Theorem lnmlssfg 43037

Description: A submodule of Noetherian module is finitely generated. (Contributed by Stefan O'Rear, 1-Jan-2015.)

Hypotheses

Ref	Expression
lnmlssfg.s	⊢ 𝑆 = (LSubSp‘𝑀)
lnmlssfg.r	⊢ 𝑅 = (𝑀 ↾s 𝑈)

Assertion

Ref	Expression
lnmlssfg	⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LFinGen)

Proof of Theorem lnmlssfg

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lnmlssfg.s	. . . 4 ⊢ 𝑆 = (LSubSp‘𝑀)
2	1	islnm 43034	. . 3 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen))
3	2	simprbi 496	. 2 ⊢ (𝑀 ∈ LNoeM → ∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen)
4		oveq2 7456	. . . . 5 ⊢ (𝑎 = 𝑈 → (𝑀 ↾s 𝑎) = (𝑀 ↾s 𝑈))
5		lnmlssfg.r	. . . . 5 ⊢ 𝑅 = (𝑀 ↾s 𝑈)
6	4, 5	eqtr4di 2798	. . . 4 ⊢ (𝑎 = 𝑈 → (𝑀 ↾s 𝑎) = 𝑅)
7	6	eleq1d 2829	. . 3 ⊢ (𝑎 = 𝑈 → ((𝑀 ↾s 𝑎) ∈ LFinGen ↔ 𝑅 ∈ LFinGen))
8	7	rspcv 3631	. 2 ⊢ (𝑈 ∈ 𝑆 → (∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen → 𝑅 ∈ LFinGen))
9	3, 8	mpan9 506	1 ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LFinGen)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ‘cfv 6573  (class class class)co 7448   ↾_s cress 17287  LModclmod 20880  LSubSpclss 20952  LFinGenclfig 43024  LNoeMclnm 43032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-lnm 43033

This theorem is referenced by:  lnmlsslnm  43038  lnmfg  43039  lnmepi  43042  lmhmlnmsplit  43044  lnrfgtr  43077