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Theorem lnmlssfg 38176
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 38173 . . 3 (𝑀 ∈ LNoeM ↔ (𝑀 ∈ LMod ∧ ∀𝑎𝑆 (𝑀s 𝑎) ∈ LFinGen))
32simprbi 484 . 2 (𝑀 ∈ LNoeM → ∀𝑎𝑆 (𝑀s 𝑎) ∈ LFinGen)
4 oveq2 6804 . . . . 5 (𝑎 = 𝑈 → (𝑀s 𝑎) = (𝑀s 𝑈))
5 lnmlssfg.r . . . . 5 𝑅 = (𝑀s 𝑈)
64, 5syl6eqr 2823 . . . 4 (𝑎 = 𝑈 → (𝑀s 𝑎) = 𝑅)
76eleq1d 2835 . . 3 (𝑎 = 𝑈 → ((𝑀s 𝑎) ∈ LFinGen ↔ 𝑅 ∈ LFinGen))
87rspcv 3456 . 2 (𝑈𝑆 → (∀𝑎𝑆 (𝑀s 𝑎) ∈ LFinGen → 𝑅 ∈ LFinGen))
93, 8mpan9 496 1 ((𝑀 ∈ LNoeM ∧ 𝑈𝑆) → 𝑅 ∈ LFinGen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  cfv 6030  (class class class)co 6796  s cress 16065  LModclmod 19073  LSubSpclss 19142  LFinGenclfig 38163  LNoeMclnm 38171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-iota 5993  df-fv 6038  df-ov 6799  df-lnm 38172
This theorem is referenced by:  lnmlsslnm  38177  lnmfg  38178  lnmepi  38181  lmhmlnmsplit  38183  lnrfgtr  38216
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