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Mirrors > Home > MPE Home > Th. List > lidlss | Structured version Visualization version GIF version |
Description: An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
lidlss.b | ⊢ 𝐵 = (Base‘𝑊) |
lidlss.i | ⊢ 𝐼 = (LIdeal‘𝑊) |
Ref | Expression |
---|---|
lidlss | ⊢ (𝑈 ∈ 𝐼 → 𝑈 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lidlss.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
2 | rlmbas 21098 | . . 3 ⊢ (Base‘𝑊) = (Base‘(ringLMod‘𝑊)) | |
3 | 1, 2 | eqtri 2753 | . 2 ⊢ 𝐵 = (Base‘(ringLMod‘𝑊)) |
4 | lidlss.i | . . 3 ⊢ 𝐼 = (LIdeal‘𝑊) | |
5 | lidlval 21118 | . . 3 ⊢ (LIdeal‘𝑊) = (LSubSp‘(ringLMod‘𝑊)) | |
6 | 4, 5 | eqtri 2753 | . 2 ⊢ 𝐼 = (LSubSp‘(ringLMod‘𝑊)) |
7 | 3, 6 | lssss 20832 | 1 ⊢ (𝑈 ∈ 𝐼 → 𝑈 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 ‘cfv 6549 Basecbs 17183 LSubSpclss 20827 ringLModcrglmod 21069 LIdealclidl 21114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-sca 17252 df-vsca 17253 df-ip 17254 df-lss 20828 df-sra 21070 df-rgmod 21071 df-lidl 21116 |
This theorem is referenced by: lidlbas 21122 lidlsubg 21131 lidl1el 21134 drngnidl 21150 2idlss 21169 2idlcpblrng 21178 rng2idl1cntr 21212 lpigen 21242 zringlpirlem1 21405 zringlpirlem3 21407 zndvds 21500 ig1peu 26154 ig1pdvds 26159 ig1prsp 26160 ply1lpir 26161 rspidlid 33187 ringlsmss1 33208 ringlsmss2 33209 lsmidl 33213 intlidl 33232 0ringidl 33233 elrspunidl 33240 elrspunsn 33241 rhmimaidl 33244 prmidl2 33253 idlmulssprm 33254 ssdifidllem 33268 ssdifidlprm 33270 mxidlprm 33282 ssmxidllem 33285 opprqusmulr 33303 opprqus1r 33304 opprqusdrng 33305 qsdrngilem 33306 qsdrngi 33307 qsdrnglem2 33308 idlsrgmulrcl 33322 idlsrgmulrss1 33323 idlsrgmulrss2 33324 dfufd2 33365 ig1pmindeg 33403 minplycl 33508 irngnminplynz 33513 zarcls1 33601 zarclsun 33602 zarclsiin 33603 zarclsint 33604 zarcmplem 33613 rhmpreimacnlem 33616 hbtlem2 42690 hbtlem4 42692 hbtlem5 42694 hbtlem6 42695 hbt 42696 lidldomn1 47479 |
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