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Theorem sspid 21301
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 3197 . . . 4  |-  ( +v
`  U )  C_  ( +v `  U )
2 ssid 3197 . . . 4  |-  ( .s
OLD `  U )  C_  ( .s OLD `  U
)
3 ssid 3197 . . . 4  |-  ( normCV `  U )  C_  ( normCV `  U )
41, 2, 33pm3.2i 1130 . . 3  |-  ( ( +v `  U ) 
C_  ( +v `  U )  /\  ( .s OLD `  U ) 
C_  ( .s OLD `  U )  /\  ( normCV `  U )  C_  ( normCV `  U ) )
54jctr 526 . 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 2283 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
7 eqid 2283 . . 3  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
8 eqid 2283 . . 3  |-  ( normCV `  U )  =  (
normCV
`  U )
9 sspid.h . . 3  |-  H  =  ( SubSp `  U )
106, 6, 7, 7, 8, 8, 9isssp 21300 . 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 223 1  |-  ( U  e.  NrmCVec  ->  U  e.  H
)
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:  hhsssh  21846
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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