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Mirrors > Home > MPE Home > Th. List > lmodfgrp | Structured version Visualization version Unicode 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 | Scalar |
Ref | Expression |
---|---|
lmodfgrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodring.1 | . . 3 Scalar | |
2 | 1 | lmodring 18871 | . 2 |
3 | ringgrp 18552 | . 2 | |
4 | 2, 3 | syl 17 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 cfv 5888 Scalarcsca 15944 cgrp 17422 crg 18547 clmod 18863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-ring 18549 df-lmod 18865 |
This theorem is referenced by: lmodacl 18874 lmodsn0 18876 lmodvneg1 18906 lssvsubcl 18944 lspsnneg 19006 lvecvscan2 19112 lspexch 19129 lspsolvlem 19142 ipsubdir 19987 ipsubdi 19988 ip2eq 19998 ocvlss 20016 lsmcss 20036 islindf4 20177 clmfgrp 22871 lflmul 34355 lkrlss 34382 eqlkr 34386 lkrlsp 34389 lshpkrlem1 34397 ldualvsubval 34444 lcfrlem1 36831 lcdvsubval 36907 lmodvsmdi 42163 ascl0 42165 lincsum 42218 lincsumcl 42220 lincext1 42243 lindslinindsimp1 42246 lindslinindimp2lem1 42247 lindslinindsimp2lem5 42251 ldepsprlem 42261 ldepspr 42262 lincresunit3lem3 42263 lincresunit3lem1 42268 lincresunit3lem2 42269 lincresunit3 42270 |
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