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Theorem sspnv 24269
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 2451 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
2 eqid 2451 . . 3  |-  ( +v
`  W )  =  ( +v `  W
)
3 eqid 2451 . . 3  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
4 eqid 2451 . . 3  |-  ( .sOLD `  W )  =  ( .sOLD `  W )
5 eqid 2451 . . 3  |-  ( normCV `  U )  =  (
normCV
`  U )
6 eqid 2451 . . 3  |-  ( normCV `  W )  =  (
normCV
`  W )
7 sspnv.h . . 3  |-  H  =  ( SubSp `  U )
81, 2, 3, 4, 5, 6, 7isssp 24267 . 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 623 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  W  e.  NrmCVec )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    C_ wss 3429   ` cfv 5519   NrmCVeccnv 24107   +vcpv 24108   .sOLDcns 24110   normCVcnmcv 24113   SubSpcss 24264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fo 5525  df-fv 5527  df-oprab 6197  df-1st 6680  df-2nd 6681  df-vc 24069  df-nv 24115  df-va 24118  df-sm 24120  df-nmcv 24123  df-ssp 24265
This theorem is referenced by:  sspg  24271  ssps  24273  sspmlem  24275  sspmval  24276  sspz  24278  sspn  24279  sspival  24281  sspimsval  24283  sspph  24400  bnsscmcl  24414  minvecolem2  24421  hhshsslem1  24813  hhshsslem2  24814
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