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Theorem lcvbr2 32504
Description: The covers relation for a left vector space (or a left module). (cvbr2 27919 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 32503 . 2  |-  ( ph  ->  ( T C U  <-> 
( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U
) ) ) )
7 iman 425 . . . . . 6  |-  ( ( ( T  C.  s  /\  s  C_  U )  ->  s  =  U )  <->  -.  ( ( T  C.  s  /\  s  C_  U )  /\  -.  s  =  U )
)
8 anass 653 . . . . . . 7  |-  ( ( ( T  C.  s  /\  s  C_  U )  /\  -.  s  =  U )  <->  ( T  C.  s  /\  ( s 
C_  U  /\  -.  s  =  U )
) )
9 dfpss2 3550 . . . . . . . 8  |-  ( s 
C.  U  <->  ( s  C_  U  /\  -.  s  =  U ) )
109anbi2i 698 . . . . . . 7  |-  ( ( T  C.  s  /\  s  C.  U )  <->  ( T  C.  s  /\  ( s 
C_  U  /\  -.  s  =  U )
) )
118, 10bitr4i 255 . . . . . 6  |-  ( ( ( T  C.  s  /\  s  C_  U )  /\  -.  s  =  U )  <->  ( T  C.  s  /\  s  C.  U ) )
127, 11xchbinx 311 . . . . 5  |-  ( ( ( T  C.  s  /\  s  C_  U )  ->  s  =  U )  <->  -.  ( T  C.  s  /\  s  C.  U ) )
1312ralbii 2856 . . . 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 2871 . . . 4  |-  ( A. s  e.  S  -.  ( T  C.  s  /\  s  C.  U )  <->  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) )
1513, 14bitri 252 . . 3  |-  ( A. s  e.  S  (
( T  C.  s  /\  s  C_  U )  ->  s  =  U )  <->  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) )
1615anbi2i 698 . 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 266 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 187    /\ wa 370    = wceq 1437    e. wcel 1868   A.wral 2775   E.wrex 2776    C_ wss 3436    C. wpss 3437   class class class wbr 4420   ` cfv 5597   LSubSpclss 18140    <oLL clcv 32500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-iota 5561  df-fun 5599  df-fv 5605  df-lcv 32501
This theorem is referenced by:  lsmcv2  32511  lsat0cv  32515
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