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Theorem sspnv 21302
Description: A subspace is a normed complex vector space. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
sspnv.h  |-  H  =  ( SubSp `  U )
Assertion
Ref Expression
sspnv  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  W  e.  NrmCVec )

Proof of Theorem sspnv
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
2 eqid 2283 . . 3  |-  ( +v
`  W )  =  ( +v `  W
)
3 eqid 2283 . . 3  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 eqid 2283 . . 3  |-  ( .s
OLD `  W )  =  ( .s OLD `  W )
5 eqid 2283 . . 3  |-  ( normCV `  U )  =  (
normCV
`  U )
6 eqid 2283 . . 3  |-  ( normCV `  W )  =  (
normCV
`  W )
7 sspnv.h . . 3  |-  H  =  ( SubSp `  U )
81, 2, 3, 4, 5, 6, 7isssp 21300 . 2  |-  ( U  e.  NrmCVec  ->  ( W  e.  H  <->  ( W  e.  NrmCVec 
/\  ( ( +v
`  W )  C_  ( +v `  U )  /\  ( .s OLD `  W )  C_  ( .s OLD `  U )  /\  ( normCV `  W
)  C_  ( normCV `  U
) ) ) ) )
98simprbda 606 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  W  e.  NrmCVec )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   ` cfv 5255   NrmCVeccnv 21140   +vcpv 21141   .s OLDcns 21143   normCVcnmcv 21146   SubSpcss 21297
This theorem is referenced by:  sspg  21304  ssps  21306  sspmlem  21308  sspmval  21309  sspz  21311  sspn  21312  sspival  21314  sspimsval  21316  sspph  21433  bnsscmcl  21447  minvecolem2  21454  hhshsslem1  21844  hhshsslem2  21845
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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