Theorem lmodfgrp 20230

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
lmodring.1	⊢ 𝐹 = (Scalar‘𝑊)

Assertion

Ref	Expression
lmodfgrp	⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp)

Proof of Theorem lmodfgrp

Step	Hyp	Ref	Expression
1		lmodring.1	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
2	1	lmodring 20229	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring)
3		ringgrp 19875	. 2 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp)
4	2, 3	syl 17	1 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  ‘cfv 6473  Scalarcsca 17054  Grpcgrp 18665  Ringcrg 19870  LModclmod 20221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-nul 5247

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rab 3404  df-v 3443  df-sbc 3727  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-iota 6425  df-fv 6481  df-ov 7332  df-ring 19872  df-lmod 20223

This theorem is referenced by:  lmodacl  20232  lmodsn0  20234  lmodvneg1  20264  lssvsubcl  20303  lspsnneg  20366  lvecvscan2  20472  lspexch  20489  lspsolvlem  20502  ipsubdir  20945  ipsubdi  20946  ip2eq  20956  ocvlss  20975  lsmcss  20995  islindf4  21143  ascl0  21186  clmfgrp  24332  lmodvslmhm  31538  lflmul  37328  lkrlss  37355  eqlkr  37359  lkrlsp  37362  lshpkrlem1  37370  ldualvsubval  37417  lcfrlem1  39803  lcdvsubval  39879  lmodvsmdi  46058  lincsum  46110  lincsumcl  46112  lincext1  46135  lindslinindsimp1  46138  lindslinindimp2lem1  46139  lindslinindsimp2lem5  46143  ldepsprlem  46153  ldepspr  46154  lincresunit3lem3  46155  lincresunit3lem1  46160  lincresunit3lem2  46161  lincresunit3  46162