lmodfgrp - Metamath Proof Explorer

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Theorem lmodfgrp 20132

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 20131	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring)
3		ringgrp 19788	. 2 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp)
4	2, 3	syl 17	1 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ‘cfv 6433 Scalarcsca 16965 Grpcgrp 18577 Ringcrg 19783 LModclmod 20123

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-nul 5230

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-iota 6391 df-fv 6441 df-ov 7278 df-ring 19785 df-lmod 20125

This theorem is referenced by: lmodacl 20134 lmodsn0 20136 lmodvneg1 20166 lssvsubcl 20205 lspsnneg 20268 lvecvscan2 20374 lspexch 20391 lspsolvlem 20404 ipsubdir 20847 ipsubdi 20848 ip2eq 20858 ocvlss 20877 lsmcss 20897 islindf4 21045 ascl0 21088 clmfgrp 24234 lmodvslmhm 31310 lflmul 37082 lkrlss 37109 eqlkr 37113 lkrlsp 37116 lshpkrlem1 37124 ldualvsubval 37171 lcfrlem1 39556 lcdvsubval 39632 lmodvsmdi 45718 lincsum 45770 lincsumcl 45772 lincext1 45795 lindslinindsimp1 45798 lindslinindimp2lem1 45799 lindslinindsimp2lem5 45803 ldepsprlem 45813 ldepspr 45814 lincresunit3lem3 45815 lincresunit3lem1 45820 lincresunit3lem2 45821 lincresunit3 45822

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