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Theorem lnmlssfg 42778
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 = (𝑀s 𝑈)
Assertion
Ref Expression
lnmlssfg ((𝑀 ∈ LNoeM ∧ 𝑈𝑆) → 𝑅 ∈ LFinGen)

Proof of Theorem lnmlssfg
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 lnmlssfg.s . . . 4 𝑆 = (LSubSp‘𝑀)
21islnm 42775 . . 3 (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎𝑆 (𝑀s 𝑎) ∈ LFinGen))
32simprbi 495 . 2 (𝑀 ∈ LNoeM → ∀𝑎𝑆 (𝑀s 𝑎) ∈ LFinGen)
4 oveq2 7424 . . . . 5 (𝑎 = 𝑈 → (𝑀s 𝑎) = (𝑀s 𝑈))
5 lnmlssfg.r . . . . 5 𝑅 = (𝑀s 𝑈)
64, 5eqtr4di 2784 . . . 4 (𝑎 = 𝑈 → (𝑀s 𝑎) = 𝑅)
76eleq1d 2811 . . 3 (𝑎 = 𝑈 → ((𝑀s 𝑎) ∈ LFinGen ↔ 𝑅 ∈ LFinGen))
87rspcv 3603 . 2 (𝑈𝑆 → (∀𝑎𝑆 (𝑀s 𝑎) ∈ LFinGen → 𝑅 ∈ LFinGen))
93, 8mpan9 505 1 ((𝑀 ∈ LNoeM ∧ 𝑈𝑆) → 𝑅 ∈ LFinGen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  wral 3051  cfv 6546  (class class class)co 7416  s cress 17237  LModclmod 20832  LSubSpclss 20904  LFinGenclfig 42765  LNoeMclnm 42773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-iota 6498  df-fv 6554  df-ov 7419  df-lnm 42774
This theorem is referenced by:  lnmlsslnm  42779  lnmfg  42780  lnmepi  42783  lmhmlnmsplit  42785  lnrfgtr  42818
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