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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islnr2 | Structured version Visualization version GIF version |
Description: Property of being a left-Noetherian ring in terms of finite generation of ideals (the usual "pure ring theory" definition). (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
islnr2.b | ⊢ 𝐵 = (Base‘𝑅) |
islnr2.u | ⊢ 𝑈 = (LIdeal‘𝑅) |
islnr2.n | ⊢ 𝑁 = (RSpan‘𝑅) |
Ref | Expression |
---|---|
islnr2 | ⊢ (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ ∀𝑖 ∈ 𝑈 ∃𝑔 ∈ (𝒫 𝐵 ∩ Fin)𝑖 = (𝑁‘𝑔))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islnr 43100 | . 2 ⊢ (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ (ringLMod‘𝑅) ∈ LNoeM)) | |
2 | rlmlmod 21228 | . . . 4 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
3 | islnr2.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
4 | rlmbas 21218 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
5 | 3, 4 | eqtri 2763 | . . . . . 6 ⊢ 𝐵 = (Base‘(ringLMod‘𝑅)) |
6 | islnr2.u | . . . . . . 7 ⊢ 𝑈 = (LIdeal‘𝑅) | |
7 | lidlval 21238 | . . . . . . 7 ⊢ (LIdeal‘𝑅) = (LSubSp‘(ringLMod‘𝑅)) | |
8 | 6, 7 | eqtri 2763 | . . . . . 6 ⊢ 𝑈 = (LSubSp‘(ringLMod‘𝑅)) |
9 | islnr2.n | . . . . . . 7 ⊢ 𝑁 = (RSpan‘𝑅) | |
10 | rspval 21239 | . . . . . . 7 ⊢ (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅)) | |
11 | 9, 10 | eqtri 2763 | . . . . . 6 ⊢ 𝑁 = (LSpan‘(ringLMod‘𝑅)) |
12 | 5, 8, 11 | islnm2 43067 | . . . . 5 ⊢ ((ringLMod‘𝑅) ∈ LNoeM ↔ ((ringLMod‘𝑅) ∈ LMod ∧ ∀𝑖 ∈ 𝑈 ∃𝑔 ∈ (𝒫 𝐵 ∩ Fin)𝑖 = (𝑁‘𝑔))) |
13 | 12 | baib 535 | . . . 4 ⊢ ((ringLMod‘𝑅) ∈ LMod → ((ringLMod‘𝑅) ∈ LNoeM ↔ ∀𝑖 ∈ 𝑈 ∃𝑔 ∈ (𝒫 𝐵 ∩ Fin)𝑖 = (𝑁‘𝑔))) |
14 | 2, 13 | syl 17 | . . 3 ⊢ (𝑅 ∈ Ring → ((ringLMod‘𝑅) ∈ LNoeM ↔ ∀𝑖 ∈ 𝑈 ∃𝑔 ∈ (𝒫 𝐵 ∩ Fin)𝑖 = (𝑁‘𝑔))) |
15 | 14 | pm5.32i 574 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (ringLMod‘𝑅) ∈ LNoeM) ↔ (𝑅 ∈ Ring ∧ ∀𝑖 ∈ 𝑈 ∃𝑔 ∈ (𝒫 𝐵 ∩ Fin)𝑖 = (𝑁‘𝑔))) |
16 | 1, 15 | bitri 275 | 1 ⊢ (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ ∀𝑖 ∈ 𝑈 ∃𝑔 ∈ (𝒫 𝐵 ∩ Fin)𝑖 = (𝑁‘𝑔))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ∩ cin 3962 𝒫 cpw 4605 ‘cfv 6563 Fincfn 8984 Basecbs 17245 Ringcrg 20251 LModclmod 20875 LSubSpclss 20947 LSpanclspn 20987 ringLModcrglmod 21189 LIdealclidl 21234 RSpancrsp 21235 LNoeMclnm 43064 LNoeRclnr 43098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-mgp 20153 df-ur 20200 df-ring 20253 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-lfig 43057 df-lnm 43065 df-lnr 43099 |
This theorem is referenced by: islnr3 43104 lnr2i 43105 lpirlnr 43106 hbt 43119 |
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