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Theorem lspssp 15955
Description: If a set of vectors is a subset of a subspace, then the span of those vectors is also contained in the subspace. (Contributed by Mario Carneiro, 4-Sep-2014.)
Hypotheses
Ref Expression
lspssp.s  |-  S  =  ( LSubSp `  W )
lspssp.n  |-  N  =  ( LSpan `  W )
Assertion
Ref Expression
lspssp  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  T  C_  U )  ->  ( N `  T )  C_  U )

Proof of Theorem lspssp
StepHypRef Expression
1 eqid 2366 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 lspssp.s . . . 4  |-  S  =  ( LSubSp `  W )
31, 2lssss 15904 . . 3  |-  ( U  e.  S  ->  U  C_  ( Base `  W
) )
4 lspssp.n . . . 4  |-  N  =  ( LSpan `  W )
51, 4lspss 15951 . . 3  |-  ( ( W  e.  LMod  /\  U  C_  ( Base `  W
)  /\  T  C_  U
)  ->  ( N `  T )  C_  ( N `  U )
)
63, 5syl3an2 1217 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  T  C_  U )  ->  ( N `  T )  C_  ( N `  U
) )
72, 4lspid 15949 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( N `  U )  =  U )
873adant3 976 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  T  C_  U )  ->  ( N `  U )  =  U )
96, 8sseqtrd 3300 1  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  T  C_  U )  ->  ( N `  T )  C_  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 935    = wceq 1647    e. wcel 1715    C_ wss 3238   ` cfv 5358   Basecbs 13356   LModclmod 15837   LSubSpclss 15899   LSpanclspn 15938
This theorem is referenced by:  lspsnss  15957  lspprss  15959  lsp0  15976  lsslsp  15982  lmhmlsp  16016  lspextmo  16023  lsmsp  16049  lsppratlem3  16112  lsppratlem4  16113  islbs3  16118  rspssp  16188  ocvlsp  16793  frlmsslsp  26754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-riota 6446  df-0g 13614  df-mnd 14577  df-grp 14699  df-lmod 15839  df-lss 15900  df-lsp 15939
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