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Theorem lmodfgrp 15636
Description: The scalar component of a left module is an additive group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypothesis
Ref Expression
lmodrng.1  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
lmodfgrp  |-  ( W  e.  LMod  ->  F  e. 
Grp )

Proof of Theorem lmodfgrp
StepHypRef Expression
1 lmodrng.1 . . 3  |-  F  =  (Scalar `  W )
21lmodrng 15635 . 2  |-  ( W  e.  LMod  ->  F  e. 
Ring )
3 rnggrp 15346 . 2  |-  ( F  e.  Ring  ->  F  e. 
Grp )
42, 3syl 15 1  |-  ( W  e.  LMod  ->  F  e. 
Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255  Scalarcsca 13211   Grpcgrp 14362   Ringcrg 15337   LModclmod 15627
This theorem is referenced by:  lmodacl  15638  lmodsn0  15640  lmodvneg1  15667  lssvsubcl  15701  lspsnneg  15763  lvecvscan2  15865  lspexch  15882  lspsolvlem  15895  ipsubdir  16546  ipsubdi  16547  ip2eq  16557  ocvlss  16572  lsmcss  16592  clmfgrp  18569  islindf4  27308  lflmul  29258  lkrlss  29285  eqlkr  29289  lkrlsp  29292  lshpkrlem1  29300  ldualvsubval  29347  lcfrlem1  31732  lcdvsubval  31808
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  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-iota 5219  df-fv 5263  df-ov 5861  df-rng 15340  df-lmod 15629
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