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Theorem lmodfgrp 15846
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 15845 . 2  |-  ( W  e.  LMod  ->  F  e. 
Ring )
3 rnggrp 15556 . 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 1647    e. wcel 1715   ` cfv 5358  Scalarcsca 13419   Grpcgrp 14572   Ringcrg 15547   LModclmod 15837
This theorem is referenced by:  lmodacl  15848  lmodsn0  15850  lmodvneg1  15877  lssvsubcl  15911  lspsnneg  15973  lvecvscan2  16075  lspexch  16092  lspsolvlem  16105  ipsubdir  16763  ipsubdi  16764  ip2eq  16774  ocvlss  16789  lsmcss  16809  clmfgrp  18784  islindf4  26814  lflmul  29329  lkrlss  29356  eqlkr  29360  lkrlsp  29363  lshpkrlem1  29371  ldualvsubval  29418  lcfrlem1  31803  lcdvsubval  31879
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-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-nul 4251
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-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  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-iota 5322  df-fv 5366  df-ov 5984  df-rng 15550  df-lmod 15839
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