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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 6103 | . . . 4 ⊢ (𝑤 = 𝑀 → (LSubSp‘𝑤) = (LSubSp‘𝑀)) | |
2 | islnm.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑀) | |
3 | 1, 2 | syl6eqr 2662 | . . 3 ⊢ (𝑤 = 𝑀 → (LSubSp‘𝑤) = 𝑆) |
4 | oveq1 6556 | . . . 4 ⊢ (𝑤 = 𝑀 → (𝑤 ↾s 𝑖) = (𝑀 ↾s 𝑖)) | |
5 | 4 | eleq1d 2672 | . . 3 ⊢ (𝑤 = 𝑀 → ((𝑤 ↾s 𝑖) ∈ LFinGen ↔ (𝑀 ↾s 𝑖) ∈ LFinGen)) |
6 | 3, 5 | raleqbidv 3129 | . 2 ⊢ (𝑤 = 𝑀 → (∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen ↔ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen)) |
7 | df-lnm 36664 | . 2 ⊢ LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen} | |
8 | 6, 7 | elrab2 3333 | 1 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ‘cfv 5804 (class class class)co 6549 ↾s cress 15696 LModclmod 18686 LSubSpclss 18753 LFinGenclfig 36655 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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-lnm 36664 |
This theorem is referenced by: islnm2 36666 lnmlmod 36667 lnmlssfg 36668 lnmlsslnm 36669 lnmepi 36673 lmhmlnmsplit 36675 |
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