![]() |
Mathbox for Stefan O'Rear |
< Previous
Next >
Nearby theorems |
|
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 6847 | . . . 4 ⊢ (𝑤 = 𝑀 → (LSubSp‘𝑤) = (LSubSp‘𝑀)) | |
2 | islnm.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑀) | |
3 | 1, 2 | eqtr4di 2795 | . . 3 ⊢ (𝑤 = 𝑀 → (LSubSp‘𝑤) = 𝑆) |
4 | oveq1 7369 | . . . 4 ⊢ (𝑤 = 𝑀 → (𝑤 ↾s 𝑖) = (𝑀 ↾s 𝑖)) | |
5 | 4 | eleq1d 2823 | . . 3 ⊢ (𝑤 = 𝑀 → ((𝑤 ↾s 𝑖) ∈ LFinGen ↔ (𝑀 ↾s 𝑖) ∈ LFinGen)) |
6 | 3, 5 | raleqbidv 3322 | . 2 ⊢ (𝑤 = 𝑀 → (∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen ↔ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen)) |
7 | df-lnm 41432 | . 2 ⊢ LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen} | |
8 | 6, 7 | elrab2 3653 | 1 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ‘cfv 6501 (class class class)co 7362 ↾s cress 17119 LModclmod 20338 LSubSpclss 20408 LFinGenclfig 41423 LNoeMclnm 41431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-iota 6453 df-fv 6509 df-ov 7365 df-lnm 41432 |
This theorem is referenced by: islnm2 41434 lnmlmod 41435 lnmlssfg 41436 lnmlsslnm 41437 lnmepi 41441 lmhmlnmsplit 41443 |
Copyright terms: Public domain | W3C validator |