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Theorem lsmfgcl 41864
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 2733 . . . . . 6 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
8 eqid 2733 . . . . . 6 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
95, 6, 7, 8islssfg2 41861 . . . . 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 585 . . . 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 41861 . . . . . . . . 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 585 . . . . . . . 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 482 . . . . . 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 4229 . . . . . . . . . . . . . . 15 (𝒫 (Baseβ€˜π‘Š) ∩ Fin) βŠ† 𝒫 (Baseβ€˜π‘Š)
2019sseli 3979 . . . . . . . . . . . . . 14 (π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) β†’ π‘Ž ∈ 𝒫 (Baseβ€˜π‘Š))
2120elpwid 4612 . . . . . . . . . . . . 13 (π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) β†’ π‘Ž βŠ† (Baseβ€˜π‘Š))
2219sseli 3979 . . . . . . . . . . . . . 14 (𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) β†’ 𝑏 ∈ 𝒫 (Baseβ€˜π‘Š))
2322elpwid 4612 . . . . . . . . . . . . 13 (𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) β†’ 𝑏 βŠ† (Baseβ€˜π‘Š))
24 lsmfgcl.p . . . . . . . . . . . . . 14 βŠ• = (LSSumβ€˜π‘Š)
258, 7, 24lsmsp2 20698 . . . . . . . . . . . . 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 1161 . . . . . . . . . . . 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 1121 . . . . . . . . . . 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 7425 . . . . . . . . . 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 482 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) ∧ 𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin))) β†’ π‘Š ∈ LMod)
30 unss 4185 . . . . . . . . . . . . . 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 597 . . . . . . . . . . . 12 ((π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) ∧ 𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)) β†’ (π‘Ž βˆͺ 𝑏) βŠ† (Baseβ€˜π‘Š))
3332adantl 483 . . . . . . . . . . 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 4230 . . . . . . . . . . . . . 14 (𝒫 (Baseβ€˜π‘Š) ∩ Fin) βŠ† Fin
3534sseli 3979 . . . . . . . . . . . . 13 (π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) β†’ π‘Ž ∈ Fin)
3634sseli 3979 . . . . . . . . . . . . 13 (𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) β†’ 𝑏 ∈ Fin)
37 unfi 9172 . . . . . . . . . . . . 13 ((π‘Ž ∈ Fin ∧ 𝑏 ∈ Fin) β†’ (π‘Ž βˆͺ 𝑏) ∈ Fin)
3835, 36, 37syl2an 597 . . . . . . . . . . . 12 ((π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) ∧ 𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin)) β†’ (π‘Ž βˆͺ 𝑏) ∈ Fin)
3938adantl 483 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin) ∧ 𝑏 ∈ (𝒫 (Baseβ€˜π‘Š) ∩ Fin))) β†’ (π‘Ž βˆͺ 𝑏) ∈ Fin)
40 eqid 2733 . . . . . . . . . . . 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 41862 . . . . . . . . . . 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 1372 . . . . . . . . . 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 2834 . . . . . . . . 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 469 . . . . . . . 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 7417 . . . . . . . . . 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 7425 . . . . . . . . 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 2819 . . . . . . . 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 3156 . . . . . 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 7416 . . . . . . 7 (((LSpanβ€˜π‘Š)β€˜π‘Ž) = 𝐴 β†’ (((LSpanβ€˜π‘Š)β€˜π‘Ž) βŠ• 𝐡) = (𝐴 βŠ• 𝐡))
5251oveq2d 7425 . . . . . 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 2819 . . . . 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 3156 . . 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 2838 1 (πœ‘ β†’ 𝐹 ∈ LFinGen)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071   βˆͺ cun 3947   ∩ cin 3948   βŠ† wss 3949  π’« cpw 4603  β€˜cfv 6544  (class class class)co 7409  Fincfn 8939  Basecbs 17144   β†Ύs cress 17173  LSSumclsm 19502  LModclmod 20471  LSubSpclss 20542  LSpanclspn 20582  LFinGenclfig 41857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-sca 17213  df-vsca 17214  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-submnd 18672  df-grp 18822  df-minusg 18823  df-sbg 18824  df-subg 19003  df-cntz 19181  df-lsm 19504  df-cmn 19650  df-abl 19651  df-mgp 19988  df-ur 20005  df-ring 20058  df-lmod 20473  df-lss 20543  df-lsp 20583  df-lfig 41858
This theorem is referenced by:  lmhmfgsplit  41876
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