Theorem lnmlssfg 43538

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 43535	. . 3 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen))
3	2	simprbi 499	. 2 ⊢ (𝑀 ∈ LNoeM → ∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen)
4		oveq2 7367	. . . . 5 ⊢ (𝑎 = 𝑈 → (𝑀 ↾s 𝑎) = (𝑀 ↾s 𝑈))
5		lnmlssfg.r	. . . . 5 ⊢ 𝑅 = (𝑀 ↾s 𝑈)
6	4, 5	eqtr4di 2794	. . . 4 ⊢ (𝑎 = 𝑈 → (𝑀 ↾s 𝑎) = 𝑅)
7	6	eleq1d 2826	. . 3 ⊢ (𝑎 = 𝑈 → ((𝑀 ↾s 𝑎) ∈ LFinGen ↔ 𝑅 ∈ LFinGen))
8	7	rspcv 3557	. 2 ⊢ (𝑈 ∈ 𝑆 → (∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen → 𝑅 ∈ LFinGen))
9	3, 8	mpan9 512	1 ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LFinGen)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ‘cfv 6488  (class class class)co 7359   ↾_s cress 17195  LModclmod 20853  LSubSpclss 20924  LFinGenclfig 43525  LNoeMclnm 43533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-iota 6444  df-fv 6496  df-ov 7362  df-lnm 43534

This theorem is referenced by:  lnmlsslnm  43539  lnmfg  43540  lnmepi  43543  lmhmlnmsplit  43545  lnrfgtr  43578