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Theorem lcvbr3 32589
Description: The covers relation for a left vector space (or a left module). (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
lcvbr3  |-  ( ph  ->  ( T C U  <-> 
( T  C.  U  /\  A. s  e.  S  ( ( T  C_  s  /\  s  C_  U
)  ->  ( s  =  T  \/  s  =  U ) ) ) ) )
Distinct variable groups:    S, s    W, s    T, s    U, s
Allowed substitution hints:    ph( s)    C( s)    X( s)

Proof of Theorem lcvbr3
StepHypRef Expression
1 lcvfbr.s . . 3  |-  S  =  ( LSubSp `  W )
2 lcvfbr.c . . 3  |-  C  =  (  <oLL  `  W )
3 lcvfbr.w . . 3  |-  ( ph  ->  W  e.  X )
4 lcvfbr.t . . 3  |-  ( ph  ->  T  e.  S )
5 lcvfbr.u . . 3  |-  ( ph  ->  U  e.  S )
61, 2, 3, 4, 5lcvbr 32587 . 2  |-  ( ph  ->  ( T C U  <-> 
( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U
) ) ) )
7 iman 426 . . . . . 6  |-  ( ( ( T  C_  s  /\  s  C_  U )  ->  ( s  =  T  \/  s  =  U ) )  <->  -.  (
( T  C_  s  /\  s  C_  U )  /\  -.  ( s  =  T  \/  s  =  U ) ) )
8 df-pss 3420 . . . . . . . . 9  |-  ( T 
C.  s  <->  ( T  C_  s  /\  T  =/=  s ) )
9 necom 2677 . . . . . . . . . 10  |-  ( T  =/=  s  <->  s  =/=  T )
109anbi2i 700 . . . . . . . . 9  |-  ( ( T  C_  s  /\  T  =/=  s )  <->  ( T  C_  s  /\  s  =/= 
T ) )
118, 10bitri 253 . . . . . . . 8  |-  ( T 
C.  s  <->  ( T  C_  s  /\  s  =/= 
T ) )
12 df-pss 3420 . . . . . . . 8  |-  ( s 
C.  U  <->  ( s  C_  U  /\  s  =/= 
U ) )
1311, 12anbi12i 703 . . . . . . 7  |-  ( ( T  C.  s  /\  s  C.  U )  <->  ( ( T  C_  s  /\  s  =/=  T )  /\  (
s  C_  U  /\  s  =/=  U ) ) )
14 an4 833 . . . . . . . 8  |-  ( ( ( T  C_  s  /\  s  =/=  T
)  /\  ( s  C_  U  /\  s  =/= 
U ) )  <->  ( ( T  C_  s  /\  s  C_  U )  /\  (
s  =/=  T  /\  s  =/=  U ) ) )
15 neanior 2716 . . . . . . . . 9  |-  ( ( s  =/=  T  /\  s  =/=  U )  <->  -.  (
s  =  T  \/  s  =  U )
)
1615anbi2i 700 . . . . . . . 8  |-  ( ( ( T  C_  s  /\  s  C_  U )  /\  ( s  =/= 
T  /\  s  =/=  U ) )  <->  ( ( T  C_  s  /\  s  C_  U )  /\  -.  ( s  =  T  \/  s  =  U ) ) )
1714, 16bitri 253 . . . . . . 7  |-  ( ( ( T  C_  s  /\  s  =/=  T
)  /\  ( s  C_  U  /\  s  =/= 
U ) )  <->  ( ( T  C_  s  /\  s  C_  U )  /\  -.  ( s  =  T  \/  s  =  U ) ) )
1813, 17bitri 253 . . . . . 6  |-  ( ( T  C.  s  /\  s  C.  U )  <->  ( ( T  C_  s  /\  s  C_  U )  /\  -.  ( s  =  T  \/  s  =  U ) ) )
197, 18xchbinxr 313 . . . . 5  |-  ( ( ( T  C_  s  /\  s  C_  U )  ->  ( s  =  T  \/  s  =  U ) )  <->  -.  ( T  C.  s  /\  s  C.  U ) )
2019ralbii 2819 . . . 4  |-  ( A. s  e.  S  (
( T  C_  s  /\  s  C_  U )  ->  ( s  =  T  \/  s  =  U ) )  <->  A. s  e.  S  -.  ( T  C.  s  /\  s  C.  U ) )
21 ralnex 2834 . . . 4  |-  ( A. s  e.  S  -.  ( T  C.  s  /\  s  C.  U )  <->  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) )
2220, 21bitri 253 . . 3  |-  ( A. s  e.  S  (
( T  C_  s  /\  s  C_  U )  ->  ( s  =  T  \/  s  =  U ) )  <->  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) )
2322anbi2i 700 . 2  |-  ( ( T  C.  U  /\  A. s  e.  S  ( ( T  C_  s  /\  s  C_  U )  ->  ( s  =  T  \/  s  =  U ) ) )  <-> 
( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U
) ) )
246, 23syl6bbr 267 1  |-  ( ph  ->  ( T C U  <-> 
( T  C.  U  /\  A. s  e.  S  ( ( T  C_  s  /\  s  C_  U
)  ->  ( s  =  T  \/  s  =  U ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738    C_ wss 3404    C. wpss 3405   class class class wbr 4402   ` cfv 5582   LSubSpclss 18155    <oLL clcv 32584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-iota 5546  df-fun 5584  df-fv 5590  df-lcv 32585
This theorem is referenced by:  lcvexchlem4  32603  lcvexchlem5  32604
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