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Theorem lindsss 27272
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 2437 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
21linds1 27258 . . . . 5  |-  ( F  e.  (LIndS `  W
)  ->  F  C_  ( Base `  W ) )
32adantl 454 . . . 4  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )
)  ->  F  C_  ( Base `  W ) )
4 sstr2 3356 . . . 4  |-  ( G 
C_  F  ->  ( F  C_  ( Base `  W
)  ->  G  C_  ( Base `  W ) ) )
53, 4syl5com 29 . . 3  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )
)  ->  ( G  C_  F  ->  G  C_  ( Base `  W ) ) )
653impia 1151 . 2  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )  /\  G  C_  F )  ->  G  C_  ( Base `  W ) )
7 simp1 958 . . . 4  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )  /\  G  C_  F )  ->  W  e.  LMod )
8 linds2 27259 . . . . 5  |-  ( F  e.  (LIndS `  W
)  ->  (  _I  |`  F ) LIndF  W )
983ad2ant2 980 . . . 4  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )  /\  G  C_  F )  ->  (  _I  |`  F ) LIndF 
W )
10 lindfres 27271 . . . 4  |-  ( ( W  e.  LMod  /\  (  _I  |`  F ) LIndF  W
)  ->  ( (  _I  |`  F )  |`  G ) LIndF  W )
117, 9, 10syl2anc 644 . . 3  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )  /\  G  C_  F )  ->  ( (  _I  |`  F )  |`  G ) LIndF 
W )
12 resabs1 5176 . . . . 5  |-  ( G 
C_  F  ->  (
(  _I  |`  F )  |`  G )  =  (  _I  |`  G )
)
1312breq1d 4223 . . . 4  |-  ( G 
C_  F  ->  (
( (  _I  |`  F )  |`  G ) LIndF  W  <->  (  _I  |`  G ) LIndF  W ) )
14133ad2ant3 981 . . 3  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )  /\  G  C_  F )  ->  ( ( (  _I  |`  F )  |`  G ) LIndF  W  <->  (  _I  |`  G ) LIndF  W ) )
1511, 14mpbid 203 . 2  |-  ( ( W  e.  LMod  /\  F  e.  (LIndS `  W )  /\  G  C_  F )  ->  (  _I  |`  G ) LIndF 
W )
161islinds 27257 . . 3  |-  ( W  e.  LMod  ->  ( G  e.  (LIndS `  W
)  <->  ( G  C_  ( Base `  W )  /\  (  _I  |`  G ) LIndF 
W ) ) )
17163ad2ant1 979 . 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 890 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 178    /\ wa 360    /\ w3a 937    e. wcel 1726    C_ wss 3321   class class class wbr 4213    _I cid 4494    |` cres 4881   ` cfv 5455   Basecbs 13470   LModclmod 15951   LIndF clindf 27252  LIndSclinds 27253
This theorem is referenced by:  islinds4  27283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-riota 6550  df-slot 13474  df-base 13475  df-0g 13728  df-mnd 14691  df-grp 14813  df-lmod 15953  df-lss 16010  df-lsp 16049  df-lindf 27254  df-linds 27255
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