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Theorem lcvbr 28584
Description: The covers relation for a left vector space (or a left module). (cvbr 22862 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 2343 . . . . . 6  |-  ( t  =  T  ->  (
t  e.  S  <->  T  e.  S ) )
43anbi1d 685 . . . . 5  |-  ( t  =  T  ->  (
( t  e.  S  /\  u  e.  S
)  <->  ( T  e.  S  /\  u  e.  S ) ) )
5 psseq1 3263 . . . . . 6  |-  ( t  =  T  ->  (
t  C.  u  <->  T  C.  u ) )
6 psseq1 3263 . . . . . . . . 9  |-  ( t  =  T  ->  (
t  C.  s  <->  T  C.  s ) )
76anbi1d 685 . . . . . . . 8  |-  ( t  =  T  ->  (
( t  C.  s  /\  s  C.  u )  <-> 
( T  C.  s  /\  s  C.  u ) ) )
87rexbidv 2564 . . . . . . 7  |-  ( t  =  T  ->  ( E. s  e.  S  ( t  C.  s  /\  s  C.  u )  <->  E. s  e.  S  ( T  C.  s  /\  s  C.  u ) ) )
98notbid 285 . . . . . 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 691 . . . . 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 691 . . . 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 2343 . . . . . 6  |-  ( u  =  U  ->  (
u  e.  S  <->  U  e.  S ) )
1312anbi2d 684 . . . . 5  |-  ( u  =  U  ->  (
( T  e.  S  /\  u  e.  S
)  <->  ( T  e.  S  /\  U  e.  S ) ) )
14 psseq2 3264 . . . . . 6  |-  ( u  =  U  ->  ( T  C.  u  <->  T  C.  U ) )
15 psseq2 3264 . . . . . . . . 9  |-  ( u  =  U  ->  (
s  C.  u  <->  s  C.  U ) )
1615anbi2d 684 . . . . . . . 8  |-  ( u  =  U  ->  (
( T  C.  s  /\  s  C.  u )  <-> 
( T  C.  s  /\  s  C.  U ) ) )
1716rexbidv 2564 . . . . . . 7  |-  ( u  =  U  ->  ( E. s  e.  S  ( T  C.  s  /\  s  C.  u )  <->  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) ) )
1817notbid 285 . . . . . 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 691 . . . . 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 691 . . . 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 2283 . . . 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 4284 . . 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 642 . 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 28583 . . 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 4034 . 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 518 . . 3  |-  ( ph  ->  ( T  e.  S  /\  U  e.  S
) )
3029biantrurd 494 . 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 276 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    C. wpss 3153   class class class wbr 4023   {copab 4076   ` cfv 5255   LSubSpclss 15689    <oLL clcv 28581
This theorem is referenced by:  lcvbr2  28585  lcvbr3  28586  lcvpss  28587  lcvnbtwn  28588  lsatcv0  28594  mapdcv  31223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-lcv 28582
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