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Theorem sspba 21319
Description: The base set of a subspace is included in the parent base set. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
sspba.x  |-  X  =  ( BaseSet `  U )
sspba.y  |-  Y  =  ( BaseSet `  W )
sspba.h  |-  H  =  ( SubSp `  U )
Assertion
Ref Expression
sspba  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  X )

Proof of Theorem sspba
StepHypRef Expression
1 eqid 2296 . . . . . 6  |-  ( +v
`  U )  =  ( +v `  U
)
2 eqid 2296 . . . . . 6  |-  ( +v
`  W )  =  ( +v `  W
)
3 eqid 2296 . . . . . 6  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 eqid 2296 . . . . . 6  |-  ( .s
OLD `  W )  =  ( .s OLD `  W )
5 eqid 2296 . . . . . 6  |-  ( normCV `  U )  =  (
normCV
`  U )
6 eqid 2296 . . . . . 6  |-  ( normCV `  W )  =  (
normCV
`  W )
7 sspba.h . . . . . 6  |-  H  =  ( SubSp `  U )
81, 2, 3, 4, 5, 6, 7isssp 21316 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( W  e.  H  <->  ( W  e.  NrmCVec 
/\  ( ( +v
`  W )  C_  ( +v `  U )  /\  ( .s OLD `  W )  C_  ( .s OLD `  U )  /\  ( normCV `  W
)  C_  ( normCV `  U
) ) ) ) )
98simplbda 607 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  (
( +v `  W
)  C_  ( +v `  U )  /\  ( .s OLD `  W ) 
C_  ( .s OLD `  U )  /\  ( normCV `  W )  C_  ( normCV `  U ) ) )
109simp1d 967 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( +v `  W )  C_  ( +v `  U ) )
11 rnss 4923 . . 3  |-  ( ( +v `  W ) 
C_  ( +v `  U )  ->  ran  ( +v `  W ) 
C_  ran  ( +v `  U ) )
1210, 11syl 15 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ran  ( +v `  W ) 
C_  ran  ( +v `  U ) )
13 sspba.y . . 3  |-  Y  =  ( BaseSet `  W )
1413, 2bafval 21176 . 2  |-  Y  =  ran  ( +v `  W )
15 sspba.x . . 3  |-  X  =  ( BaseSet `  U )
1615, 1bafval 21176 . 2  |-  X  =  ran  ( +v `  U )
1712, 14, 163sstr4g 3232 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   ran crn 4706   ` cfv 5271   NrmCVeccnv 21156   +vcpv 21157   BaseSetcba 21158   .s
OLDcns 21159   normCVcnmcv 21162   SubSpcss 21313
This theorem is referenced by:  sspg  21320  ssps  21322  sspmlem  21324  sspmval  21325  sspz  21327  sspn  21328  sspival  21330  sspimsval  21332  sspph  21449  minvecolem1  21469  minvecolem2  21470  minvecolem3  21471  minvecolem4b  21473  minvecolem4  21475  minvecolem5  21476  minvecolem6  21477  minvecolem7  21478
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-ba 21168  df-sm 21169  df-nmcv 21172  df-ssp 21314
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