Theorem lnmlssfg 43835

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 43832	. . 3 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen))
3	2	simprbi 502	. 2 ⊢ (𝑀 ∈ LNoeM → ∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen)
4		oveq2 7420	. . . . 5 ⊢ (𝑎 = 𝑈 → (𝑀 ↾s 𝑎) = (𝑀 ↾s 𝑈))
5		lnmlssfg.r	. . . . 5 ⊢ 𝑅 = (𝑀 ↾s 𝑈)
6	4, 5	eqtr4di 2815	. . . 4 ⊢ (𝑎 = 𝑈 → (𝑀 ↾s 𝑎) = 𝑅)
7	6	eleq1d 2847	. . 3 ⊢ (𝑎 = 𝑈 → ((𝑀 ↾s 𝑎) ∈ LFinGen ↔ 𝑅 ∈ LFinGen))
8	7	rspcv 3576	. 2 ⊢ (𝑈 ∈ 𝑆 → (∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen → 𝑅 ∈ LFinGen))
9	3, 8	mpan9 515	1 ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LFinGen)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  ‘cfv 6536  (class class class)co 7412   ↾_s cress 17296  LModclmod 20992  LSubSpclss 21063  LFinGenclfig 43822  LNoeMclnm 43830

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-ov 7415  df-lnm 43831

This theorem is used by:  lnmlsslnm  43836  lnmfg  43837  lnmepi  43840  lmhmlnmsplit  43842  lnrfgtr  43875