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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 21201 | . . 3 ⊢ (Base‘𝑊) = (Base‘(ringLMod‘𝑊)) | |
| 3 | 1, 2 | eqtri 2764 | . 2 ⊢ 𝐵 = (Base‘(ringLMod‘𝑊)) | 
| 4 | lidlss.i | . . 3 ⊢ 𝐼 = (LIdeal‘𝑊) | |
| 5 | lidlval 21221 | . . 3 ⊢ (LIdeal‘𝑊) = (LSubSp‘(ringLMod‘𝑊)) | |
| 6 | 4, 5 | eqtri 2764 | . 2 ⊢ 𝐼 = (LSubSp‘(ringLMod‘𝑊)) | 
| 7 | 3, 6 | lssss 20935 | 1 ⊢ (𝑈 ∈ 𝐼 → 𝑈 ⊆ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ‘cfv 6560 Basecbs 17248 LSubSpclss 20930 ringLModcrglmod 21172 LIdealclidl 21217 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-sca 17314 df-vsca 17315 df-ip 17316 df-lss 20931 df-sra 21173 df-rgmod 21174 df-lidl 21219 | 
| This theorem is referenced by: lidlbas 21225 lidlsubg 21234 lidl1el 21237 drngnidl 21254 2idlss 21273 2idlcpblrng 21282 rng2idl1cntr 21316 lpigen 21346 zringlpirlem1 21474 zringlpirlem3 21476 zndvds 21569 ig1peu 26215 ig1pdvds 26220 ig1prsp 26221 ply1lpir 26222 rspidlid 33404 ringlsmss1 33425 ringlsmss2 33426 lsmidl 33430 intlidl 33449 0ringidl 33450 elrspunidl 33457 elrspunsn 33458 rhmimaidl 33461 prmidl2 33470 idlmulssprm 33471 ssdifidllem 33485 ssdifidlprm 33487 mxidlprm 33499 ssmxidllem 33502 opprqusmulr 33520 opprqus1r 33521 opprqusdrng 33522 qsdrngilem 33523 qsdrngi 33524 qsdrnglem2 33525 idlsrgmulrcl 33539 idlsrgmulrss1 33540 idlsrgmulrss2 33541 dfufd2 33579 ig1pmindeg 33623 minplycl 33750 irngnminplynz 33756 zarcls1 33869 zarclsun 33870 zarclsiin 33871 zarclsint 33872 zarcmplem 33881 rhmpreimacnlem 33884 rspssbasd 35646 hbtlem2 43141 hbtlem4 43143 hbtlem5 43145 hbtlem6 43146 hbt 43147 lidldomn1 48152 | 
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