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Theorem isssp 24122
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 24121 . . 3  |-  ( U  e.  NrmCVec  ->  H  =  {
w  e.  NrmCVec  |  ( ( +v `  w
)  C_  G  /\  ( .sOLD `  w
)  C_  S  /\  ( normCV `  w )  C_  N ) } )
65eleq2d 2510 . 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 5691 . . . . . 6  |-  ( w  =  W  ->  ( +v `  w )  =  ( +v `  W
) )
8 isssp.f . . . . . 6  |-  F  =  ( +v `  W
)
97, 8syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  ( +v `  w )  =  F )
109sseq1d 3383 . . . 4  |-  ( w  =  W  ->  (
( +v `  w
)  C_  G  <->  F  C_  G
) )
11 fveq2 5691 . . . . . 6  |-  ( w  =  W  ->  ( .sOLD `  w )  =  ( .sOLD `  W ) )
12 isssp.r . . . . . 6  |-  R  =  ( .sOLD `  W )
1311, 12syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  ( .sOLD `  w )  =  R )
1413sseq1d 3383 . . . 4  |-  ( w  =  W  ->  (
( .sOLD `  w )  C_  S  <->  R 
C_  S ) )
15 fveq2 5691 . . . . . 6  |-  ( w  =  W  ->  ( normCV `  w )  =  (
normCV
`  W ) )
16 isssp.m . . . . . 6  |-  M  =  ( normCV `  W )
1715, 16syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  ( normCV `  w )  =  M )
1817sseq1d 3383 . . . 4  |-  ( w  =  W  ->  (
( normCV `  w )  C_  N 
<->  M  C_  N )
)
1910, 14, 183anbi123d 1289 . . 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 3117 . 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 965    = wceq 1369    e. wcel 1756   {crab 2719    C_ wss 3328   ` cfv 5418   NrmCVeccnv 23962   +vcpv 23963   .sOLDcns 23965   normCVcnmcv 23968   SubSpcss 24119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fo 5424  df-fv 5426  df-oprab 6095  df-1st 6577  df-2nd 6578  df-vc 23924  df-nv 23970  df-va 23973  df-sm 23975  df-nmcv 23978  df-ssp 24120
This theorem is referenced by:  sspid  24123  sspnv  24124  sspba  24125  sspg  24126  ssps  24128  sspn  24134  hhsst  24667  hhsssh2  24671
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