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Theorem isssp 22225
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  =  ( .s OLD `  U
)
isssp.r  |-  R  =  ( .s OLD `  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  =  ( .s OLD `  U
)
3 isssp.n . . . 4  |-  N  =  ( normCV `  U )
4 isssp.h . . . 4  |-  H  =  ( SubSp `  U )
51, 2, 3, 4sspval 22224 . . 3  |-  ( U  e.  NrmCVec  ->  H  =  {
w  e.  NrmCVec  |  ( ( +v `  w
)  C_  G  /\  ( .s OLD `  w
)  C_  S  /\  ( normCV `  w )  C_  N ) } )
65eleq2d 2505 . 2  |-  ( U  e.  NrmCVec  ->  ( W  e.  H  <->  W  e.  { w  e.  NrmCVec  |  ( ( +v `  w ) 
C_  G  /\  ( .s OLD `  w ) 
C_  S  /\  ( normCV `  w )  C_  N
) } ) )
7 fveq2 5730 . . . . . 6  |-  ( w  =  W  ->  ( +v `  w )  =  ( +v `  W
) )
8 isssp.f . . . . . 6  |-  F  =  ( +v `  W
)
97, 8syl6eqr 2488 . . . . 5  |-  ( w  =  W  ->  ( +v `  w )  =  F )
109sseq1d 3377 . . . 4  |-  ( w  =  W  ->  (
( +v `  w
)  C_  G  <->  F  C_  G
) )
11 fveq2 5730 . . . . . 6  |-  ( w  =  W  ->  ( .s OLD `  w )  =  ( .s OLD `  W ) )
12 isssp.r . . . . . 6  |-  R  =  ( .s OLD `  W
)
1311, 12syl6eqr 2488 . . . . 5  |-  ( w  =  W  ->  ( .s OLD `  w )  =  R )
1413sseq1d 3377 . . . 4  |-  ( w  =  W  ->  (
( .s OLD `  w
)  C_  S  <->  R  C_  S
) )
15 fveq2 5730 . . . . . 6  |-  ( w  =  W  ->  ( normCV `  w )  =  (
normCV
`  W ) )
16 isssp.m . . . . . 6  |-  M  =  ( normCV `  W )
1715, 16syl6eqr 2488 . . . . 5  |-  ( w  =  W  ->  ( normCV `  w )  =  M )
1817sseq1d 3377 . . . 4  |-  ( w  =  W  ->  (
( normCV `  w )  C_  N 
<->  M  C_  N )
)
1910, 14, 183anbi123d 1255 . . 3  |-  ( w  =  W  ->  (
( ( +v `  w )  C_  G  /\  ( .s OLD `  w
)  C_  S  /\  ( normCV `  w )  C_  N )  <->  ( F  C_  G  /\  R  C_  S  /\  M  C_  N
) ) )
2019elrab 3094 . 2  |-  ( W  e.  { w  e.  NrmCVec  |  ( ( +v
`  w )  C_  G  /\  ( .s OLD `  w )  C_  S  /\  ( normCV `  w )  C_  N ) }  <->  ( W  e.  NrmCVec  /\  ( F  C_  G  /\  R  C_  S  /\  M  C_  N
) ) )
216, 20syl6bb 254 1  |-  ( U  e.  NrmCVec  ->  ( W  e.  H  <->  ( W  e.  NrmCVec 
/\  ( F  C_  G  /\  R  C_  S  /\  M  C_  N ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {crab 2711    C_ wss 3322   ` cfv 5456   NrmCVeccnv 22065   +vcpv 22066   .s OLDcns 22068   normCVcnmcv 22071   SubSpcss 22222
This theorem is referenced by:  sspid  22226  sspnv  22227  sspba  22228  sspg  22229  ssps  22231  sspn  22237  hhsst  22768  hhsssh2  22772
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 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464  df-oprab 6087  df-1st 6351  df-2nd 6352  df-vc 22027  df-nv 22073  df-va 22076  df-sm 22078  df-nmcv 22081  df-ssp 22223
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