lmodfgrp - Metamath Proof Explorer

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Theorem lmodfgrp 19645

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 19644	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring)
3		ringgrp 19304	. 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 1537 ∈ wcel 2114 ‘cfv 6357 Scalarcsca 16570 Grpcgrp 18105 Ringcrg 19299 LModclmod 19636

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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-ring 19301 df-lmod 19638

This theorem is referenced by: lmodacl 19647 lmodsn0 19649 lmodvneg1 19679 lssvsubcl 19717 lspsnneg 19780 lvecvscan2 19886 lspexch 19903 lspsolvlem 19916 ascl0 20115 ipsubdir 20788 ipsubdi 20789 ip2eq 20799 ocvlss 20818 lsmcss 20838 islindf4 20984 clmfgrp 23677 lmodvslmhm 30690 lflmul 36206 lkrlss 36233 eqlkr 36237 lkrlsp 36240 lshpkrlem1 36248 ldualvsubval 36295 lcfrlem1 38680 lcdvsubval 38756 lmodvsmdi 44437 lincsum 44491 lincsumcl 44493 lincext1 44516 lindslinindsimp1 44519 lindslinindimp2lem1 44520 lindslinindsimp2lem5 44524 ldepsprlem 44534 ldepspr 44535 lincresunit3lem3 44536 lincresunit3lem1 44541 lincresunit3lem2 44542 lincresunit3 44543

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