MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sspid Structured version   Unicode version

Theorem sspid 22216
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 3359 . . . 4  |-  ( +v
`  U )  C_  ( +v `  U )
2 ssid 3359 . . . 4  |-  ( .s
OLD `  U )  C_  ( .s OLD `  U
)
3 ssid 3359 . . . 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 2435 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
7 eqid 2435 . . 3  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
8 eqid 2435 . . 3  |-  ( normCV `  U )  =  (
normCV
`  U )
9 sspid.h . . 3  |-  H  =  ( SubSp `  U )
106, 6, 7, 7, 8, 8, 9isssp 22215 . 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 1652    e. wcel 1725    C_ wss 3312   ` cfv 5446   NrmCVeccnv 22055   +vcpv 22056   .s OLDcns 22058   normCVcnmcv 22061   SubSpcss 22212
This theorem is referenced by:  hhsssh  22761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-oprab 6077  df-1st 6341  df-2nd 6342  df-vc 22017  df-nv 22063  df-va 22066  df-sm 22068  df-nmcv 22071  df-ssp 22213
  Copyright terms: Public domain W3C validator