Theorem lnmlssfg 43359

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 43356	. . 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 7366	. . . . 5 ⊢ (𝑎 = 𝑈 → (𝑀 ↾s 𝑎) = (𝑀 ↾s 𝑈))
5		lnmlssfg.r	. . . . 5 ⊢ 𝑅 = (𝑀 ↾s 𝑈)
6	4, 5	eqtr4di 2788	. . . 4 ⊢ (𝑎 = 𝑈 → (𝑀 ↾s 𝑎) = 𝑅)
7	6	eleq1d 2820	. . 3 ⊢ (𝑎 = 𝑈 → ((𝑀 ↾s 𝑎) ∈ LFinGen ↔ 𝑅 ∈ LFinGen))
8	7	rspcv 3571	. 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 1542   ∈ wcel 2114  ∀wral 3050  ‘cfv 6491  (class class class)co 7358   ↾_s cress 17159  LModclmod 20813  LSubSpclss 20884  LFinGenclfig 43346  LNoeMclnm 43354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6447  df-fv 6499  df-ov 7361  df-lnm 43355

This theorem is referenced by:  lnmlsslnm  43360  lnmfg  43361  lnmepi  43364  lmhmlnmsplit  43366  lnrfgtr  43399