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Mirrors > Home > MPE Home > Th. List > Mathboxes > filnm | Structured version Visualization version GIF version |
Description: Finite left modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
filnm.b | ⊢ 𝐵 = (Base‘𝑊) |
Ref | Expression |
---|---|
filnm | ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) → 𝑊 ∈ LNoeM) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 472 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) → 𝑊 ∈ LMod) | |
2 | filnm.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑊) | |
3 | eqid 2610 | . . . . . . . 8 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
4 | 2, 3 | lssss 18758 | . . . . . . 7 ⊢ (𝑎 ∈ (LSubSp‘𝑊) → 𝑎 ⊆ 𝐵) |
5 | 4 | adantl 481 | . . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) ∧ 𝑎 ∈ (LSubSp‘𝑊)) → 𝑎 ⊆ 𝐵) |
6 | selpw 4115 | . . . . . 6 ⊢ (𝑎 ∈ 𝒫 𝐵 ↔ 𝑎 ⊆ 𝐵) | |
7 | 5, 6 | sylibr 223 | . . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) ∧ 𝑎 ∈ (LSubSp‘𝑊)) → 𝑎 ∈ 𝒫 𝐵) |
8 | simplr 788 | . . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) ∧ 𝑎 ∈ (LSubSp‘𝑊)) → 𝐵 ∈ Fin) | |
9 | ssfi 8065 | . . . . . 6 ⊢ ((𝐵 ∈ Fin ∧ 𝑎 ⊆ 𝐵) → 𝑎 ∈ Fin) | |
10 | 8, 5, 9 | syl2anc 691 | . . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) ∧ 𝑎 ∈ (LSubSp‘𝑊)) → 𝑎 ∈ Fin) |
11 | 7, 10 | elind 3760 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) ∧ 𝑎 ∈ (LSubSp‘𝑊)) → 𝑎 ∈ (𝒫 𝐵 ∩ Fin)) |
12 | eqid 2610 | . . . . . . 7 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
13 | 3, 12 | lspid 18803 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑎 ∈ (LSubSp‘𝑊)) → ((LSpan‘𝑊)‘𝑎) = 𝑎) |
14 | 13 | adantlr 747 | . . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) ∧ 𝑎 ∈ (LSubSp‘𝑊)) → ((LSpan‘𝑊)‘𝑎) = 𝑎) |
15 | 14 | eqcomd 2616 | . . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) ∧ 𝑎 ∈ (LSubSp‘𝑊)) → 𝑎 = ((LSpan‘𝑊)‘𝑎)) |
16 | fveq2 6103 | . . . . . 6 ⊢ (𝑏 = 𝑎 → ((LSpan‘𝑊)‘𝑏) = ((LSpan‘𝑊)‘𝑎)) | |
17 | 16 | eqeq2d 2620 | . . . . 5 ⊢ (𝑏 = 𝑎 → (𝑎 = ((LSpan‘𝑊)‘𝑏) ↔ 𝑎 = ((LSpan‘𝑊)‘𝑎))) |
18 | 17 | rspcev 3282 | . . . 4 ⊢ ((𝑎 ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝑎 = ((LSpan‘𝑊)‘𝑎)) → ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)𝑎 = ((LSpan‘𝑊)‘𝑏)) |
19 | 11, 15, 18 | syl2anc 691 | . . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) ∧ 𝑎 ∈ (LSubSp‘𝑊)) → ∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)𝑎 = ((LSpan‘𝑊)‘𝑏)) |
20 | 19 | ralrimiva 2949 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) → ∀𝑎 ∈ (LSubSp‘𝑊)∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)𝑎 = ((LSpan‘𝑊)‘𝑏)) |
21 | 2, 3, 12 | islnm2 36666 | . 2 ⊢ (𝑊 ∈ LNoeM ↔ (𝑊 ∈ LMod ∧ ∀𝑎 ∈ (LSubSp‘𝑊)∃𝑏 ∈ (𝒫 𝐵 ∩ Fin)𝑎 = ((LSpan‘𝑊)‘𝑏))) |
22 | 1, 20, 21 | sylanbrc 695 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ Fin) → 𝑊 ∈ LNoeM) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 ‘cfv 5804 Fincfn 7841 Basecbs 15695 LModclmod 18686 LSubSpclss 18753 LSpanclspn 18792 LNoeMclnm 36663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-sca 15784 df-vsca 15785 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-mgp 18313 df-ur 18325 df-ring 18372 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lfig 36656 df-lnm 36664 |
This theorem is referenced by: pwslnmlem0 36679 |
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