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Theorem clmfgrp 19096
Description: The scalar ring of a complex module is a group. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypothesis
Ref Expression
clm0.f  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
clmfgrp  |-  ( W  e. CMod  ->  F  e.  Grp )

Proof of Theorem clmfgrp
StepHypRef Expression
1 clmlmod 19092 . 2  |-  ( W  e. CMod  ->  W  e.  LMod )
2 clm0.f . . 3  |-  F  =  (Scalar `  W )
32lmodfgrp 15959 . 2  |-  ( W  e.  LMod  ->  F  e. 
Grp )
41, 3syl 16 1  |-  ( W  e. CMod  ->  F  e.  Grp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5454  Scalarcsca 13532   Grpcgrp 14685   LModclmod 15950  CModcclm 19087
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-rng 15663  df-lmod 15952  df-clm 19088
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