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Theorem lmodbn0 15637
Description: The base set of a left module is nonempty. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypothesis
Ref Expression
lmodbn0.b  |-  B  =  ( Base `  W
)
Assertion
Ref Expression
lmodbn0  |-  ( W  e.  LMod  ->  B  =/=  (/) )

Proof of Theorem lmodbn0
StepHypRef Expression
1 lmodgrp 15634 . 2  |-  ( W  e.  LMod  ->  W  e. 
Grp )
2 lmodbn0.b . . 3  |-  B  =  ( Base `  W
)
32grpbn0 14511 . 2  |-  ( W  e.  Grp  ->  B  =/=  (/) )
41, 3syl 15 1  |-  ( W  e.  LMod  ->  B  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   ` cfv 5255   Basecbs 13148   Grpcgrp 14362   LModclmod 15627
This theorem is referenced by:  lss1  15696
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
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-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  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-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-lmod 15629
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