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Theorem lcvbr2 35144
Description: The covers relation for a left vector space (or a left module). (cvbr2 27400 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
lcvbr2  |-  ( ph  ->  ( T C U  <-> 
( T  C.  U  /\  A. s  e.  S  ( ( T  C.  s  /\  s  C_  U
)  ->  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 lcvbr2
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 35143 . 2  |-  ( ph  ->  ( T C U  <-> 
( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U
) ) ) )
7 iman 422 . . . . . 6  |-  ( ( ( T  C.  s  /\  s  C_  U )  ->  s  =  U )  <->  -.  ( ( T  C.  s  /\  s  C_  U )  /\  -.  s  =  U )
)
8 anass 647 . . . . . . 7  |-  ( ( ( T  C.  s  /\  s  C_  U )  /\  -.  s  =  U )  <->  ( T  C.  s  /\  ( s 
C_  U  /\  -.  s  =  U )
) )
9 dfpss2 3575 . . . . . . . 8  |-  ( s 
C.  U  <->  ( s  C_  U  /\  -.  s  =  U ) )
109anbi2i 692 . . . . . . 7  |-  ( ( T  C.  s  /\  s  C.  U )  <->  ( T  C.  s  /\  ( s 
C_  U  /\  -.  s  =  U )
) )
118, 10bitr4i 252 . . . . . 6  |-  ( ( ( T  C.  s  /\  s  C_  U )  /\  -.  s  =  U )  <->  ( T  C.  s  /\  s  C.  U ) )
127, 11xchbinx 308 . . . . 5  |-  ( ( ( T  C.  s  /\  s  C_  U )  ->  s  =  U )  <->  -.  ( T  C.  s  /\  s  C.  U ) )
1312ralbii 2885 . . . 4  |-  ( A. s  e.  S  (
( T  C.  s  /\  s  C_  U )  ->  s  =  U )  <->  A. s  e.  S  -.  ( T  C.  s  /\  s  C.  U ) )
14 ralnex 2900 . . . 4  |-  ( A. s  e.  S  -.  ( T  C.  s  /\  s  C.  U )  <->  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) )
1513, 14bitri 249 . . 3  |-  ( A. s  e.  S  (
( T  C.  s  /\  s  C_  U )  ->  s  =  U )  <->  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) )
1615anbi2i 692 . 2  |-  ( ( T  C.  U  /\  A. s  e.  S  ( ( T  C.  s  /\  s  C_  U )  ->  s  =  U ) )  <->  ( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) ) )
176, 16syl6bbr 263 1  |-  ( ph  ->  ( T C U  <-> 
( T  C.  U  /\  A. s  e.  S  ( ( T  C.  s  /\  s  C_  U
)  ->  s  =  U ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805    C_ wss 3461    C. wpss 3462   class class class wbr 4439   ` cfv 5570   LSubSpclss 17773    <oLL clcv 35140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-lcv 35141
This theorem is referenced by:  lsmcv2  35151  lsat0cv  35155
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