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Mirrors > Home > MPE Home > Th. List > Mathboxes > islnm | Structured version Visualization version GIF version |
Description: Property of being a Noetherian left module. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
Ref | Expression |
---|---|
islnm.s | ⊢ 𝑆 = (LSubSp‘𝑀) |
Ref | Expression |
---|---|
islnm | ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6673 | . . . 4 ⊢ (𝑤 = 𝑀 → (LSubSp‘𝑤) = (LSubSp‘𝑀)) | |
2 | islnm.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑀) | |
3 | 1, 2 | syl6eqr 2877 | . . 3 ⊢ (𝑤 = 𝑀 → (LSubSp‘𝑤) = 𝑆) |
4 | oveq1 7166 | . . . 4 ⊢ (𝑤 = 𝑀 → (𝑤 ↾s 𝑖) = (𝑀 ↾s 𝑖)) | |
5 | 4 | eleq1d 2900 | . . 3 ⊢ (𝑤 = 𝑀 → ((𝑤 ↾s 𝑖) ∈ LFinGen ↔ (𝑀 ↾s 𝑖) ∈ LFinGen)) |
6 | 3, 5 | raleqbidv 3404 | . 2 ⊢ (𝑤 = 𝑀 → (∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen ↔ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen)) |
7 | df-lnm 39682 | . 2 ⊢ LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen} | |
8 | 6, 7 | elrab2 3686 | 1 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ‘cfv 6358 (class class class)co 7159 ↾s cress 16487 LModclmod 19637 LSubSpclss 19706 LFinGenclfig 39673 LNoeMclnm 39681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 df-lnm 39682 |
This theorem is referenced by: islnm2 39684 lnmlmod 39685 lnmlssfg 39686 lnmlsslnm 39687 lnmepi 39691 lmhmlnmsplit 39693 |
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