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Theorem lcvbr3 32971
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 32969 . 2  |-  ( ph  ->  ( T C U  <-> 
( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U
) ) ) )
7 iman 424 . . . . . 6  |-  ( ( ( T  C_  s  /\  s  C_  U )  ->  ( s  =  T  \/  s  =  U ) )  <->  -.  (
( T  C_  s  /\  s  C_  U )  /\  -.  ( s  =  T  \/  s  =  U ) ) )
8 df-pss 3439 . . . . . . . . 9  |-  ( T 
C.  s  <->  ( T  C_  s  /\  T  =/=  s ) )
9 necom 2715 . . . . . . . . . 10  |-  ( T  =/=  s  <->  s  =/=  T )
109anbi2i 694 . . . . . . . . 9  |-  ( ( T  C_  s  /\  T  =/=  s )  <->  ( T  C_  s  /\  s  =/= 
T ) )
118, 10bitri 249 . . . . . . . 8  |-  ( T 
C.  s  <->  ( T  C_  s  /\  s  =/= 
T ) )
12 df-pss 3439 . . . . . . . 8  |-  ( s 
C.  U  <->  ( s  C_  U  /\  s  =/= 
U ) )
1311, 12anbi12i 697 . . . . . . 7  |-  ( ( T  C.  s  /\  s  C.  U )  <->  ( ( T  C_  s  /\  s  =/=  T )  /\  (
s  C_  U  /\  s  =/=  U ) ) )
14 an4 820 . . . . . . . 8  |-  ( ( ( T  C_  s  /\  s  =/=  T
)  /\  ( s  C_  U  /\  s  =/= 
U ) )  <->  ( ( T  C_  s  /\  s  C_  U )  /\  (
s  =/=  T  /\  s  =/=  U ) ) )
15 neanior 2771 . . . . . . . . 9  |-  ( ( s  =/=  T  /\  s  =/=  U )  <->  -.  (
s  =  T  \/  s  =  U )
)
1615anbi2i 694 . . . . . . . 8  |-  ( ( ( T  C_  s  /\  s  C_  U )  /\  ( s  =/= 
T  /\  s  =/=  U ) )  <->  ( ( T  C_  s  /\  s  C_  U )  /\  -.  ( s  =  T  \/  s  =  U ) ) )
1714, 16bitri 249 . . . . . . 7  |-  ( ( ( T  C_  s  /\  s  =/=  T
)  /\  ( s  C_  U  /\  s  =/= 
U ) )  <->  ( ( T  C_  s  /\  s  C_  U )  /\  -.  ( s  =  T  \/  s  =  U ) ) )
1813, 17bitri 249 . . . . . 6  |-  ( ( T  C.  s  /\  s  C.  U )  <->  ( ( T  C_  s  /\  s  C_  U )  /\  -.  ( s  =  T  \/  s  =  U ) ) )
197, 18xchbinxr 311 . . . . 5  |-  ( ( ( T  C_  s  /\  s  C_  U )  ->  ( s  =  T  \/  s  =  U ) )  <->  -.  ( T  C.  s  /\  s  C.  U ) )
2019ralbii 2828 . . . 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 2861 . . . 4  |-  ( A. s  e.  S  -.  ( T  C.  s  /\  s  C.  U )  <->  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) )
2220, 21bitri 249 . . 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 694 . 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 263 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 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2642   A.wral 2793   E.wrex 2794    C_ wss 3423    C. wpss 3424   class class class wbr 4387   ` cfv 5513   LSubSpclss 17116    <oLL clcv 32966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-iota 5476  df-fun 5515  df-fv 5521  df-lcv 32967
This theorem is referenced by:  lcvexchlem4  32985  lcvexchlem5  32986
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