Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lnrfgtr | Structured version Visualization version GIF version |
Description: A submodule of a finitely generated module over a Noetherian ring is finitely generated. Often taken as the definition of Noetherian ring. (Contributed by Stefan O'Rear, 7-Feb-2015.) |
Ref | Expression |
---|---|
lnrfg.s | ⊢ 𝑆 = (Scalar‘𝑀) |
lnrfgtr.u | ⊢ 𝑈 = (LSubSp‘𝑀) |
lnrfgtr.n | ⊢ 𝑁 = (𝑀 ↾s 𝑃) |
Ref | Expression |
---|---|
lnrfgtr | ⊢ ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR ∧ 𝑃 ∈ 𝑈) → 𝑁 ∈ LFinGen) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lnrfg.s | . . 3 ⊢ 𝑆 = (Scalar‘𝑀) | |
2 | 1 | lnrfg 40933 | . 2 ⊢ ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) → 𝑀 ∈ LNoeM) |
3 | lnrfgtr.u | . . 3 ⊢ 𝑈 = (LSubSp‘𝑀) | |
4 | lnrfgtr.n | . . 3 ⊢ 𝑁 = (𝑀 ↾s 𝑃) | |
5 | 3, 4 | lnmlssfg 40894 | . 2 ⊢ ((𝑀 ∈ LNoeM ∧ 𝑃 ∈ 𝑈) → 𝑁 ∈ LFinGen) |
6 | 2, 5 | stoic3 1783 | 1 ⊢ ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR ∧ 𝑃 ∈ 𝑈) → 𝑁 ∈ LFinGen) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ‘cfv 6431 (class class class)co 7269 ↾s cress 16931 Scalarcsca 16955 LSubSpclss 20183 LFinGenclfig 40881 LNoeMclnm 40889 LNoeRclnr 40923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10920 ax-resscn 10921 ax-1cn 10922 ax-icn 10923 ax-addcl 10924 ax-addrcl 10925 ax-mulcl 10926 ax-mulrcl 10927 ax-mulcom 10928 ax-addass 10929 ax-mulass 10930 ax-distr 10931 ax-i2m1 10932 ax-1ne0 10933 ax-1rid 10934 ax-rnegex 10935 ax-rrecex 10936 ax-cnre 10937 ax-pre-lttri 10938 ax-pre-lttrn 10939 ax-pre-ltadd 10940 ax-pre-mulgt0 10941 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-om 7702 df-1st 7818 df-2nd 7819 df-supp 7963 df-frecs 8082 df-wrecs 8113 df-recs 8187 df-rdg 8226 df-1o 8282 df-er 8473 df-map 8592 df-ixp 8661 df-en 8709 df-dom 8710 df-sdom 8711 df-fin 8712 df-fsupp 9099 df-sup 9171 df-oi 9239 df-card 9690 df-pnf 11004 df-mnf 11005 df-xr 11006 df-ltxr 11007 df-le 11008 df-sub 11199 df-neg 11200 df-nn 11966 df-2 12028 df-3 12029 df-4 12030 df-5 12031 df-6 12032 df-7 12033 df-8 12034 df-9 12035 df-n0 12226 df-z 12312 df-dec 12429 df-uz 12574 df-fz 13231 df-fzo 13374 df-seq 13712 df-hash 14035 df-struct 16838 df-sets 16855 df-slot 16873 df-ndx 16885 df-base 16903 df-ress 16932 df-plusg 16965 df-mulr 16966 df-sca 16968 df-vsca 16969 df-ip 16970 df-tset 16971 df-ple 16972 df-ds 16974 df-hom 16976 df-cco 16977 df-0g 17142 df-gsum 17143 df-prds 17148 df-pws 17150 df-mre 17285 df-mrc 17286 df-acs 17288 df-mgm 18316 df-sgrp 18365 df-mnd 18376 df-mhm 18420 df-submnd 18421 df-grp 18570 df-minusg 18571 df-sbg 18572 df-mulg 18691 df-subg 18742 df-ghm 18822 df-cntz 18913 df-lsm 19231 df-cmn 19378 df-abl 19379 df-mgp 19711 df-ur 19728 df-ring 19775 df-subrg 20012 df-lmod 20115 df-lss 20184 df-lsp 20224 df-lmhm 20274 df-lmim 20275 df-lmic 20276 df-lbs 20327 df-sra 20424 df-rgmod 20425 df-nzr 20519 df-dsmm 20929 df-frlm 20944 df-uvc 20980 df-lfig 40882 df-lnm 40890 df-lnr 40924 |
This theorem is referenced by: (None) |
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