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Theorem lnmlssfg 42397
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 42394 . . 3 (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎𝑆 (𝑀s 𝑎) ∈ LFinGen))
32simprbi 496 . 2 (𝑀 ∈ LNoeM → ∀𝑎𝑆 (𝑀s 𝑎) ∈ LFinGen)
4 oveq2 7413 . . . . 5 (𝑎 = 𝑈 → (𝑀s 𝑎) = (𝑀s 𝑈))
5 lnmlssfg.r . . . . 5 𝑅 = (𝑀s 𝑈)
64, 5eqtr4di 2784 . . . 4 (𝑎 = 𝑈 → (𝑀s 𝑎) = 𝑅)
76eleq1d 2812 . . 3 (𝑎 = 𝑈 → ((𝑀s 𝑎) ∈ LFinGen ↔ 𝑅 ∈ LFinGen))
87rspcv 3602 . 2 (𝑈𝑆 → (∀𝑎𝑆 (𝑀s 𝑎) ∈ LFinGen → 𝑅 ∈ LFinGen))
93, 8mpan9 506 1 ((𝑀 ∈ LNoeM ∧ 𝑈𝑆) → 𝑅 ∈ LFinGen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3055  cfv 6537  (class class class)co 7405  s cress 17182  LModclmod 20706  LSubSpclss 20778  LFinGenclfig 42384  LNoeMclnm 42392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545  df-ov 7408  df-lnm 42393
This theorem is referenced by:  lnmlsslnm  42398  lnmfg  42399  lnmepi  42402  lmhmlnmsplit  42404  lnrfgtr  42437
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