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Mirrors > Home > MPE Home > Th. List > rlmlmod | Structured version Visualization version GIF version |
Description: The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.) |
Ref | Expression |
---|---|
rlmlmod | ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlmval 19510 | . 2 ⊢ (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘(Base‘𝑅)) | |
2 | eqid 2797 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 2 | subrgid 19096 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ∈ (SubRing‘𝑅)) |
4 | eqid 2797 | . . . 4 ⊢ ((subringAlg ‘𝑅)‘(Base‘𝑅)) = ((subringAlg ‘𝑅)‘(Base‘𝑅)) | |
5 | 4 | sralmod 19506 | . . 3 ⊢ ((Base‘𝑅) ∈ (SubRing‘𝑅) → ((subringAlg ‘𝑅)‘(Base‘𝑅)) ∈ LMod) |
6 | 3, 5 | syl 17 | . 2 ⊢ (𝑅 ∈ Ring → ((subringAlg ‘𝑅)‘(Base‘𝑅)) ∈ LMod) |
7 | 1, 6 | syl5eqel 2880 | 1 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2157 ‘cfv 6099 Basecbs 16180 Ringcrg 18859 SubRingcsubrg 19090 LModclmod 19177 subringAlg csra 19487 ringLModcrglmod 19488 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-ndx 16183 df-slot 16184 df-base 16186 df-sets 16187 df-ress 16188 df-plusg 16276 df-mulr 16277 df-sca 16279 df-vsca 16280 df-ip 16281 df-0g 16413 df-mgm 17553 df-sgrp 17595 df-mnd 17606 df-grp 17737 df-subg 17900 df-mgp 18802 df-ur 18814 df-ring 18861 df-subrg 19092 df-lmod 19179 df-sra 19491 df-rgmod 19492 |
This theorem is referenced by: rlmlvec 19525 lidl0cl 19531 lidlacl 19532 lidlnegcl 19533 lidlmcl 19536 lidl0 19538 lidl1 19539 lidlacs 19540 rspcl 19541 rspssid 19542 rsp0 19544 rspssp 19545 mrcrsp 19546 rspsn 19573 isphld 20319 frlmlmod 20414 frlmlss 20416 frlm0 20419 frlmsubgval 20429 frlmgsum 20432 frlmsplit2 20433 cnrlmod 23266 recvs 23269 qcvs 23270 zclmncvs 23271 islnr2 38456 |
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