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Mirrors > Home > MPE Home > Th. List > lmodring | Structured version Visualization version GIF version |
Description: The scalar component of a left module is a ring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmodring.1 | β’ πΉ = (Scalarβπ) |
Ref | Expression |
---|---|
lmodring | β’ (π β LMod β πΉ β Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
2 | eqid 2736 | . . 3 β’ (+gβπ) = (+gβπ) | |
3 | eqid 2736 | . . 3 β’ ( Β·π βπ) = ( Β·π βπ) | |
4 | lmodring.1 | . . 3 β’ πΉ = (Scalarβπ) | |
5 | eqid 2736 | . . 3 β’ (BaseβπΉ) = (BaseβπΉ) | |
6 | eqid 2736 | . . 3 β’ (+gβπΉ) = (+gβπΉ) | |
7 | eqid 2736 | . . 3 β’ (.rβπΉ) = (.rβπΉ) | |
8 | eqid 2736 | . . 3 β’ (1rβπΉ) = (1rβπΉ) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | islmod 20326 | . 2 β’ (π β LMod β (π β Grp β§ πΉ β Ring β§ βπ β (BaseβπΉ)βπ β (BaseβπΉ)βπ₯ β (Baseβπ)βπ€ β (Baseβπ)(((π( Β·π βπ)π€) β (Baseβπ) β§ (π( Β·π βπ)(π€(+gβπ)π₯)) = ((π( Β·π βπ)π€)(+gβπ)(π( Β·π βπ)π₯)) β§ ((π(+gβπΉ)π)( Β·π βπ)π€) = ((π( Β·π βπ)π€)(+gβπ)(π( Β·π βπ)π€))) β§ (((π(.rβπΉ)π)( Β·π βπ)π€) = (π( Β·π βπ)(π( Β·π βπ)π€)) β§ ((1rβπΉ)( Β·π βπ)π€) = π€)))) |
10 | 9 | simp2bi 1146 | 1 β’ (π β LMod β πΉ β Ring) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βwral 3064 βcfv 6496 (class class class)co 7357 Basecbs 17083 +gcplusg 17133 .rcmulr 17134 Scalarcsca 17136 Β·π cvsca 17137 Grpcgrp 18748 1rcur 19913 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-lmod 20324 |
This theorem is referenced by: lmodfgrp 20331 lmodmcl 20334 lmod0cl 20348 lmod1cl 20349 lmod0vs 20355 lmodvs0 20356 lmodvsmmulgdi 20357 lmodvsneg 20366 lmodsubvs 20378 lmodsubdi 20379 lmodsubdir 20380 lssvnegcl 20417 islss3 20420 pwslmod 20431 lmodvsinv 20497 islmhm2 20499 lbsind2 20542 lspsneq 20583 lspexch 20590 ip2subdi 21048 isphld 21058 ocvlss 21076 frlmup1 21204 frlmup2 21205 frlmup3 21206 frlmup4 21207 islindf5 21245 lmisfree 21248 asclghm 21286 ascl1 21288 ascldimul 21291 tlmtgp 23547 clmring 24433 lmodslmd 32039 imaslmod 32145 linds2eq 32168 lindsadd 36071 lfl0 37527 lfladd 37528 lflsub 37529 lfl0f 37531 lfladdcl 37533 lfladdcom 37534 lfladdass 37535 lfladd0l 37536 lflnegcl 37537 lflnegl 37538 lflvscl 37539 lflvsdi1 37540 lflvsdi2 37541 lflvsass 37543 lfl0sc 37544 lflsc0N 37545 lfl1sc 37546 lkrlss 37557 eqlkr 37561 eqlkr3 37563 lkrlsp 37564 ldualvsass 37603 lduallmodlem 37614 ldualvsubcl 37618 ldualvsubval 37619 lkrin 37626 dochfl1 39939 lcfl7lem 39962 lclkrlem2m 39982 lclkrlem2o 39984 lclkrlem2p 39985 lcfrlem1 40005 lcfrlem2 40006 lcfrlem3 40007 lcfrlem29 40034 lcfrlem33 40038 lcdvsubval 40081 mapdpglem30 40165 baerlem3lem1 40170 baerlem5alem1 40171 baerlem5blem1 40172 baerlem5blem2 40175 hgmapval1 40356 hdmapinvlem3 40383 hdmapinvlem4 40384 hdmapglem5 40385 hgmapvvlem1 40386 hdmapglem7b 40391 hdmapglem7 40392 lvecring 40713 prjspertr 40929 lmod0rng 46156 linc0scn0 46494 linc1 46496 lincscm 46501 lincscmcl 46503 el0ldep 46537 lindsrng01 46539 lindszr 46540 ldepsprlem 46543 ldepspr 46544 lincresunit3lem3 46545 lincresunitlem1 46546 lincresunitlem2 46547 lincresunit2 46549 lincresunit3lem1 46550 |
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