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Theorem lindsss 27294
Description: Any subset of an independent set is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Assertion
Ref Expression
lindsss  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )  /\  G  C_  F )  ->  G  e.  (LIndS `  W ) )

Proof of Theorem lindsss
StepHypRef Expression
1 eqid 2283 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
21linds1 27280 . . . . 5  |-  ( F  e.  (LIndS `  W
)  ->  F  C_  ( Base `  W ) )
32adantl 452 . . . 4  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )
)  ->  F  C_  ( Base `  W ) )
4 sstr2 3186 . . . 4  |-  ( G 
C_  F  ->  ( F  C_  ( Base `  W
)  ->  G  C_  ( Base `  W ) ) )
53, 4syl5com 26 . . 3  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )
)  ->  ( G  C_  F  ->  G  C_  ( Base `  W ) ) )
653impia 1148 . 2  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )  /\  G  C_  F )  ->  G  C_  ( Base `  W ) )
7 simp1 955 . . . 4  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )  /\  G  C_  F )  ->  W  e.  LMod )
8 linds2 27281 . . . . 5  |-  ( F  e.  (LIndS `  W
)  ->  (  _I  |`  F ) LIndF  W )
983ad2ant2 977 . . . 4  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )  /\  G  C_  F )  ->  (  _I  |`  F ) LIndF 
W )
10 lindfres 27293 . . . 4  |-  ( ( W  e.  LMod  /\  (  _I  |`  F ) LIndF  W
)  ->  ( (  _I  |`  F )  |`  G ) LIndF  W )
117, 9, 10syl2anc 642 . . 3  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )  /\  G  C_  F )  ->  ( (  _I  |`  F )  |`  G ) LIndF 
W )
12 resabs1 4984 . . . . 5  |-  ( G 
C_  F  ->  (
(  _I  |`  F )  |`  G )  =  (  _I  |`  G )
)
1312breq1d 4033 . . . 4  |-  ( G 
C_  F  ->  (
( (  _I  |`  F )  |`  G ) LIndF  W  <->  (  _I  |`  G ) LIndF  W ) )
14133ad2ant3 978 . . 3  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )  /\  G  C_  F )  ->  ( ( (  _I  |`  F )  |`  G ) LIndF  W  <->  (  _I  |`  G ) LIndF  W ) )
1511, 14mpbid 201 . 2  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )  /\  G  C_  F )  ->  (  _I  |`  G ) LIndF 
W )
161islinds 27279 . . 3  |-  ( W  e.  LMod  ->  ( G  e.  (LIndS `  W
)  <->  ( G  C_  ( Base `  W )  /\  (  _I  |`  G ) LIndF 
W ) ) )
17163ad2ant1 976 . 2  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )  /\  G  C_  F )  ->  ( G  e.  (LIndS `  W )  <->  ( G  C_  ( Base `  W )  /\  (  _I  |`  G ) LIndF  W
) ) )
186, 15, 17mpbir2and 888 1  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )  /\  G  C_  F )  ->  G  e.  (LIndS `  W ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684    C_ wss 3152   class class class wbr 4023    _I cid 4304    |` cres 4691   ` cfv 5255   Basecbs 13148   LModclmod 15627   LIndF clindf 27274  LIndSclinds 27275
This theorem is referenced by:  islinds4  27305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-slot 13152  df-base 13153  df-0g 13404  df-mnd 14367  df-grp 14489  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lindf 27276  df-linds 27277
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