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Theorem lcvbr 32020
Description: The covers relation for a left vector space (or a left module). (cvbr 27494 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 2474 . . . . . 6  |-  ( t  =  T  ->  (
t  e.  S  <->  T  e.  S ) )
43anbi1d 703 . . . . 5  |-  ( t  =  T  ->  (
( t  e.  S  /\  u  e.  S
)  <->  ( T  e.  S  /\  u  e.  S ) ) )
5 psseq1 3529 . . . . . 6  |-  ( t  =  T  ->  (
t  C.  u  <->  T  C.  u
) )
6 psseq1 3529 . . . . . . . . 9  |-  ( t  =  T  ->  (
t  C.  s  <->  T  C.  s
) )
76anbi1d 703 . . . . . . . 8  |-  ( t  =  T  ->  (
( t  C.  s  /\  s  C.  u )  <-> 
( T  C.  s  /\  s  C.  u ) ) )
87rexbidv 2917 . . . . . . 7  |-  ( t  =  T  ->  ( E. s  e.  S  ( t  C.  s  /\  s  C.  u )  <->  E. s  e.  S  ( T  C.  s  /\  s  C.  u ) ) )
98notbid 292 . . . . . 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 709 . . . . 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 709 . . . 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 2474 . . . . . 6  |-  ( u  =  U  ->  (
u  e.  S  <->  U  e.  S ) )
1312anbi2d 702 . . . . 5  |-  ( u  =  U  ->  (
( T  e.  S  /\  u  e.  S
)  <->  ( T  e.  S  /\  U  e.  S ) ) )
14 psseq2 3530 . . . . . 6  |-  ( u  =  U  ->  ( T  C.  u  <->  T  C.  U
) )
15 psseq2 3530 . . . . . . . . 9  |-  ( u  =  U  ->  (
s  C.  u  <->  s  C.  U
) )
1615anbi2d 702 . . . . . . . 8  |-  ( u  =  U  ->  (
( T  C.  s  /\  s  C.  u )  <-> 
( T  C.  s  /\  s  C.  U ) ) )
1716rexbidv 2917 . . . . . . 7  |-  ( u  =  U  ->  ( E. s  e.  S  ( T  C.  s  /\  s  C.  u )  <->  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) ) )
1817notbid 292 . . . . . 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 709 . . . . 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 709 . . . 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 2402 . . . 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 4708 . . 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 659 . 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 32019 . . 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 4405 . 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 530 . . 3  |-  ( ph  ->  ( T  e.  S  /\  U  e.  S
) )
3029biantrurd 506 . 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 367    = wceq 1405    e. wcel 1842   E.wrex 2754    C. wpss 3414   class class class wbr 4394   {copab 4451   ` cfv 5525   LSubSpclss 17790    <oLL clcv 32017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-iota 5489  df-fun 5527  df-fv 5533  df-lcv 32018
This theorem is referenced by:  lcvbr2  32021  lcvbr3  32022  lcvpss  32023  lcvnbtwn  32024  lsatcv0  32030  mapdcv  34661
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