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Mirrors > Home > MPE Home > Th. List > lmodfgrp | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
lmodring.1 | β’ πΉ = (Scalarβπ) |
Ref | Expression |
---|---|
lmodfgrp | β’ (π β LMod β πΉ β Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodring.1 | . . 3 β’ πΉ = (Scalarβπ) | |
2 | 1 | lmodring 20622 | . 2 β’ (π β LMod β πΉ β Ring) |
3 | ringgrp 20132 | . 2 β’ (πΉ β Ring β πΉ β Grp) | |
4 | 2, 3 | syl 17 | 1 β’ (π β LMod β πΉ β Grp) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1539 β wcel 2104 βcfv 6542 Scalarcsca 17204 Grpcgrp 18855 Ringcrg 20127 LModclmod 20614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-nul 5305 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rab 3431 df-v 3474 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6494 df-fv 6550 df-ov 7414 df-ring 20129 df-lmod 20616 |
This theorem is referenced by: lmodacl 20626 lmodsn0 20628 lmodvneg1 20659 lssvsubcl 20698 lspsnneg 20761 lvecvscan2 20870 lspexch 20887 lspsolvlem 20900 ipsubdir 21414 ipsubdi 21415 ip2eq 21425 ocvlss 21444 lsmcss 21464 islindf4 21612 ascl0 21657 clmfgrp 24818 lmodvslmhm 32472 lflmul 38241 lkrlss 38268 eqlkr 38272 lkrlsp 38275 lshpkrlem1 38283 ldualvsubval 38330 lcfrlem1 40716 lcdvsubval 40792 lmodvsmdi 47146 lincsum 47197 lincsumcl 47199 lincext1 47222 lindslinindsimp1 47225 lindslinindimp2lem1 47226 lindslinindsimp2lem5 47230 ldepsprlem 47240 ldepspr 47241 lincresunit3lem3 47242 lincresunit3lem1 47247 lincresunit3lem2 47248 lincresunit3 47249 |
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