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Theorem lnmlssfg 39558
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 39555 . . 3 (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎𝑆 (𝑀s 𝑎) ∈ LFinGen))
32simprbi 497 . 2 (𝑀 ∈ LNoeM → ∀𝑎𝑆 (𝑀s 𝑎) ∈ LFinGen)
4 oveq2 7153 . . . . 5 (𝑎 = 𝑈 → (𝑀s 𝑎) = (𝑀s 𝑈))
5 lnmlssfg.r . . . . 5 𝑅 = (𝑀s 𝑈)
64, 5syl6eqr 2871 . . . 4 (𝑎 = 𝑈 → (𝑀s 𝑎) = 𝑅)
76eleq1d 2894 . . 3 (𝑎 = 𝑈 → ((𝑀s 𝑎) ∈ LFinGen ↔ 𝑅 ∈ LFinGen))
87rspcv 3615 . 2 (𝑈𝑆 → (∀𝑎𝑆 (𝑀s 𝑎) ∈ LFinGen → 𝑅 ∈ LFinGen))
93, 8mpan9 507 1 ((𝑀 ∈ LNoeM ∧ 𝑈𝑆) → 𝑅 ∈ LFinGen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  cfv 6348  (class class class)co 7145  s cress 16472  LModclmod 19563  LSubSpclss 19632  LFinGenclfig 39545  LNoeMclnm 39553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-lnm 39554
This theorem is referenced by:  lnmlsslnm  39559  lnmfg  39560  lnmepi  39563  lmhmlnmsplit  39565  lnrfgtr  39598
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