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Theorem lmhmfgsplit 41828
Description: If the kernel and range of a homomorphism of left modules are finitely generated, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypotheses
Ref Expression
lmhmfgsplit.z 0 = (0gβ€˜π‘‡)
lmhmfgsplit.k 𝐾 = (◑𝐹 β€œ { 0 })
lmhmfgsplit.u π‘ˆ = (𝑆 β†Ύs 𝐾)
lmhmfgsplit.v 𝑉 = (𝑇 β†Ύs ran 𝐹)
Assertion
Ref Expression
lmhmfgsplit ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) β†’ 𝑆 ∈ LFinGen)

Proof of Theorem lmhmfgsplit
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 1139 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) β†’ 𝑉 ∈ LFinGen)
2 lmhmlmod2 20643 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝑇 ∈ LMod)
323ad2ant1 1134 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) β†’ 𝑇 ∈ LMod)
4 lmhmrnlss 20661 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ ran 𝐹 ∈ (LSubSpβ€˜π‘‡))
543ad2ant1 1134 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) β†’ ran 𝐹 ∈ (LSubSpβ€˜π‘‡))
6 lmhmfgsplit.v . . . . 5 𝑉 = (𝑇 β†Ύs ran 𝐹)
7 eqid 2733 . . . . 5 (LSubSpβ€˜π‘‡) = (LSubSpβ€˜π‘‡)
8 eqid 2733 . . . . 5 (LSpanβ€˜π‘‡) = (LSpanβ€˜π‘‡)
96, 7, 8islssfg 41812 . . . 4 ((𝑇 ∈ LMod ∧ ran 𝐹 ∈ (LSubSpβ€˜π‘‡)) β†’ (𝑉 ∈ LFinGen ↔ βˆƒπ‘Ž ∈ 𝒫 ran 𝐹(π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹)))
103, 5, 9syl2anc 585 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) β†’ (𝑉 ∈ LFinGen ↔ βˆƒπ‘Ž ∈ 𝒫 ran 𝐹(π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹)))
111, 10mpbid 231 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) β†’ βˆƒπ‘Ž ∈ 𝒫 ran 𝐹(π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))
12 simpl1 1192 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) β†’ 𝐹 ∈ (𝑆 LMHom 𝑇))
13 eqid 2733 . . . . . 6 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
14 eqid 2733 . . . . . 6 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
1513, 14lmhmf 20645 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
16 ffn 6718 . . . . 5 (𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡) β†’ 𝐹 Fn (Baseβ€˜π‘†))
1712, 15, 163syl 18 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) β†’ 𝐹 Fn (Baseβ€˜π‘†))
18 elpwi 4610 . . . . 5 (π‘Ž ∈ 𝒫 ran 𝐹 β†’ π‘Ž βŠ† ran 𝐹)
1918ad2antrl 727 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) β†’ π‘Ž βŠ† ran 𝐹)
20 simprrl 780 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) β†’ π‘Ž ∈ Fin)
21 fipreima 9358 . . . 4 ((𝐹 Fn (Baseβ€˜π‘†) ∧ π‘Ž βŠ† ran 𝐹 ∧ π‘Ž ∈ Fin) β†’ βˆƒπ‘ ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin)(𝐹 β€œ 𝑏) = π‘Ž)
2217, 19, 20, 21syl3anc 1372 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) β†’ βˆƒπ‘ ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin)(𝐹 β€œ 𝑏) = π‘Ž)
23 eqid 2733 . . . . . . 7 (LSubSpβ€˜π‘†) = (LSubSpβ€˜π‘†)
24 eqid 2733 . . . . . . 7 (LSSumβ€˜π‘†) = (LSSumβ€˜π‘†)
25 lmhmfgsplit.z . . . . . . 7 0 = (0gβ€˜π‘‡)
26 lmhmfgsplit.k . . . . . . 7 𝐾 = (◑𝐹 β€œ { 0 })
27 simpll1 1213 . . . . . . 7 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ 𝐹 ∈ (𝑆 LMHom 𝑇))
28 lmhmlmod1 20644 . . . . . . . . . 10 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝑆 ∈ LMod)
29283ad2ant1 1134 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) β†’ 𝑆 ∈ LMod)
3029ad2antrr 725 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ 𝑆 ∈ LMod)
31 inss1 4229 . . . . . . . . . . 11 (𝒫 (Baseβ€˜π‘†) ∩ Fin) βŠ† 𝒫 (Baseβ€˜π‘†)
3231sseli 3979 . . . . . . . . . 10 (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) β†’ 𝑏 ∈ 𝒫 (Baseβ€˜π‘†))
33 elpwi 4610 . . . . . . . . . 10 (𝑏 ∈ 𝒫 (Baseβ€˜π‘†) β†’ 𝑏 βŠ† (Baseβ€˜π‘†))
3432, 33syl 17 . . . . . . . . 9 (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) β†’ 𝑏 βŠ† (Baseβ€˜π‘†))
3534ad2antrl 727 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ 𝑏 βŠ† (Baseβ€˜π‘†))
36 eqid 2733 . . . . . . . . 9 (LSpanβ€˜π‘†) = (LSpanβ€˜π‘†)
3713, 23, 36lspcl 20587 . . . . . . . 8 ((𝑆 ∈ LMod ∧ 𝑏 βŠ† (Baseβ€˜π‘†)) β†’ ((LSpanβ€˜π‘†)β€˜π‘) ∈ (LSubSpβ€˜π‘†))
3830, 35, 37syl2anc 585 . . . . . . 7 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ ((LSpanβ€˜π‘†)β€˜π‘) ∈ (LSubSpβ€˜π‘†))
3913, 36, 8lmhmlsp 20660 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑏 βŠ† (Baseβ€˜π‘†)) β†’ (𝐹 β€œ ((LSpanβ€˜π‘†)β€˜π‘)) = ((LSpanβ€˜π‘‡)β€˜(𝐹 β€œ 𝑏)))
4027, 35, 39syl2anc 585 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ (𝐹 β€œ ((LSpanβ€˜π‘†)β€˜π‘)) = ((LSpanβ€˜π‘‡)β€˜(𝐹 β€œ 𝑏)))
41 fveq2 6892 . . . . . . . . 9 ((𝐹 β€œ 𝑏) = π‘Ž β†’ ((LSpanβ€˜π‘‡)β€˜(𝐹 β€œ 𝑏)) = ((LSpanβ€˜π‘‡)β€˜π‘Ž))
4241ad2antll 728 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ ((LSpanβ€˜π‘‡)β€˜(𝐹 β€œ 𝑏)) = ((LSpanβ€˜π‘‡)β€˜π‘Ž))
43 simp2rr 1244 . . . . . . . . 9 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹)) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹)
44433expa 1119 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹)
4540, 42, 443eqtrd 2777 . . . . . . 7 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ (𝐹 β€œ ((LSpanβ€˜π‘†)β€˜π‘)) = ran 𝐹)
4623, 24, 25, 26, 13, 27, 38, 45kercvrlsm 41825 . . . . . 6 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ (𝐾(LSSumβ€˜π‘†)((LSpanβ€˜π‘†)β€˜π‘)) = (Baseβ€˜π‘†))
4746oveq2d 7425 . . . . 5 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ (𝑆 β†Ύs (𝐾(LSSumβ€˜π‘†)((LSpanβ€˜π‘†)β€˜π‘))) = (𝑆 β†Ύs (Baseβ€˜π‘†)))
4813ressid 17189 . . . . . . 7 (𝑆 ∈ LMod β†’ (𝑆 β†Ύs (Baseβ€˜π‘†)) = 𝑆)
4929, 48syl 17 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) β†’ (𝑆 β†Ύs (Baseβ€˜π‘†)) = 𝑆)
5049ad2antrr 725 . . . . 5 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ (𝑆 β†Ύs (Baseβ€˜π‘†)) = 𝑆)
5147, 50eqtr2d 2774 . . . 4 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ 𝑆 = (𝑆 β†Ύs (𝐾(LSSumβ€˜π‘†)((LSpanβ€˜π‘†)β€˜π‘))))
52 lmhmfgsplit.u . . . . 5 π‘ˆ = (𝑆 β†Ύs 𝐾)
53 eqid 2733 . . . . 5 (𝑆 β†Ύs ((LSpanβ€˜π‘†)β€˜π‘)) = (𝑆 β†Ύs ((LSpanβ€˜π‘†)β€˜π‘))
54 eqid 2733 . . . . 5 (𝑆 β†Ύs (𝐾(LSSumβ€˜π‘†)((LSpanβ€˜π‘†)β€˜π‘))) = (𝑆 β†Ύs (𝐾(LSSumβ€˜π‘†)((LSpanβ€˜π‘†)β€˜π‘)))
5526, 25, 23lmhmkerlss 20662 . . . . . . 7 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐾 ∈ (LSubSpβ€˜π‘†))
56553ad2ant1 1134 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) β†’ 𝐾 ∈ (LSubSpβ€˜π‘†))
5756ad2antrr 725 . . . . 5 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ 𝐾 ∈ (LSubSpβ€˜π‘†))
58 simpll2 1214 . . . . 5 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ π‘ˆ ∈ LFinGen)
59 inss2 4230 . . . . . . . 8 (𝒫 (Baseβ€˜π‘†) ∩ Fin) βŠ† Fin
6059sseli 3979 . . . . . . 7 (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) β†’ 𝑏 ∈ Fin)
6160ad2antrl 727 . . . . . 6 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ 𝑏 ∈ Fin)
6236, 13, 53islssfgi 41814 . . . . . 6 ((𝑆 ∈ LMod ∧ 𝑏 βŠ† (Baseβ€˜π‘†) ∧ 𝑏 ∈ Fin) β†’ (𝑆 β†Ύs ((LSpanβ€˜π‘†)β€˜π‘)) ∈ LFinGen)
6330, 35, 61, 62syl3anc 1372 . . . . 5 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ (𝑆 β†Ύs ((LSpanβ€˜π‘†)β€˜π‘)) ∈ LFinGen)
6423, 24, 52, 53, 54, 30, 57, 38, 58, 63lsmfgcl 41816 . . . 4 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ (𝑆 β†Ύs (𝐾(LSSumβ€˜π‘†)((LSpanβ€˜π‘†)β€˜π‘))) ∈ LFinGen)
6551, 64eqeltrd 2834 . . 3 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) ∧ (𝐹 β€œ 𝑏) = π‘Ž)) β†’ 𝑆 ∈ LFinGen)
6622, 65rexlimddv 3162 . 2 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (π‘Ž ∈ 𝒫 ran 𝐹 ∧ (π‘Ž ∈ Fin ∧ ((LSpanβ€˜π‘‡)β€˜π‘Ž) = ran 𝐹))) β†’ 𝑆 ∈ LFinGen)
6711, 66rexlimddv 3162 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) β†’ 𝑆 ∈ LFinGen)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071   ∩ cin 3948   βŠ† wss 3949  π’« cpw 4603  {csn 4629  β—‘ccnv 5676  ran crn 5678   β€œ cima 5680   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  Fincfn 8939  Basecbs 17144   β†Ύs cress 17173  0gc0g 17385  LSSumclsm 19502  LModclmod 20471  LSubSpclss 20542  LSpanclspn 20582   LMHom clmhm 20630  LFinGenclfig 41809
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-1o 8466  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-ghm 19090  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-lmhm 20633  df-lfig 41810
This theorem is referenced by:  lmhmlnmsplit  41829
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