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Theorem sspnv 25837
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 2454 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
2 eqid 2454 . . 3  |-  ( +v
`  W )  =  ( +v `  W
)
3 eqid 2454 . . 3  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
4 eqid 2454 . . 3  |-  ( .sOLD `  W )  =  ( .sOLD `  W )
5 eqid 2454 . . 3  |-  ( normCV `  U )  =  (
normCV
`  U )
6 eqid 2454 . . 3  |-  ( normCV `  W )  =  (
normCV
`  W )
7 sspnv.h . . 3  |-  H  =  ( SubSp `  U )
81, 2, 3, 4, 5, 6, 7isssp 25835 . 2  |-  ( U  e.  NrmCVec  ->  ( W  e.  H  <->  ( W  e.  NrmCVec 
/\  ( ( +v
`  W )  C_  ( +v `  U )  /\  ( .sOLD `  W )  C_  ( .sOLD `  U )  /\  ( normCV `  W
)  C_  ( normCV `  U
) ) ) ) )
98simprbda 621 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  W  e.  NrmCVec )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    C_ wss 3461   ` cfv 5570   NrmCVeccnv 25675   +vcpv 25676   .sOLDcns 25678   normCVcnmcv 25681   SubSpcss 25832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-oprab 6274  df-1st 6773  df-2nd 6774  df-vc 25637  df-nv 25683  df-va 25686  df-sm 25688  df-nmcv 25691  df-ssp 25833
This theorem is referenced by:  sspg  25839  ssps  25841  sspmlem  25843  sspmval  25844  sspz  25846  sspn  25847  sspival  25849  sspimsval  25851  sspph  25968  bnsscmcl  25982  minvecolem2  25989  hhshsslem1  26381  hhshsslem2  26382
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