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Theorem lcvbr3 29722
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 29720 . 2  |-  ( ph  ->  ( T C U  <-> 
( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U
) ) ) )
7 iman 414 . . . . . 6  |-  ( ( ( T  C_  s  /\  s  C_  U )  ->  ( s  =  T  \/  s  =  U ) )  <->  -.  (
( T  C_  s  /\  s  C_  U )  /\  -.  ( s  =  T  \/  s  =  U ) ) )
8 df-pss 3328 . . . . . . . . 9  |-  ( T 
C.  s  <->  ( T  C_  s  /\  T  =/=  s ) )
9 necom 2679 . . . . . . . . . 10  |-  ( T  =/=  s  <->  s  =/=  T )
109anbi2i 676 . . . . . . . . 9  |-  ( ( T  C_  s  /\  T  =/=  s )  <->  ( T  C_  s  /\  s  =/= 
T ) )
118, 10bitri 241 . . . . . . . 8  |-  ( T 
C.  s  <->  ( T  C_  s  /\  s  =/= 
T ) )
12 df-pss 3328 . . . . . . . 8  |-  ( s 
C.  U  <->  ( s  C_  U  /\  s  =/= 
U ) )
1311, 12anbi12i 679 . . . . . . 7  |-  ( ( T  C.  s  /\  s  C.  U )  <->  ( ( T  C_  s  /\  s  =/=  T )  /\  (
s  C_  U  /\  s  =/=  U ) ) )
14 an4 798 . . . . . . . 8  |-  ( ( ( T  C_  s  /\  s  =/=  T
)  /\  ( s  C_  U  /\  s  =/= 
U ) )  <->  ( ( T  C_  s  /\  s  C_  U )  /\  (
s  =/=  T  /\  s  =/=  U ) ) )
15 neanior 2683 . . . . . . . . 9  |-  ( ( s  =/=  T  /\  s  =/=  U )  <->  -.  (
s  =  T  \/  s  =  U )
)
1615anbi2i 676 . . . . . . . 8  |-  ( ( ( T  C_  s  /\  s  C_  U )  /\  ( s  =/= 
T  /\  s  =/=  U ) )  <->  ( ( T  C_  s  /\  s  C_  U )  /\  -.  ( s  =  T  \/  s  =  U ) ) )
1714, 16bitri 241 . . . . . . 7  |-  ( ( ( T  C_  s  /\  s  =/=  T
)  /\  ( s  C_  U  /\  s  =/= 
U ) )  <->  ( ( T  C_  s  /\  s  C_  U )  /\  -.  ( s  =  T  \/  s  =  U ) ) )
1813, 17bitri 241 . . . . . 6  |-  ( ( T  C.  s  /\  s  C.  U )  <->  ( ( T  C_  s  /\  s  C_  U )  /\  -.  ( s  =  T  \/  s  =  U ) ) )
197, 18xchbinxr 303 . . . . 5  |-  ( ( ( T  C_  s  /\  s  C_  U )  ->  ( s  =  T  \/  s  =  U ) )  <->  -.  ( T  C.  s  /\  s  C.  U ) )
2019ralbii 2721 . . . 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 2707 . . . 4  |-  ( A. s  e.  S  -.  ( T  C.  s  /\  s  C.  U )  <->  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) )
2220, 21bitri 241 . . 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 676 . 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 255 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698    C_ wss 3312    C. wpss 3313   class class class wbr 4204   ` cfv 5446   LSubSpclss 15998    <oLL clcv 29717
This theorem is referenced by:  lcvexchlem4  29736  lcvexchlem5  29737
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-lcv 29718
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