MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isssp Structured version   Unicode version

Theorem isssp 25460
Description: The predicate "is a subspace." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
isssp.g  |-  G  =  ( +v `  U
)
isssp.f  |-  F  =  ( +v `  W
)
isssp.s  |-  S  =  ( .sOLD `  U )
isssp.r  |-  R  =  ( .sOLD `  W )
isssp.n  |-  N  =  ( normCV `  U )
isssp.m  |-  M  =  ( normCV `  W )
isssp.h  |-  H  =  ( SubSp `  U )
Assertion
Ref Expression
isssp  |-  ( U  e.  NrmCVec  ->  ( W  e.  H  <->  ( W  e.  NrmCVec 
/\  ( F  C_  G  /\  R  C_  S  /\  M  C_  N ) ) ) )

Proof of Theorem isssp
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 isssp.g . . . 4  |-  G  =  ( +v `  U
)
2 isssp.s . . . 4  |-  S  =  ( .sOLD `  U )
3 isssp.n . . . 4  |-  N  =  ( normCV `  U )
4 isssp.h . . . 4  |-  H  =  ( SubSp `  U )
51, 2, 3, 4sspval 25459 . . 3  |-  ( U  e.  NrmCVec  ->  H  =  {
w  e.  NrmCVec  |  ( ( +v `  w
)  C_  G  /\  ( .sOLD `  w
)  C_  S  /\  ( normCV `  w )  C_  N ) } )
65eleq2d 2537 . 2  |-  ( U  e.  NrmCVec  ->  ( W  e.  H  <->  W  e.  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  G  /\  ( .sOLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
) } ) )
7 fveq2 5872 . . . . . 6  |-  ( w  =  W  ->  ( +v `  w )  =  ( +v `  W
) )
8 isssp.f . . . . . 6  |-  F  =  ( +v `  W
)
97, 8syl6eqr 2526 . . . . 5  |-  ( w  =  W  ->  ( +v `  w )  =  F )
109sseq1d 3536 . . . 4  |-  ( w  =  W  ->  (
( +v `  w
)  C_  G  <->  F  C_  G
) )
11 fveq2 5872 . . . . . 6  |-  ( w  =  W  ->  ( .sOLD `  w )  =  ( .sOLD `  W ) )
12 isssp.r . . . . . 6  |-  R  =  ( .sOLD `  W )
1311, 12syl6eqr 2526 . . . . 5  |-  ( w  =  W  ->  ( .sOLD `  w )  =  R )
1413sseq1d 3536 . . . 4  |-  ( w  =  W  ->  (
( .sOLD `  w )  C_  S  <->  R 
C_  S ) )
15 fveq2 5872 . . . . . 6  |-  ( w  =  W  ->  ( normCV `  w )  =  (
normCV
`  W ) )
16 isssp.m . . . . . 6  |-  M  =  ( normCV `  W )
1715, 16syl6eqr 2526 . . . . 5  |-  ( w  =  W  ->  ( normCV `  w )  =  M )
1817sseq1d 3536 . . . 4  |-  ( w  =  W  ->  (
( normCV `  w )  C_  N 
<->  M  C_  N )
)
1910, 14, 183anbi123d 1299 . . 3  |-  ( w  =  W  ->  (
( ( +v `  w )  C_  G  /\  ( .sOLD `  w )  C_  S  /\  ( normCV `  w )  C_  N )  <->  ( F  C_  G  /\  R  C_  S  /\  M  C_  N
) ) )
2019elrab 3266 . 2  |-  ( W  e.  { w  e.  NrmCVec  |  ( ( +v
`  w )  C_  G  /\  ( .sOLD `  w )  C_  S  /\  ( normCV `  w )  C_  N ) }  <->  ( W  e.  NrmCVec  /\  ( F  C_  G  /\  R  C_  S  /\  M  C_  N
) ) )
216, 20syl6bb 261 1  |-  ( U  e.  NrmCVec  ->  ( W  e.  H  <->  ( W  e.  NrmCVec 
/\  ( F  C_  G  /\  R  C_  S  /\  M  C_  N ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2821    C_ wss 3481   ` cfv 5594   NrmCVeccnv 25300   +vcpv 25301   .sOLDcns 25303   normCVcnmcv 25306   SubSpcss 25457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-oprab 6299  df-1st 6795  df-2nd 6796  df-vc 25262  df-nv 25308  df-va 25311  df-sm 25313  df-nmcv 25316  df-ssp 25458
This theorem is referenced by:  sspid  25461  sspnv  25462  sspba  25463  sspg  25464  ssps  25466  sspn  25472  hhsst  26005  hhsssh2  26009
  Copyright terms: Public domain W3C validator