Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > islnr3 | Structured version Visualization version GIF version |
Description: Relate left-Noetherian rings to Noetherian-type closure property of the left ideal system. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
Ref | Expression |
---|---|
islnr3.b | ⊢ 𝐵 = (Base‘𝑅) |
islnr3.u | ⊢ 𝑈 = (LIdeal‘𝑅) |
Ref | Expression |
---|---|
islnr3 | ⊢ (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ 𝑈 ∈ (NoeACS‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islnr3.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
2 | islnr3.u | . . 3 ⊢ 𝑈 = (LIdeal‘𝑅) | |
3 | eqid 2823 | . . 3 ⊢ (RSpan‘𝑅) = (RSpan‘𝑅) | |
4 | 1, 2, 3 | islnr2 39721 | . 2 ⊢ (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ ∀𝑥 ∈ 𝑈 ∃𝑦 ∈ (𝒫 𝐵 ∩ Fin)𝑥 = ((RSpan‘𝑅)‘𝑦))) |
5 | eqid 2823 | . . . . . . . . . 10 ⊢ (mrCls‘𝑈) = (mrCls‘𝑈) | |
6 | 2, 3, 5 | mrcrsp 20002 | . . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (RSpan‘𝑅) = (mrCls‘𝑈)) |
7 | 6 | fveq1d 6674 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → ((RSpan‘𝑅)‘𝑦) = ((mrCls‘𝑈)‘𝑦)) |
8 | 7 | eqeq2d 2834 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝑥 = ((RSpan‘𝑅)‘𝑦) ↔ 𝑥 = ((mrCls‘𝑈)‘𝑦))) |
9 | 8 | rexbidv 3299 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (∃𝑦 ∈ (𝒫 𝐵 ∩ Fin)𝑥 = ((RSpan‘𝑅)‘𝑦) ↔ ∃𝑦 ∈ (𝒫 𝐵 ∩ Fin)𝑥 = ((mrCls‘𝑈)‘𝑦))) |
10 | 9 | ralbidv 3199 | . . . . 5 ⊢ (𝑅 ∈ Ring → (∀𝑥 ∈ 𝑈 ∃𝑦 ∈ (𝒫 𝐵 ∩ Fin)𝑥 = ((RSpan‘𝑅)‘𝑦) ↔ ∀𝑥 ∈ 𝑈 ∃𝑦 ∈ (𝒫 𝐵 ∩ Fin)𝑥 = ((mrCls‘𝑈)‘𝑦))) |
11 | 1, 2 | lidlacs 19996 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (ACS‘𝐵)) |
12 | 11 | biantrurd 535 | . . . . 5 ⊢ (𝑅 ∈ Ring → (∀𝑥 ∈ 𝑈 ∃𝑦 ∈ (𝒫 𝐵 ∩ Fin)𝑥 = ((mrCls‘𝑈)‘𝑦) ↔ (𝑈 ∈ (ACS‘𝐵) ∧ ∀𝑥 ∈ 𝑈 ∃𝑦 ∈ (𝒫 𝐵 ∩ Fin)𝑥 = ((mrCls‘𝑈)‘𝑦)))) |
13 | 10, 12 | bitrd 281 | . . . 4 ⊢ (𝑅 ∈ Ring → (∀𝑥 ∈ 𝑈 ∃𝑦 ∈ (𝒫 𝐵 ∩ Fin)𝑥 = ((RSpan‘𝑅)‘𝑦) ↔ (𝑈 ∈ (ACS‘𝐵) ∧ ∀𝑥 ∈ 𝑈 ∃𝑦 ∈ (𝒫 𝐵 ∩ Fin)𝑥 = ((mrCls‘𝑈)‘𝑦)))) |
14 | 5 | isnacs 39308 | . . . 4 ⊢ (𝑈 ∈ (NoeACS‘𝐵) ↔ (𝑈 ∈ (ACS‘𝐵) ∧ ∀𝑥 ∈ 𝑈 ∃𝑦 ∈ (𝒫 𝐵 ∩ Fin)𝑥 = ((mrCls‘𝑈)‘𝑦))) |
15 | 13, 14 | syl6bbr 291 | . . 3 ⊢ (𝑅 ∈ Ring → (∀𝑥 ∈ 𝑈 ∃𝑦 ∈ (𝒫 𝐵 ∩ Fin)𝑥 = ((RSpan‘𝑅)‘𝑦) ↔ 𝑈 ∈ (NoeACS‘𝐵))) |
16 | 15 | pm5.32i 577 | . 2 ⊢ ((𝑅 ∈ Ring ∧ ∀𝑥 ∈ 𝑈 ∃𝑦 ∈ (𝒫 𝐵 ∩ Fin)𝑥 = ((RSpan‘𝑅)‘𝑦)) ↔ (𝑅 ∈ Ring ∧ 𝑈 ∈ (NoeACS‘𝐵))) |
17 | 4, 16 | bitri 277 | 1 ⊢ (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ 𝑈 ∈ (NoeACS‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ∩ cin 3937 𝒫 cpw 4541 ‘cfv 6357 Fincfn 8511 Basecbs 16485 mrClscmrc 16856 ACScacs 16858 Ringcrg 19299 LIdealclidl 19944 RSpancrsp 19945 NoeACScnacs 39306 LNoeRclnr 39716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-0g 16717 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-mgp 19242 df-ur 19254 df-ring 19301 df-subrg 19535 df-lmod 19638 df-lss 19706 df-lsp 19746 df-sra 19946 df-rgmod 19947 df-lidl 19948 df-rsp 19949 df-nacs 39307 df-lfig 39675 df-lnm 39683 df-lnr 39717 |
This theorem is referenced by: hbt 39737 |
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