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Mathbox for Stefan O'Rear |
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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 6499 | . . . 4 ⊢ (𝑤 = 𝑀 → (LSubSp‘𝑤) = (LSubSp‘𝑀)) | |
2 | islnm.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑀) | |
3 | 1, 2 | syl6eqr 2833 | . . 3 ⊢ (𝑤 = 𝑀 → (LSubSp‘𝑤) = 𝑆) |
4 | oveq1 6983 | . . . 4 ⊢ (𝑤 = 𝑀 → (𝑤 ↾s 𝑖) = (𝑀 ↾s 𝑖)) | |
5 | 4 | eleq1d 2851 | . . 3 ⊢ (𝑤 = 𝑀 → ((𝑤 ↾s 𝑖) ∈ LFinGen ↔ (𝑀 ↾s 𝑖) ∈ LFinGen)) |
6 | 3, 5 | raleqbidv 3342 | . 2 ⊢ (𝑤 = 𝑀 → (∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen ↔ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen)) |
7 | df-lnm 39069 | . 2 ⊢ LNoeM = {𝑤 ∈ LMod ∣ ∀𝑖 ∈ (LSubSp‘𝑤)(𝑤 ↾s 𝑖) ∈ LFinGen} | |
8 | 6, 7 | elrab2 3600 | 1 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑖 ∈ 𝑆 (𝑀 ↾s 𝑖) ∈ LFinGen)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∀wral 3089 ‘cfv 6188 (class class class)co 6976 ↾s cress 16340 LModclmod 19356 LSubSpclss 19425 LFinGenclfig 39060 LNoeMclnm 39068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2751 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3418 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-iota 6152 df-fv 6196 df-ov 6979 df-lnm 39069 |
This theorem is referenced by: islnm2 39071 lnmlmod 39072 lnmlssfg 39073 lnmlsslnm 39074 lnmepi 39078 lmhmlnmsplit 39080 |
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