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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 | ⊢ 𝐹 = (Scalar‘𝑊) |
| Ref | Expression |
|---|---|
| lmodring | ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 2 | eqid 2737 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
| 3 | eqid 2737 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
| 4 | lmodring.1 | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
| 5 | eqid 2737 | . . 3 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
| 6 | eqid 2737 | . . 3 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
| 7 | eqid 2737 | . . 3 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
| 8 | eqid 2737 | . . 3 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
| 9 | 1, 2, 3, 4, 5, 6, 7, 8 | islmod 20862 | . 2 ⊢ (𝑊 ∈ LMod ↔ (𝑊 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ (Base‘𝐹)∀𝑟 ∈ (Base‘𝐹)∀𝑥 ∈ (Base‘𝑊)∀𝑤 ∈ (Base‘𝑊)(((𝑟( ·𝑠 ‘𝑊)𝑤) ∈ (Base‘𝑊) ∧ (𝑟( ·𝑠 ‘𝑊)(𝑤(+g‘𝑊)𝑥)) = ((𝑟( ·𝑠 ‘𝑊)𝑤)(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑥)) ∧ ((𝑞(+g‘𝐹)𝑟)( ·𝑠 ‘𝑊)𝑤) = ((𝑞( ·𝑠 ‘𝑊)𝑤)(+g‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑤))) ∧ (((𝑞(.r‘𝐹)𝑟)( ·𝑠 ‘𝑊)𝑤) = (𝑞( ·𝑠 ‘𝑊)(𝑟( ·𝑠 ‘𝑊)𝑤)) ∧ ((1r‘𝐹)( ·𝑠 ‘𝑊)𝑤) = 𝑤)))) |
| 10 | 9 | simp2bi 1147 | 1 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 Grpcgrp 18951 1rcur 20178 Ringcrg 20230 LModclmod 20858 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-lmod 20860 |
| This theorem is referenced by: lmodfgrp 20867 lmodmcl 20871 lmod0cl 20886 lmod1cl 20887 lmod0vs 20893 lmodvs0 20894 lmodvsmmulgdi 20895 lmodvsneg 20904 lmodsubvs 20916 lmodsubdi 20917 lmodsubdir 20918 lssvnegcl 20954 islss3 20957 pwslmod 20968 lmodvsinv 21035 islmhm2 21037 lbsind2 21080 lspsneq 21124 lspexch 21131 ip2subdi 21662 isphld 21672 ocvlss 21690 frlmup1 21818 frlmup2 21819 frlmup3 21820 frlmup4 21821 islindf5 21859 lmisfree 21862 assasca 21882 asclghm 21903 ascl1 21905 ascldimul 21908 tlmtgp 24204 clmring 25103 lmodslmd 33210 imaslmod 33381 linds2eq 33409 lindsadd 37620 lfl0 39066 lfladd 39067 lflsub 39068 lfl0f 39070 lfladdcl 39072 lfladdcom 39073 lfladdass 39074 lfladd0l 39075 lflnegcl 39076 lflnegl 39077 lflvscl 39078 lflvsdi1 39079 lflvsdi2 39080 lflvsass 39082 lfl0sc 39083 lflsc0N 39084 lfl1sc 39085 lkrlss 39096 eqlkr 39100 eqlkr3 39102 lkrlsp 39103 ldualvsass 39142 lduallmodlem 39153 ldualvsubcl 39157 ldualvsubval 39158 lkrin 39165 dochfl1 41478 lcfl7lem 41501 lclkrlem2m 41521 lclkrlem2o 41523 lclkrlem2p 41524 lcfrlem1 41544 lcfrlem2 41545 lcfrlem3 41546 lcfrlem29 41573 lcfrlem33 41577 lcdvsubval 41620 mapdpglem30 41704 baerlem3lem1 41709 baerlem5alem1 41710 baerlem5blem1 41711 baerlem5blem2 41714 hgmapval1 41895 hdmapinvlem3 41922 hdmapinvlem4 41923 hdmapglem5 41924 hgmapvvlem1 41925 hdmapglem7b 41930 hdmapglem7 41931 lvecring 42548 prjspertr 42615 lmod0rng 48145 linc0scn0 48340 linc1 48342 lincscm 48347 lincscmcl 48349 el0ldep 48383 lindsrng01 48385 lindszr 48386 ldepsprlem 48389 ldepspr 48390 lincresunit3lem3 48391 lincresunitlem1 48392 lincresunitlem2 48393 lincresunit2 48395 lincresunit3lem1 48396 |
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