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Theorem lcvbr 29757
Description: The covers relation for a left vector space (or a left module). (cvbr 23778 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 2496 . . . . . 6  |-  ( t  =  T  ->  (
t  e.  S  <->  T  e.  S ) )
43anbi1d 686 . . . . 5  |-  ( t  =  T  ->  (
( t  e.  S  /\  u  e.  S
)  <->  ( T  e.  S  /\  u  e.  S ) ) )
5 psseq1 3427 . . . . . 6  |-  ( t  =  T  ->  (
t  C.  u  <->  T  C.  u ) )
6 psseq1 3427 . . . . . . . . 9  |-  ( t  =  T  ->  (
t  C.  s  <->  T  C.  s ) )
76anbi1d 686 . . . . . . . 8  |-  ( t  =  T  ->  (
( t  C.  s  /\  s  C.  u )  <-> 
( T  C.  s  /\  s  C.  u ) ) )
87rexbidv 2719 . . . . . . 7  |-  ( t  =  T  ->  ( E. s  e.  S  ( t  C.  s  /\  s  C.  u )  <->  E. s  e.  S  ( T  C.  s  /\  s  C.  u ) ) )
98notbid 286 . . . . . 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 692 . . . . 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 692 . . . 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 2496 . . . . . 6  |-  ( u  =  U  ->  (
u  e.  S  <->  U  e.  S ) )
1312anbi2d 685 . . . . 5  |-  ( u  =  U  ->  (
( T  e.  S  /\  u  e.  S
)  <->  ( T  e.  S  /\  U  e.  S ) ) )
14 psseq2 3428 . . . . . 6  |-  ( u  =  U  ->  ( T  C.  u  <->  T  C.  U ) )
15 psseq2 3428 . . . . . . . . 9  |-  ( u  =  U  ->  (
s  C.  u  <->  s  C.  U ) )
1615anbi2d 685 . . . . . . . 8  |-  ( u  =  U  ->  (
( T  C.  s  /\  s  C.  u )  <-> 
( T  C.  s  /\  s  C.  U ) ) )
1716rexbidv 2719 . . . . . . 7  |-  ( u  =  U  ->  ( E. s  e.  S  ( T  C.  s  /\  s  C.  u )  <->  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) ) )
1817notbid 286 . . . . . 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 692 . . . . 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 692 . . . 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 2436 . . . 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 4467 . . 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 643 . 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 29756 . . 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 4216 . 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 519 . . 3  |-  ( ph  ->  ( T  e.  S  /\  U  e.  S
) )
3029biantrurd 495 . 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 277 1  |-  ( ph  ->  ( T C U  <-> 
( 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    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2699    C. wpss 3314   class class class wbr 4205   {copab 4258   ` cfv 5447   LSubSpclss 16001    <oLL clcv 29754
This theorem is referenced by:  lcvbr2  29758  lcvbr3  29759  lcvpss  29760  lcvnbtwn  29761  lsatcv0  29767  mapdcv  32396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-iota 5411  df-fun 5449  df-fv 5455  df-lcv 29755
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