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Theorem lmhmfgsplit 41916
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 1138 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) β†’ 𝑉 ∈ LFinGen)
2 lmhmlmod2 20648 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝑇 ∈ LMod)
323ad2ant1 1133 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) β†’ 𝑇 ∈ LMod)
4 lmhmrnlss 20666 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ ran 𝐹 ∈ (LSubSpβ€˜π‘‡))
543ad2ant1 1133 . . . 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 2732 . . . . 5 (LSubSpβ€˜π‘‡) = (LSubSpβ€˜π‘‡)
8 eqid 2732 . . . . 5 (LSpanβ€˜π‘‡) = (LSpanβ€˜π‘‡)
96, 7, 8islssfg 41900 . . . 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 584 . . 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 1191 . . . . 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 2732 . . . . . 6 (Baseβ€˜π‘†) = (Baseβ€˜π‘†)
14 eqid 2732 . . . . . 6 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
1513, 14lmhmf 20650 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐹:(Baseβ€˜π‘†)⟢(Baseβ€˜π‘‡))
16 ffn 6717 . . . . 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 4609 . . . . 5 (π‘Ž ∈ 𝒫 ran 𝐹 β†’ π‘Ž βŠ† ran 𝐹)
1918ad2antrl 726 . . . 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 779 . . . 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 9360 . . . 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 1371 . . 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 2732 . . . . . . 7 (LSubSpβ€˜π‘†) = (LSubSpβ€˜π‘†)
24 eqid 2732 . . . . . . 7 (LSSumβ€˜π‘†) = (LSSumβ€˜π‘†)
25 lmhmfgsplit.z . . . . . . 7 0 = (0gβ€˜π‘‡)
26 lmhmfgsplit.k . . . . . . 7 𝐾 = (◑𝐹 β€œ { 0 })
27 simpll1 1212 . . . . . . 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 20649 . . . . . . . . . 10 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝑆 ∈ LMod)
29283ad2ant1 1133 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) β†’ 𝑆 ∈ LMod)
3029ad2antrr 724 . . . . . . . 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 4228 . . . . . . . . . . 11 (𝒫 (Baseβ€˜π‘†) ∩ Fin) βŠ† 𝒫 (Baseβ€˜π‘†)
3231sseli 3978 . . . . . . . . . 10 (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) β†’ 𝑏 ∈ 𝒫 (Baseβ€˜π‘†))
33 elpwi 4609 . . . . . . . . . 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 726 . . . . . . . 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 2732 . . . . . . . . 9 (LSpanβ€˜π‘†) = (LSpanβ€˜π‘†)
3713, 23, 36lspcl 20592 . . . . . . . 8 ((𝑆 ∈ LMod ∧ 𝑏 βŠ† (Baseβ€˜π‘†)) β†’ ((LSpanβ€˜π‘†)β€˜π‘) ∈ (LSubSpβ€˜π‘†))
3830, 35, 37syl2anc 584 . . . . . . 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 20665 . . . . . . . . 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 584 . . . . . . . 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 6891 . . . . . . . . 9 ((𝐹 β€œ 𝑏) = π‘Ž β†’ ((LSpanβ€˜π‘‡)β€˜(𝐹 β€œ 𝑏)) = ((LSpanβ€˜π‘‡)β€˜π‘Ž))
4241ad2antll 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†’ ((LSpanβ€˜π‘‡)β€˜(𝐹 β€œ 𝑏)) = ((LSpanβ€˜π‘‡)β€˜π‘Ž))
43 simp2rr 1243 . . . . . . . . 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 1118 . . . . . . . 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 2776 . . . . . . 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 41913 . . . . . 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 7427 . . . . 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 17191 . . . . . . 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 724 . . . . 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 2773 . . . 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 2732 . . . . 5 (𝑆 β†Ύs ((LSpanβ€˜π‘†)β€˜π‘)) = (𝑆 β†Ύs ((LSpanβ€˜π‘†)β€˜π‘))
54 eqid 2732 . . . . 5 (𝑆 β†Ύs (𝐾(LSSumβ€˜π‘†)((LSpanβ€˜π‘†)β€˜π‘))) = (𝑆 β†Ύs (𝐾(LSSumβ€˜π‘†)((LSpanβ€˜π‘†)β€˜π‘)))
5526, 25, 23lmhmkerlss 20667 . . . . . . 7 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐾 ∈ (LSubSpβ€˜π‘†))
56553ad2ant1 1133 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘ˆ ∈ LFinGen ∧ 𝑉 ∈ LFinGen) β†’ 𝐾 ∈ (LSubSpβ€˜π‘†))
5756ad2antrr 724 . . . . 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 1213 . . . . 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 4229 . . . . . . . 8 (𝒫 (Baseβ€˜π‘†) ∩ Fin) βŠ† Fin
6059sseli 3978 . . . . . . 7 (𝑏 ∈ (𝒫 (Baseβ€˜π‘†) ∩ Fin) β†’ 𝑏 ∈ Fin)
6160ad2antrl 726 . . . . . 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 41902 . . . . . 6 ((𝑆 ∈ LMod ∧ 𝑏 βŠ† (Baseβ€˜π‘†) ∧ 𝑏 ∈ Fin) β†’ (𝑆 β†Ύs ((LSpanβ€˜π‘†)β€˜π‘)) ∈ LFinGen)
6330, 35, 61, 62syl3anc 1371 . . . . 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 41904 . . . 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 2833 . . 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 3161 . 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 3161 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  {csn 4628  β—‘ccnv 5675  ran crn 5677   β€œ cima 5679   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  Fincfn 8941  Basecbs 17146   β†Ύs cress 17175  0gc0g 17387  LSSumclsm 19504  LModclmod 20475  LSubSpclss 20547  LSpanclspn 20587   LMHom clmhm 20635  LFinGenclfig 41897
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 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
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 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-sets 17099  df-slot 17117  df-ndx 17129  df-base 17147  df-ress 17176  df-plusg 17212  df-sca 17215  df-vsca 17216  df-0g 17389  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-submnd 18674  df-grp 18824  df-minusg 18825  df-sbg 18826  df-subg 19005  df-ghm 19092  df-cntz 19183  df-lsm 19506  df-cmn 19652  df-abl 19653  df-mgp 19990  df-ur 20007  df-ring 20060  df-lmod 20477  df-lss 20548  df-lsp 20588  df-lmhm 20638  df-lfig 41898
This theorem is referenced by:  lmhmlnmsplit  41917
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