lcvbr
Dummy variables  t  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lcvfbr.t . . 3  |-  ( ph  ->  T  e.  S )
2 lcvfbr.u . . 3  |-  ( ph  ->  U  e.  S )
3 eleq1 2526 . . . . . 6  |-  ( t  =  T  ->  (
t  e.  S  <->  T  e.  S ) )
43anbi1d 704 . . . . 5  |-  ( t  =  T  ->  (
( t  e.  S  /\  u  e.  S
)  <->  ( T  e.  S  /\  u  e.  S ) ) )
5 psseq1 3554 . . . . . 6  |-  ( t  =  T  ->  (
t  C.  u  <->  T  C.  u
) )
6 psseq1 3554 . . . . . . . . 9  |-  ( t  =  T  ->  (
t  C.  s  <->  T  C.  s
) )
76anbi1d 704 . . . . . . . 8  |-  ( t  =  T  ->  (
( t  C.  s  /\  s  C.  u )  <-> 
( T  C.  s  /\  s  C.  u ) ) )
87rexbidv 2868 . . . . . . 7  |-  ( t  =  T  ->  ( E. s  e.  S  ( t  C.  s  /\  s  C.  u )  <->  E. s  e.  S  ( T  C.  s  /\  s  C.  u ) ) )
98notbid 294 . . . . . 6  |-  ( t  =  T  ->  ( -.  E. s  e.  S  ( t  C.  s  /\  s  C.  u )  <->  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  u ) ) )
105, 9anbi12d 710 . . . . 5  |-  ( t  =  T  ->  (
( t  C.  u  /\  -.  E. s  e.  S  ( t  C.  s  /\  s  C.  u
) )  <->  ( T  C.  u  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  u ) ) ) )
114, 10anbi12d 710 . . . 4  |-  ( t  =  T  ->  (
( ( t  e.  S  /\  u  e.  S )  /\  (
t  C.  u  /\  -.  E. s  e.  S  ( 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
) ) ) ) )
12 eleq1 2526 . . . . . 6  |-  ( u  =  U  ->  (
u  e.  S  <->  U  e.  S ) )
1312anbi2d 703 . . . . 5  |-  ( u  =  U  ->  (
( T  e.  S  /\  u  e.  S
)  <->  ( T  e.  S  /\  U  e.  S ) ) )
14 psseq2 3555 . . . . . 6  |-  ( u  =  U  ->  ( T  C.  u  <->  T  C.  U
) )
15 psseq2 3555 . . . . . . . . 9  |-  ( u  =  U  ->  (
s  C.  u  <->  s  C.  U
) )
1615anbi2d 703 . . . . . . . 8  |-  ( u  =  U  ->  (
( T  C.  s  /\  s  C.  u )  <-> 
( T  C.  s  /\  s  C.  U ) ) )
1716rexbidv 2868 . . . . . . 7  |-  ( u  =  U  ->  ( E. s  e.  S  ( T  C.  s  /\  s  C.  u )  <->  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) ) )
1817notbid 294 . . . . . 6  |-  ( u  =  U  ->  ( -.  E. s  e.  S  ( T  C.  s  /\  s  C.  u )  <->  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) ) )
1914, 18anbi12d 710 . . . . 5  |-  ( u  =  U  ->  (
( T  C.  u  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  u
) )  <->  ( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) ) ) )
2013, 19anbi12d 710 . . . 4  |-  ( u  =  U  ->  (
( ( T  e.  S  /\  u  e.  S )  /\  ( T  C.  u  /\  -.  E. s  e.  S  ( 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 ) ) ) ) )
21 eqid 2454 . . . 4  |-  { <. t ,  u >.  |  ( ( t  e.  S  /\  u  e.  S
)  /\  ( t  C.  u  /\  -.  E. s  e.  S  (
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 ) ) ) }
2211, 20, 21brabg 4719 . . 3  |-  ( ( T  e.  S  /\  U  e.  S )  ->  ( T { <. t ,  u >.  |  ( ( t  e.  S  /\  u  e.  S
)  /\  ( t  C.  u  /\  -.  E. s  e.  S  (
t  C.  s  /\  s  C.  u ) ) ) } U  <->  ( ( T  e.  S  /\  U  e.  S )  /\  ( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U
) ) ) ) )
231, 2, 22syl2anc 661 . 2  |-  ( ph  ->  ( T { <. t ,  u >.  |  ( ( t  e.  S  /\  u  e.  S
)  /\  ( t  C.  u  /\  -.  E. s  e.  S  (
t  C.  s  /\  s  C.  u ) ) ) } U  <->  ( ( T  e.  S  /\  U  e.  S )  /\  ( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U
) ) ) ) )
24 lcvfbr.s . . . 4  |-  S  =  ( LSubSp `  W )
25 lcvfbr.c . . . 4  |-  C  =  (  <oLL  `  W )
26 lcvfbr.w . . . 4  |-  ( ph  ->  W  e.  X )
2724, 25, 26lcvfbr 33028 . . 3  |-  ( ph  ->  C  =  { <. t ,  u >.  |  ( ( t  e.  S  /\  u  e.  S
)  /\  ( t  C.  u  /\  -.  E. s  e.  S  (
t  C.  s  /\  s  C.  u ) ) ) } )
2827breqd 4414 . 2  |-  ( ph  ->  ( T C U  <-> 
T { <. t ,  u >.  |  (
( t  e.  S  /\  u  e.  S
)  /\  ( t  C.  u  /\  -.  E. s  e.  S  (
t  C.  s  /\  s  C.  u ) ) ) } U ) )
291, 2jca 532 . . 3  |-  ( ph  ->  ( T  e.  S  /\  U  e.  S
) )
3029biantrurd 508 . 2  |-  ( ph  ->  ( ( T  C.  U  /\  -.  E. s  e.  S  ( 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
) ) ) ) )
3123, 28, 303bitr4d 285 1  |-  ( ph  ->  ( T C U  <-> 
( 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    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2800    C. wpss 3440   class class class wbr 4403   {copab 4460   ` cfv 5529   LSubSpclss 17146    <oLL clcv 33026
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-lcv 33027
This theorem is referenced by:  lcvbr2  33030  lcvbr3  33031  lcvpss  33032  lcvnbtwn  33033  lsatcv0  33039  mapdcv  35668
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