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Theorem isssp 21300
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 21299 . . 3  |-  ( U  e.  NrmCVec  ->  H  =  {
w  e.  NrmCVec  |  ( ( +v `  w
)  C_  G  /\  ( .s OLD `  w
)  C_  S  /\  ( normCV `  w )  C_  N ) } )
65eleq2d 2350 . 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 5525 . . . . . 6  |-  ( w  =  W  ->  ( +v `  w )  =  ( +v `  W
) )
8 isssp.f . . . . . 6  |-  F  =  ( +v `  W
)
97, 8syl6eqr 2333 . . . . 5  |-  ( w  =  W  ->  ( +v `  w )  =  F )
109sseq1d 3205 . . . 4  |-  ( w  =  W  ->  (
( +v `  w
)  C_  G  <->  F  C_  G
) )
11 fveq2 5525 . . . . . 6  |-  ( w  =  W  ->  ( .s OLD `  w )  =  ( .s OLD `  W ) )
12 isssp.r . . . . . 6  |-  R  =  ( .s OLD `  W
)
1311, 12syl6eqr 2333 . . . . 5  |-  ( w  =  W  ->  ( .s OLD `  w )  =  R )
1413sseq1d 3205 . . . 4  |-  ( w  =  W  ->  (
( .s OLD `  w
)  C_  S  <->  R  C_  S
) )
15 fveq2 5525 . . . . . 6  |-  ( w  =  W  ->  ( normCV `  w )  =  (
normCV
`  W ) )
16 isssp.m . . . . . 6  |-  M  =  ( normCV `  W )
1715, 16syl6eqr 2333 . . . . 5  |-  ( w  =  W  ->  ( normCV `  w )  =  M )
1817sseq1d 3205 . . . 4  |-  ( w  =  W  ->  (
( normCV `  w )  C_  N 
<->  M  C_  N )
)
1910, 14, 183anbi123d 1252 . . 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 2923 . 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 252 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   ` cfv 5255   NrmCVeccnv 21140   +vcpv 21141   .s OLDcns 21143   normCVcnmcv 21146   SubSpcss 21297
This theorem is referenced by:  sspid  21301  sspnv  21302  sspba  21303  sspg  21304  ssps  21306  sspn  21312  hhsst  21843  hhsssh2  21847
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-oprab 5862  df-1st 6122  df-2nd 6123  df-vc 21102  df-nv 21148  df-va 21151  df-sm 21153  df-nmcv 21156  df-ssp 21298
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