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Theorem lsmfgcl 41806
Description: The sum of two finitely generated submodules is finitely generated. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypotheses
Ref Expression
lsmfgcl.u π‘ˆ = (LSubSpβ€˜π‘Š)
lsmfgcl.p βŠ• = (LSSumβ€˜π‘Š)
lsmfgcl.d 𝐷 = (π‘Š β†Ύs 𝐴)
lsmfgcl.e 𝐸 = (π‘Š β†Ύs 𝐡)
lsmfgcl.f 𝐹 = (π‘Š β†Ύs (𝐴 βŠ• 𝐡))
lsmfgcl.w (πœ‘ β†’ π‘Š ∈ LMod)
lsmfgcl.a (πœ‘ β†’ 𝐴 ∈ π‘ˆ)
lsmfgcl.b (πœ‘ β†’ 𝐡 ∈ π‘ˆ)
lsmfgcl.df (πœ‘ β†’ 𝐷 ∈ LFinGen)
lsmfgcl.ef (πœ‘ β†’ 𝐸 ∈ LFinGen)
Assertion
Ref Expression
lsmfgcl (πœ‘ β†’ 𝐹 ∈ LFinGen)
Proof of Theorem lsmfgcl
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lsmfgcl.f . 2 𝐹 = (π‘Š β†Ύs (𝐴 βŠ• 𝐡))
2 lsmfgcl.df . . . 4 (πœ‘ β†’ 𝐷 ∈ LFinGen)
3 lsmfgcl.w . . . . 5 (πœ‘ β†’ π‘Š ∈ LMod)
4 lsmfgcl.a . . . . 5 (πœ‘ β†’ 𝐴 ∈ π‘ˆ)
5 lsmfgcl.d . . . . . 6 𝐷 = (π‘Š β†Ύs 𝐴)
6 lsmfgcl.u . . . . . 6 π‘ˆ = (LSubSpβ€˜π‘Š)
7 eqid 2732 . . . . . 6 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
8 eqid 2732 . . . . . 6 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
95, 6, 7, 8islssfg2 41803 . . . . 5 ((π‘Š ∈ LMod ∧ 𝐴 ∈ π‘ˆ) β†’ (𝐷 ∈ LFinGen ↔ βˆƒπ‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)((LSpanβ€˜π‘Š)β€˜π‘Ž) = 𝐴))
103, 4, 9syl2anc 584 . . . 4 (πœ‘ β†’ (𝐷 ∈ LFinGen ↔ βˆƒπ‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)((LSpanβ€˜π‘Š)β€˜π‘Ž) = 𝐴))
112, 10mpbid 231 . . 3 (πœ‘ β†’ βˆƒπ‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)((LSpanβ€˜π‘Š)β€˜π‘Ž) = 𝐴)
12 lsmfgcl.ef . . . . . . . 8 (πœ‘ β†’ 𝐸 ∈ LFinGen)
13 lsmfgcl.b . . . . . . . . 9 (πœ‘ β†’ 𝐡 ∈ π‘ˆ)
14 lsmfgcl.e . . . . . . . . . 10 𝐸 = (π‘Š β†Ύs 𝐡)
1514, 6, 7, 8islssfg2 41803 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝐡 ∈ π‘ˆ) β†’ (𝐸 ∈ LFinGen ↔ βˆƒπ‘ ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)((LSpanβ€˜π‘Š)β€˜π‘) = 𝐡))
163, 13, 15syl2anc 584 . . . . . . . 8 (πœ‘ β†’ (𝐸 ∈ LFinGen ↔ βˆƒπ‘ ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)((LSpanβ€˜π‘Š)β€˜π‘) = 𝐡))
1712, 16mpbid 231 . . . . . . 7 (πœ‘ β†’ βˆƒπ‘ ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)((LSpanβ€˜π‘Š)β€˜π‘) = 𝐡)
1817adantr 481 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)) β†’ βˆƒπ‘ ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)((LSpanβ€˜π‘Š)β€˜π‘) = 𝐡)
19 inss1 4228 . . . . . . . . . . . . . . 15 (𝒫 (Baseβ€˜π‘Š) ∩ Fin) βŠ† 𝒫 (Baseβ€˜π‘Š)
2019sseli 3978 . . . . . . . . . . . . . 14 (π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) β†’ π‘Ž ∈ 𝒫 (Baseβ€˜π‘Š))
2120elpwid 4611 . . . . . . . . . . . . 13 (π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) β†’ π‘Ž βŠ† (Baseβ€˜π‘Š))
2219sseli 3978 . . . . . . . . . . . . . 14 (𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) β†’ 𝑏 ∈ 𝒫 (Baseβ€˜π‘Š))
2322elpwid 4611 . . . . . . . . . . . . 13 (𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) β†’ 𝑏 βŠ† (Baseβ€˜π‘Š))
24 lsmfgcl.p . . . . . . . . . . . . . 14 βŠ• = (LSSumβ€˜π‘Š)
258, 7, 24lsmsp2 20697 . . . . . . . . . . . . 13 ((π‘Š ∈ LMod ∧ π‘Ž βŠ† (Baseβ€˜π‘Š) ∧ 𝑏 βŠ† (Baseβ€˜π‘Š)) β†’ (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• ((LSpanβ€˜π‘Š)β€˜π‘)) = ((LSpanβ€˜π‘Š)β€˜(π‘Ž βˆͺ 𝑏)))
263, 21, 23, 25syl3an 1160 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) ∧ 𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)) β†’ (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• ((LSpanβ€˜π‘Š)β€˜π‘)) = ((LSpanβ€˜π‘Š)β€˜(π‘Ž βˆͺ 𝑏)))
27263expb 1120 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) ∧ 𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin))) β†’ (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• ((LSpanβ€˜π‘Š)β€˜π‘)) = ((LSpanβ€˜π‘Š)β€˜(π‘Ž βˆͺ 𝑏)))
2827oveq2d 7424 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) ∧ 𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin))) β†’ (π‘Š β†Ύs (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• ((LSpanβ€˜π‘Š)β€˜π‘))) = (π‘Š β†Ύs ((LSpanβ€˜π‘Š)β€˜(π‘Ž βˆͺ 𝑏))))
293adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) ∧ 𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin))) β†’ π‘Š ∈ LMod)
30 unss 4184 . . . . . . . . . . . . . 14 ((π‘Ž βŠ† (Baseβ€˜π‘Š) ∧ 𝑏 βŠ† (Baseβ€˜π‘Š)) ↔ (π‘Ž βˆͺ 𝑏) βŠ† (Baseβ€˜π‘Š))
3130biimpi 215 . . . . . . . . . . . . 13 ((π‘Ž βŠ† (Baseβ€˜π‘Š) ∧ 𝑏 βŠ† (Baseβ€˜π‘Š)) β†’ (π‘Ž βˆͺ 𝑏) βŠ† (Baseβ€˜π‘Š))
3221, 23, 31syl2an 596 . . . . . . . . . . . 12 ((π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) ∧ 𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)) β†’ (π‘Ž βˆͺ 𝑏) βŠ† (Baseβ€˜π‘Š))
3332adantl 482 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) ∧ 𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin))) β†’ (π‘Ž βˆͺ 𝑏) βŠ† (Baseβ€˜π‘Š))
34 inss2 4229 . . . . . . . . . . . . . 14 (𝒫 (Baseβ€˜π‘Š) ∩ Fin) βŠ† Fin
3534sseli 3978 . . . . . . . . . . . . 13 (π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) β†’ π‘Ž ∈ Fin)
3634sseli 3978 . . . . . . . . . . . . 13 (𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) β†’ 𝑏 ∈ Fin)
37 unfi 9171 . . . . . . . . . . . . 13 ((π‘Ž ∈ Fin ∧ 𝑏 ∈ Fin) β†’ (π‘Ž βˆͺ 𝑏) ∈ Fin)
3835, 36, 37syl2an 596 . . . . . . . . . . . 12 ((π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) ∧ 𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)) β†’ (π‘Ž βˆͺ 𝑏) ∈ Fin)
3938adantl 482 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) ∧ 𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin))) β†’ (π‘Ž βˆͺ 𝑏) ∈ Fin)
40 eqid 2732 . . . . . . . . . . . 12 (π‘Š β†Ύs ((LSpanβ€˜π‘Š)β€˜(π‘Ž βˆͺ 𝑏))) = (π‘Š β†Ύs ((LSpanβ€˜π‘Š)β€˜(π‘Ž βˆͺ 𝑏)))
417, 8, 40islssfgi 41804 . . . . . . . . . . 11 ((π‘Š ∈ LMod ∧ (π‘Ž βˆͺ 𝑏) βŠ† (Baseβ€˜π‘Š) ∧ (π‘Ž βˆͺ 𝑏) ∈ Fin) β†’ (π‘Š β†Ύs ((LSpanβ€˜π‘Š)β€˜(π‘Ž βˆͺ 𝑏))) ∈ LFinGen)
4229, 33, 39, 41syl3anc 1371 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) ∧ 𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin))) β†’ (π‘Š β†Ύs ((LSpanβ€˜π‘Š)β€˜(π‘Ž βˆͺ 𝑏))) ∈ LFinGen)
4328, 42eqeltrd 2833 . . . . . . . . 9 ((πœ‘ ∧ (π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) ∧ 𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin))) β†’ (π‘Š β†Ύs (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• ((LSpanβ€˜π‘Š)β€˜π‘))) ∈ LFinGen)
4443anassrs 468 . . . . . . . 8 (((πœ‘ ∧ π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)) ∧ 𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)) β†’ (π‘Š β†Ύs (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• ((LSpanβ€˜π‘Š)β€˜π‘))) ∈ LFinGen)
45 oveq2 7416 . . . . . . . . . 10 (((LSpanβ€˜π‘Š)β€˜π‘) = 𝐡 β†’ (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• ((LSpanβ€˜π‘Š)β€˜π‘)) = (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• 𝐡))
4645oveq2d 7424 . . . . . . . . 9 (((LSpanβ€˜π‘Š)β€˜π‘) = 𝐡 β†’ (π‘Š β†Ύs (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• ((LSpanβ€˜π‘Š)β€˜π‘))) = (π‘Š β†Ύs (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• 𝐡)))
4746eleq1d 2818 . . . . . . . 8 (((LSpanβ€˜π‘Š)β€˜π‘) = 𝐡 β†’ ((π‘Š β†Ύs (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• ((LSpanβ€˜π‘Š)β€˜π‘))) ∈ LFinGen ↔ (π‘Š β†Ύs (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• 𝐡)) ∈ LFinGen))
4844, 47syl5ibcom 244 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)) ∧ 𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)) β†’ (((LSpanβ€˜π‘Š)β€˜π‘) = 𝐡 β†’ (π‘Š β†Ύs (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• 𝐡)) ∈ LFinGen))
4948rexlimdva 3155 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)) β†’ (βˆƒπ‘ ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)((LSpanβ€˜π‘Š)β€˜π‘) = 𝐡 β†’ (π‘Š β†Ύs (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• 𝐡)) ∈ LFinGen))
5018, 49mpd 15 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)) β†’ (π‘Š β†Ύs (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• 𝐡)) ∈ LFinGen)
51 oveq1 7415 . . . . . . 7 (((LSpanβ€˜π‘Š)β€˜π‘Ž) = 𝐴 β†’ (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• 𝐡) = (𝐴 βŠ• 𝐡))
5251oveq2d 7424 . . . . . 6 (((LSpanβ€˜π‘Š)β€˜π‘Ž) = 𝐴 β†’ (π‘Š β†Ύs (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• 𝐡)) = (π‘Š β†Ύs (𝐴 βŠ• 𝐡)))
5352eleq1d 2818 . . . . 5 (((LSpanβ€˜π‘Š)β€˜π‘Ž) = 𝐴 β†’ ((π‘Š β†Ύs (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• 𝐡)) ∈ LFinGen ↔ (π‘Š β†Ύs (𝐴 βŠ• 𝐡)) ∈ LFinGen))
5450, 53syl5ibcom 244 . . . 4 ((πœ‘ ∧ π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)) β†’ (((LSpanβ€˜π‘Š)β€˜π‘Ž) = 𝐴 β†’ (π‘Š β†Ύs (𝐴 βŠ• 𝐡)) ∈ LFinGen))
5554rexlimdva 3155 . . 3 (πœ‘ β†’ (βˆƒπ‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)((LSpanβ€˜π‘Š)β€˜π‘Ž) = 𝐴 β†’ (π‘Š β†Ύs (𝐴 βŠ• 𝐡)) ∈ LFinGen))
5611, 55mpd 15 . 2 (πœ‘ β†’ (π‘Š β†Ύs (𝐴 βŠ• 𝐡)) ∈ LFinGen)
571, 56eqeltrid 2837 1 (πœ‘ β†’ 𝐹 ∈ LFinGen)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  β€˜cfv 6543  (class class class)co 7408  Fincfn 8938  Basecbs 17143   β†Ύs cress 17172  LSSumclsm 19501  LModclmod 20470  LSubSpclss 20541  LSpanclspn 20581  LFinGenclfig 41799
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-sca 17212  df-vsca 17213  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-submnd 18671  df-grp 18821  df-minusg 18822  df-sbg 18823  df-subg 19002  df-cntz 19180  df-lsm 19503  df-cmn 19649  df-abl 19650  df-mgp 19987  df-ur 20004  df-ring 20057  df-lmod 20472  df-lss 20542  df-lsp 20582  df-lfig 41800
This theorem is referenced by:  lmhmfgsplit  41818
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