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Theorem lmod0cl 15866
Description: The ring zero in a left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
lmod0cl.f  |-  F  =  (Scalar `  W )
lmod0cl.k  |-  K  =  ( Base `  F
)
lmod0cl.z  |-  .0.  =  ( 0g `  F )
Assertion
Ref Expression
lmod0cl  |-  ( W  e.  LMod  ->  .0.  e.  K )

Proof of Theorem lmod0cl
StepHypRef Expression
1 lmod0cl.f . . 3  |-  F  =  (Scalar `  W )
21lmodrng 15845 . 2  |-  ( W  e.  LMod  ->  F  e. 
Ring )
3 lmod0cl.k . . 3  |-  K  =  ( Base `  F
)
4 lmod0cl.z . . 3  |-  .0.  =  ( 0g `  F )
53, 4rng0cl 15572 . 2  |-  ( F  e.  Ring  ->  .0.  e.  K )
62, 5syl 15 1  |-  ( W  e.  LMod  ->  .0.  e.  K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    e. wcel 1715   ` cfv 5358   Basecbs 13356  Scalarcsca 13419   0gc0g 13610   Ringcrg 15547   LModclmod 15837
This theorem is referenced by:  lss1d  15930  lspsolvlem  16105  iporthcom  16756  lfl0f  29330  lfl1dim  29382  lfl1dim2N  29383  lkrss2N  29430  baerlem5blem1  31970  hdmap14lem2a  32131  hdmap14lem4a  32135  hdmap14lem6  32137  hgmapval0  32156  hgmapeq0  32168
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-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-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  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-iota 5322  df-fun 5360  df-fv 5366  df-ov 5984  df-riota 6446  df-0g 13614  df-mnd 14577  df-grp 14699  df-rng 15550  df-lmod 15839
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