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

Theorem sspba 24297
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 2454 . . . . . 6  |-  ( +v
`  U )  =  ( +v `  U
)
2 eqid 2454 . . . . . 6  |-  ( +v
`  W )  =  ( +v `  W
)
3 eqid 2454 . . . . . 6  |-  ( .sOLD `  U )  =  ( .sOLD `  U )
4 eqid 2454 . . . . . 6  |-  ( .sOLD `  W )  =  ( .sOLD `  W )
5 eqid 2454 . . . . . 6  |-  ( normCV `  U )  =  (
normCV
`  U )
6 eqid 2454 . . . . . 6  |-  ( normCV `  W )  =  (
normCV
`  W )
7 sspba.h . . . . . 6  |-  H  =  ( SubSp `  U )
81, 2, 3, 4, 5, 6, 7isssp 24294 . . . . 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 1000 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( +v `  W )  C_  ( +v `  U ) )
11 rnss 5179 . . 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 24154 . 2  |-  Y  =  ran  ( +v `  W )
15 sspba.x . . 3  |-  X  =  ( BaseSet `  U )
1615, 1bafval 24154 . 2  |-  X  =  ran  ( +v `  U )
1712, 14, 163sstr4g 3508 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  Y  C_  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    C_ wss 3439   ran crn 4952   ` cfv 5529   NrmCVeccnv 24134   +vcpv 24135   BaseSetcba 24136   .sOLDcns 24137   normCVcnmcv 24140   SubSpcss 24291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fo 5535  df-fv 5537  df-oprab 6207  df-1st 6690  df-2nd 6691  df-vc 24096  df-nv 24142  df-va 24145  df-ba 24146  df-sm 24147  df-nmcv 24150  df-ssp 24292
This theorem is referenced by:  sspg  24298  ssps  24300  sspmlem  24302  sspmval  24303  sspz  24305  sspn  24306  sspival  24308  sspimsval  24310  sspph  24427  minvecolem1  24447  minvecolem2  24448  minvecolem3  24449  minvecolem4b  24451  minvecolem4  24453  minvecolem5  24454  minvecolem6  24455  minvecolem7  24456
  Copyright terms: Public domain W3C validator