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Theorem lcvbr2 29821
Description: The covers relation for a left vector space (or a left module). (cvbr2 23787 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 29820 . 2  |-  ( ph  ->  ( T C U  <-> 
( T  C.  U  /\  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U
) ) ) )
7 iman 415 . . . . . 6  |-  ( ( ( T  C.  s  /\  s  C_  U )  ->  s  =  U )  <->  -.  ( ( T  C.  s  /\  s  C_  U )  /\  -.  s  =  U )
)
8 anass 632 . . . . . . 7  |-  ( ( ( T  C.  s  /\  s  C_  U )  /\  -.  s  =  U )  <->  ( T  C.  s  /\  (
s  C_  U  /\  -.  s  =  U
) ) )
9 dfpss2 3433 . . . . . . . 8  |-  ( s 
C.  U  <->  ( s  C_  U  /\  -.  s  =  U ) )
109anbi2i 677 . . . . . . 7  |-  ( ( T  C.  s  /\  s  C.  U )  <->  ( T  C.  s  /\  (
s  C_  U  /\  -.  s  =  U
) ) )
118, 10bitr4i 245 . . . . . 6  |-  ( ( ( T  C.  s  /\  s  C_  U )  /\  -.  s  =  U )  <->  ( T  C.  s  /\  s  C.  U ) )
127, 11xchbinx 303 . . . . 5  |-  ( ( ( T  C.  s  /\  s  C_  U )  ->  s  =  U )  <->  -.  ( T  C.  s  /\  s  C.  U ) )
1312ralbii 2730 . . . 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 2716 . . . 4  |-  ( A. s  e.  S  -.  ( T  C.  s  /\  s  C.  U )  <->  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) )
1513, 14bitri 242 . . 3  |-  ( A. s  e.  S  (
( T  C.  s  /\  s  C_  U )  ->  s  =  U )  <->  -.  E. s  e.  S  ( T  C.  s  /\  s  C.  U ) )
1615anbi2i 677 . 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 256 1  |-  ( ph  ->  ( T C U  <-> 
( T  C.  U  /\  A. s  e.  S  ( ( T  C.  s  /\  s  C_  U
)  ->  s  =  U ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2706   E.wrex 2707    C_ wss 3321    C. wpss 3322   class class class wbr 4213   ` cfv 5455   LSubSpclss 16009    <oLL clcv 29817
This theorem is referenced by:  lsmcv2  29828  lsat0cv  29832
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-lcv 29818
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