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Theorem sspba 25413
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 2467 . . . . . 6  |-  ( +v
`  U )  =  ( +v `  U
)
2 eqid 2467 . . . . . 6  |-  ( +v
`  W )  =  ( +v `  W
)
3 eqid 2467 . . . . . 6  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
4 eqid 2467 . . . . . 6  |-  ( .sOLD `  W )  =  ( .sOLD `  W )
5 eqid 2467 . . . . . 6  |-  ( normCV `  U )  =  (
normCV
`  U )
6 eqid 2467 . . . . . 6  |-  ( normCV `  W )  =  (
normCV
`  W )
7 sspba.h . . . . . 6  |-  H  =  ( SubSp `  U )
81, 2, 3, 4, 5, 6, 7isssp 25410 . . . . 5  |-  ( U  e.  NrmCVec  ->  ( W  e.  H  <->  ( W  e.  NrmCVec 
/\  ( ( +v
`  W )  C_  ( +v `  U )  /\  ( .sOLD `  W )  C_  ( .sOLD `  U )  /\  ( normCV `  W
)  C_  ( normCV `  U
) ) ) ) )
98simplbda 624 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  (
( +v `  W
)  C_  ( +v `  U )  /\  ( .sOLD `  W ) 
C_  ( .sOLD `  U )  /\  ( normCV `  W )  C_  ( normCV `  U ) ) )
109simp1d 1008 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( +v `  W )  C_  ( +v `  U ) )
11 rnss 5231 . . 3  |-  ( ( +v `  W ) 
C_  ( +v `  U )  ->  ran  ( +v `  W ) 
C_  ran  ( +v `  U ) )
1210, 11syl 16 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ran  ( +v `  W ) 
C_  ran  ( +v `  U ) )
13 sspba.y . . 3  |-  Y  =  ( BaseSet `  W )
1413, 2bafval 25270 . 2  |-  Y  =  ran  ( +v `  W )
15 sspba.x . . 3  |-  X  =  ( BaseSet `  U )
1615, 1bafval 25270 . 2  |-  X  =  ran  ( +v `  U )
1712, 14, 163sstr4g 3545 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3476   ran crn 5000   ` cfv 5588   NrmCVeccnv 25250   +vcpv 25251   BaseSetcba 25252   .sOLDcns 25253   normCVcnmcv 25256   SubSpcss 25407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fo 5594  df-fv 5596  df-oprab 6289  df-1st 6785  df-2nd 6786  df-vc 25212  df-nv 25258  df-va 25261  df-ba 25262  df-sm 25263  df-nmcv 25266  df-ssp 25408
This theorem is referenced by:  sspg  25414  ssps  25416  sspmlem  25418  sspmval  25419  sspz  25421  sspn  25422  sspival  25424  sspimsval  25426  sspph  25543  minvecolem1  25563  minvecolem2  25564  minvecolem3  25565  minvecolem4b  25567  minvecolem4  25569  minvecolem5  25570  minvecolem6  25571  minvecolem7  25572
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