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Theorem sspid 21317
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 3210 . . . 4  |-  ( +v
`  U )  C_  ( +v `  U )
2 ssid 3210 . . . 4  |-  ( .s
OLD `  U )  C_  ( .s OLD `  U
)
3 ssid 3210 . . . 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 2296 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
7 eqid 2296 . . 3  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
8 eqid 2296 . . 3  |-  ( normCV `  U )  =  (
normCV
`  U )
9 sspid.h . . 3  |-  H  =  ( SubSp `  U )
106, 6, 7, 7, 8, 8, 9isssp 21316 . 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 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271   NrmCVeccnv 21156   +vcpv 21157   .s OLDcns 21159   normCVcnmcv 21162   SubSpcss 21313
This theorem is referenced by:  hhsssh  21862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-oprab 5878  df-1st 6138  df-2nd 6139  df-vc 21118  df-nv 21164  df-va 21167  df-sm 21169  df-nmcv 21172  df-ssp 21314
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