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Mirrors > Home > MPE Home > Th. List > rlmbas | Structured version Visualization version GIF version |
Description: Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
Ref | Expression |
---|---|
rlmbas | ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlmval 20805 | . . . 4 ⊢ (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘(Base‘𝑅)) | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → (ringLMod‘𝑅) = ((subringAlg ‘𝑅)‘(Base‘𝑅))) |
3 | ssidd 4004 | . . 3 ⊢ (⊤ → (Base‘𝑅) ⊆ (Base‘𝑅)) | |
4 | 2, 3 | srabase 20784 | . 2 ⊢ (⊤ → (Base‘𝑅) = (Base‘(ringLMod‘𝑅))) |
5 | 4 | mptru 1548 | 1 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ⊤wtru 1542 ‘cfv 6540 Basecbs 17140 subringAlg csra 20773 ringLModcrglmod 20774 |
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-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-sca 17209 df-vsca 17210 df-ip 17211 df-sra 20777 df-rgmod 20778 |
This theorem is referenced by: rlmsub 20812 rlmlsm 20821 rlmvneg 20822 rlmscaf 20823 ixpsnbasval 20824 lidlss 20825 islidl 20826 lidl1 20837 lidlacs 20838 rspcl 20839 rspssid 20840 lidlrsppropd 20847 rspsn 20884 ipcl 21177 isphld 21198 phlpropd 21199 frlmbas 21301 frlmsubgval 21311 frlmgsum 21318 rlmnm 24197 cnrbas 24649 rspsnel 32472 elrsp 32474 lsmidl 32499 mxidlprm 32574 idlsrgmulrss1 32613 idlsrgmulrss2 32614 frlmsnic 41107 mhphf2 41167 islnr2 41841 |
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