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Theorem sspid 22073
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 3311 . . . 4  |-  ( +v
`  U )  C_  ( +v `  U )
2 ssid 3311 . . . 4  |-  ( .s
OLD `  U )  C_  ( .s OLD `  U
)
3 ssid 3311 . . . 4  |-  ( normCV `  U )  C_  ( normCV `  U )
41, 2, 33pm3.2i 1132 . . 3  |-  ( ( +v `  U ) 
C_  ( +v `  U )  /\  ( .s OLD `  U ) 
C_  ( .s OLD `  U )  /\  ( normCV `  U )  C_  ( normCV `  U ) )
54jctr 527 . 2  |-  ( U  e.  NrmCVec  ->  ( U  e.  NrmCVec 
/\  ( ( +v
`  U )  C_  ( +v `  U )  /\  ( .s OLD `  U )  C_  ( .s OLD `  U )  /\  ( normCV `  U
)  C_  ( normCV `  U
) ) ) )
6 eqid 2388 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
7 eqid 2388 . . 3  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
8 eqid 2388 . . 3  |-  ( normCV `  U )  =  (
normCV
`  U )
9 sspid.h . . 3  |-  H  =  ( SubSp `  U )
106, 6, 7, 7, 8, 8, 9isssp 22072 . 2  |-  ( U  e.  NrmCVec  ->  ( U  e.  H  <->  ( U  e.  NrmCVec 
/\  ( ( +v
`  U )  C_  ( +v `  U )  /\  ( .s OLD `  U )  C_  ( .s OLD `  U )  /\  ( normCV `  U
)  C_  ( normCV `  U
) ) ) ) )
115, 10mpbird 224 1  |-  ( U  e.  NrmCVec  ->  U  e.  H
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    C_ wss 3264   ` cfv 5395   NrmCVeccnv 21912   +vcpv 21913   .s OLDcns 21915   normCVcnmcv 21918   SubSpcss 22069
This theorem is referenced by:  hhsssh  22618
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fo 5401  df-fv 5403  df-oprab 6025  df-1st 6289  df-2nd 6290  df-vc 21874  df-nv 21920  df-va 21923  df-sm 21925  df-nmcv 21928  df-ssp 22070
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