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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lnrfg | Structured version Visualization version GIF version |
Description: Finitely-generated modules over a Noetherian ring, being homomorphic images of free modules, are Noetherian. (Contributed by Stefan O'Rear, 7-Feb-2015.) |
Ref | Expression |
---|---|
lnrfg.s | ⊢ 𝑆 = (Scalar‘𝑀) |
Ref | Expression |
---|---|
lnrfg | ⊢ ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) → 𝑀 ∈ LNoeM) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . . 4 ⊢ (𝑆 freeLMod 𝑎) = (𝑆 freeLMod 𝑎) | |
2 | eqid 2735 | . . . 4 ⊢ (Base‘(𝑆 freeLMod 𝑎)) = (Base‘(𝑆 freeLMod 𝑎)) | |
3 | eqid 2735 | . . . 4 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
4 | eqid 2735 | . . . 4 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀) | |
5 | eqid 2735 | . . . 4 ⊢ (𝑏 ∈ (Base‘(𝑆 freeLMod 𝑎)) ↦ (𝑀 Σg (𝑏 ∘f ( ·𝑠 ‘𝑀)( I ↾ 𝑎)))) = (𝑏 ∈ (Base‘(𝑆 freeLMod 𝑎)) ↦ (𝑀 Σg (𝑏 ∘f ( ·𝑠 ‘𝑀)( I ↾ 𝑎)))) | |
6 | fglmod 43062 | . . . . 5 ⊢ (𝑀 ∈ LFinGen → 𝑀 ∈ LMod) | |
7 | 6 | ad3antrrr 730 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → 𝑀 ∈ LMod) |
8 | vex 3482 | . . . . 5 ⊢ 𝑎 ∈ V | |
9 | 8 | a1i 11 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → 𝑎 ∈ V) |
10 | lnrfg.s | . . . . 5 ⊢ 𝑆 = (Scalar‘𝑀) | |
11 | 10 | a1i 11 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → 𝑆 = (Scalar‘𝑀)) |
12 | f1oi 6887 | . . . . . . 7 ⊢ ( I ↾ 𝑎):𝑎–1-1-onto→𝑎 | |
13 | f1of 6849 | . . . . . . 7 ⊢ (( I ↾ 𝑎):𝑎–1-1-onto→𝑎 → ( I ↾ 𝑎):𝑎⟶𝑎) | |
14 | 12, 13 | ax-mp 5 | . . . . . 6 ⊢ ( I ↾ 𝑎):𝑎⟶𝑎 |
15 | elpwi 4612 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 (Base‘𝑀) → 𝑎 ⊆ (Base‘𝑀)) | |
16 | fss 6753 | . . . . . 6 ⊢ ((( I ↾ 𝑎):𝑎⟶𝑎 ∧ 𝑎 ⊆ (Base‘𝑀)) → ( I ↾ 𝑎):𝑎⟶(Base‘𝑀)) | |
17 | 14, 15, 16 | sylancr 587 | . . . . 5 ⊢ (𝑎 ∈ 𝒫 (Base‘𝑀) → ( I ↾ 𝑎):𝑎⟶(Base‘𝑀)) |
18 | 17 | ad2antlr 727 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → ( I ↾ 𝑎):𝑎⟶(Base‘𝑀)) |
19 | 1, 2, 3, 4, 5, 7, 9, 11, 18 | frlmup1 21836 | . . 3 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → (𝑏 ∈ (Base‘(𝑆 freeLMod 𝑎)) ↦ (𝑀 Σg (𝑏 ∘f ( ·𝑠 ‘𝑀)( I ↾ 𝑎)))) ∈ ((𝑆 freeLMod 𝑎) LMHom 𝑀)) |
20 | simpllr 776 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → 𝑆 ∈ LNoeR) | |
21 | simprl 771 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → 𝑎 ∈ Fin) | |
22 | 1 | lnrfrlm 43107 | . . . 4 ⊢ ((𝑆 ∈ LNoeR ∧ 𝑎 ∈ Fin) → (𝑆 freeLMod 𝑎) ∈ LNoeM) |
23 | 20, 21, 22 | syl2anc 584 | . . 3 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → (𝑆 freeLMod 𝑎) ∈ LNoeM) |
24 | eqid 2735 | . . . . 5 ⊢ (LSpan‘𝑀) = (LSpan‘𝑀) | |
25 | 1, 2, 3, 4, 5, 7, 9, 11, 18, 24 | frlmup3 21838 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → ran (𝑏 ∈ (Base‘(𝑆 freeLMod 𝑎)) ↦ (𝑀 Σg (𝑏 ∘f ( ·𝑠 ‘𝑀)( I ↾ 𝑎)))) = ((LSpan‘𝑀)‘ran ( I ↾ 𝑎))) |
26 | rnresi 6095 | . . . . . 6 ⊢ ran ( I ↾ 𝑎) = 𝑎 | |
27 | 26 | fveq2i 6910 | . . . . 5 ⊢ ((LSpan‘𝑀)‘ran ( I ↾ 𝑎)) = ((LSpan‘𝑀)‘𝑎) |
28 | simprr 773 | . . . . 5 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀)) | |
29 | 27, 28 | eqtrid 2787 | . . . 4 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → ((LSpan‘𝑀)‘ran ( I ↾ 𝑎)) = (Base‘𝑀)) |
30 | 25, 29 | eqtrd 2775 | . . 3 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → ran (𝑏 ∈ (Base‘(𝑆 freeLMod 𝑎)) ↦ (𝑀 Σg (𝑏 ∘f ( ·𝑠 ‘𝑀)( I ↾ 𝑎)))) = (Base‘𝑀)) |
31 | 3 | lnmepi 43074 | . . 3 ⊢ (((𝑏 ∈ (Base‘(𝑆 freeLMod 𝑎)) ↦ (𝑀 Σg (𝑏 ∘f ( ·𝑠 ‘𝑀)( I ↾ 𝑎)))) ∈ ((𝑆 freeLMod 𝑎) LMHom 𝑀) ∧ (𝑆 freeLMod 𝑎) ∈ LNoeM ∧ ran (𝑏 ∈ (Base‘(𝑆 freeLMod 𝑎)) ↦ (𝑀 Σg (𝑏 ∘f ( ·𝑠 ‘𝑀)( I ↾ 𝑎)))) = (Base‘𝑀)) → 𝑀 ∈ LNoeM) |
32 | 19, 23, 30, 31 | syl3anc 1370 | . 2 ⊢ ((((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) ∧ 𝑎 ∈ 𝒫 (Base‘𝑀)) ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) → 𝑀 ∈ LNoeM) |
33 | 3, 24 | islmodfg 43058 | . . . . 5 ⊢ (𝑀 ∈ LMod → (𝑀 ∈ LFinGen ↔ ∃𝑎 ∈ 𝒫 (Base‘𝑀)(𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀)))) |
34 | 6, 33 | syl 17 | . . . 4 ⊢ (𝑀 ∈ LFinGen → (𝑀 ∈ LFinGen ↔ ∃𝑎 ∈ 𝒫 (Base‘𝑀)(𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀)))) |
35 | 34 | ibi 267 | . . 3 ⊢ (𝑀 ∈ LFinGen → ∃𝑎 ∈ 𝒫 (Base‘𝑀)(𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) |
36 | 35 | adantr 480 | . 2 ⊢ ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) → ∃𝑎 ∈ 𝒫 (Base‘𝑀)(𝑎 ∈ Fin ∧ ((LSpan‘𝑀)‘𝑎) = (Base‘𝑀))) |
37 | 32, 36 | r19.29a 3160 | 1 ⊢ ((𝑀 ∈ LFinGen ∧ 𝑆 ∈ LNoeR) → 𝑀 ∈ LNoeM) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 Vcvv 3478 ⊆ wss 3963 𝒫 cpw 4605 ↦ cmpt 5231 I cid 5582 ran crn 5690 ↾ cres 5691 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 Fincfn 8984 Basecbs 17245 Scalarcsca 17301 ·𝑠 cvsca 17302 Σg cgsu 17487 LModclmod 20875 LSpanclspn 20987 LMHom clmhm 21036 freeLMod cfrlm 21784 LFinGenclfig 43056 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-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 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-se 5642 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-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 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-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 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-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-nzr 20530 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lmhm 21039 df-lmim 21040 df-lmic 21041 df-lbs 21092 df-sra 21190 df-rgmod 21191 df-dsmm 21770 df-frlm 21785 df-uvc 21821 df-lfig 43057 df-lnm 43065 df-lnr 43099 |
This theorem is referenced by: lnrfgtr 43109 |
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