Theorem lnmlssfg 43923

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 43920	. . 3 ⊢ (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen))
3	2	simprbi 503	. 2 ⊢ (𝑀 ∈ LNoeM → ∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen)
4		oveq2 7424	. . . . 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 3575	. 2 ⊢ (𝑈 ∈ 𝑆 → (∀𝑎 ∈ 𝑆 (𝑀 ↾s 𝑎) ∈ LFinGen → 𝑅 ∈ LFinGen))
9	3, 8	mpan9 516	1 ⊢ ((𝑀 ∈ LNoeM ∧ 𝑈 ∈ 𝑆) → 𝑅 ∈ LFinGen)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ‘cfv 6537  (class class class)co 7416   ↾_s cress 17326  LModclmod 21048  LSubSpclss 21119  LFinGenclfig 43910  LNoeMclnm 43918

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-ov 7419  df-lnm 43919

This theorem is used by:  lnmlsslnm  43924  lnmfg  43925  lnmepi  43928  lmhmlnmsplit  43930  lnrfgtr  43963