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Theorem lcvbr3 29835
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 29833 . 2  |-  ( ph  ->  ( T C U  <-> 
( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U
) ) ) )
7 iman 413 . . . . . 6  |-  ( ( ( T  C_  s  /\  s  C_  U )  ->  ( s  =  T  \/  s  =  U ) )  <->  -.  (
( T  C_  s  /\  s  C_  U )  /\  -.  ( s  =  T  \/  s  =  U ) ) )
8 df-pss 3181 . . . . . . . . 9  |-  ( T 
C.  s  <->  ( T  C_  s  /\  T  =/=  s ) )
9 necom 2540 . . . . . . . . . 10  |-  ( T  =/=  s  <->  s  =/=  T )
109anbi2i 675 . . . . . . . . 9  |-  ( ( T  C_  s  /\  T  =/=  s )  <->  ( T  C_  s  /\  s  =/= 
T ) )
118, 10bitri 240 . . . . . . . 8  |-  ( T 
C.  s  <->  ( T  C_  s  /\  s  =/= 
T ) )
12 df-pss 3181 . . . . . . . 8  |-  ( s 
C.  U  <->  ( s  C_  U  /\  s  =/= 
U ) )
1311, 12anbi12i 678 . . . . . . 7  |-  ( ( T  C.  s  /\  s  C.  U )  <->  ( ( T  C_  s  /\  s  =/=  T )  /\  (
s  C_  U  /\  s  =/=  U ) ) )
14 an4 797 . . . . . . . 8  |-  ( ( ( T  C_  s  /\  s  =/=  T
)  /\  ( s  C_  U  /\  s  =/= 
U ) )  <->  ( ( T  C_  s  /\  s  C_  U )  /\  (
s  =/=  T  /\  s  =/=  U ) ) )
15 neanior 2544 . . . . . . . . 9  |-  ( ( s  =/=  T  /\  s  =/=  U )  <->  -.  (
s  =  T  \/  s  =  U )
)
1615anbi2i 675 . . . . . . . 8  |-  ( ( ( T  C_  s  /\  s  C_  U )  /\  ( s  =/= 
T  /\  s  =/=  U ) )  <->  ( ( T  C_  s  /\  s  C_  U )  /\  -.  ( s  =  T  \/  s  =  U ) ) )
1714, 16bitri 240 . . . . . . 7  |-  ( ( ( T  C_  s  /\  s  =/=  T
)  /\  ( s  C_  U  /\  s  =/= 
U ) )  <->  ( ( T  C_  s  /\  s  C_  U )  /\  -.  ( s  =  T  \/  s  =  U ) ) )
1813, 17bitri 240 . . . . . 6  |-  ( ( T  C.  s  /\  s  C.  U )  <->  ( ( T  C_  s  /\  s  C_  U )  /\  -.  ( s  =  T  \/  s  =  U ) ) )
197, 18xchbinxr 302 . . . . 5  |-  ( ( ( T  C_  s  /\  s  C_  U )  ->  ( s  =  T  \/  s  =  U ) )  <->  -.  ( T  C.  s  /\  s  C.  U ) )
2019ralbii 2580 . . . 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 2566 . . . 4  |-  ( A. s  e.  S  -.  ( T  C.  s  /\  s  C.  U )  <->  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) )
2220, 21bitri 240 . . 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 675 . 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 254 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 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    C_ wss 3165    C. wpss 3166   class class class wbr 4039   ` cfv 5271   LSubSpclss 15705    <oLL clcv 29830
This theorem is referenced by:  lcvexchlem4  29849  lcvexchlem5  29850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-lcv 29831
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