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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 20330 | . 2 β’ (π β LMod β πΉ β Ring) |
3 | ringgrp 19969 | . 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 1541 β wcel 2106 βcfv 6496 Scalarcsca 17136 Grpcgrp 18748 Ringcrg 19964 LModclmod 20322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-nul 5263 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2944 df-ral 3065 df-rab 3408 df-v 3447 df-sbc 3740 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-iota 6448 df-fv 6504 df-ov 7360 df-ring 19966 df-lmod 20324 |
This theorem is referenced by: lmodacl 20333 lmodsn0 20335 lmodvneg1 20365 lssvsubcl 20404 lspsnneg 20467 lvecvscan2 20573 lspexch 20590 lspsolvlem 20603 ipsubdir 21046 ipsubdi 21047 ip2eq 21057 ocvlss 21076 lsmcss 21096 islindf4 21244 ascl0 21287 clmfgrp 24434 lmodvslmhm 31892 lflmul 37530 lkrlss 37557 eqlkr 37561 lkrlsp 37564 lshpkrlem1 37572 ldualvsubval 37619 lcfrlem1 40005 lcdvsubval 40081 lmodvsmdi 46448 lincsum 46500 lincsumcl 46502 lincext1 46525 lindslinindsimp1 46528 lindslinindimp2lem1 46529 lindslinindsimp2lem5 46533 ldepsprlem 46543 ldepspr 46544 lincresunit3lem3 46545 lincresunit3lem1 46550 lincresunit3lem2 46551 lincresunit3 46552 |
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