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Theorem sspid 24270
Description: A normed complex vector space is a subspace of itself. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
sspid.h  |-  H  =  ( SubSp `  U )
Assertion
Ref Expression
sspid  |-  ( U  e.  NrmCVec  ->  U  e.  H
)

Proof of Theorem sspid
StepHypRef Expression
1 ssid 3478 . . . 4  |-  ( +v
`  U )  C_  ( +v `  U )
2 ssid 3478 . . . 4  |-  ( .sOLD `  U ) 
C_  ( .sOLD `  U )
3 ssid 3478 . . . 4  |-  ( normCV `  U )  C_  ( normCV `  U )
41, 2, 33pm3.2i 1166 . . 3  |-  ( ( +v `  U ) 
C_  ( +v `  U )  /\  ( .sOLD `  U ) 
C_  ( .sOLD `  U )  /\  ( normCV `  U )  C_  ( normCV `  U ) )
54jctr 542 . 2  |-  ( U  e.  NrmCVec  ->  ( U  e.  NrmCVec 
/\  ( ( +v
`  U )  C_  ( +v `  U )  /\  ( .sOLD `  U )  C_  ( .sOLD `  U )  /\  ( normCV `  U
)  C_  ( normCV `  U
) ) ) )
6 eqid 2452 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
7 eqid 2452 . . 3  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
8 eqid 2452 . . 3  |-  ( normCV `  U )  =  (
normCV
`  U )
9 sspid.h . . 3  |-  H  =  ( SubSp `  U )
106, 6, 7, 7, 8, 8, 9isssp 24269 . 2  |-  ( U  e.  NrmCVec  ->  ( U  e.  H  <->  ( U  e.  NrmCVec 
/\  ( ( +v
`  U )  C_  ( +v `  U )  /\  ( .sOLD `  U )  C_  ( .sOLD `  U )  /\  ( normCV `  U
)  C_  ( normCV `  U
) ) ) ) )
115, 10mpbird 232 1  |-  ( U  e.  NrmCVec  ->  U  e.  H
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    C_ wss 3431   ` cfv 5521   NrmCVeccnv 24109   +vcpv 24110   .sOLDcns 24112   normCVcnmcv 24115   SubSpcss 24266
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 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-fo 5527  df-fv 5529  df-oprab 6199  df-1st 6682  df-2nd 6683  df-vc 24071  df-nv 24117  df-va 24120  df-sm 24122  df-nmcv 24125  df-ssp 24267
This theorem is referenced by:  hhsssh  24817
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