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Theorem lmhmfgsplit 43043
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 = (0g𝑇)
lmhmfgsplit.k 𝐾 = (𝐹 “ { 0 })
lmhmfgsplit.u 𝑈 = (𝑆s 𝐾)
lmhmfgsplit.v 𝑉 = (𝑇s ran 𝐹)
Assertion
Ref Expression
lmhmfgsplit ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑆 ∈ LFinGen)

Proof of Theorem lmhmfgsplit
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 1138 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑉 ∈ LFinGen)
2 lmhmlmod2 21054 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
323ad2ant1 1133 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑇 ∈ LMod)
4 lmhmrnlss 21072 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ (LSubSp‘𝑇))
543ad2ant1 1133 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → ran 𝐹 ∈ (LSubSp‘𝑇))
6 lmhmfgsplit.v . . . . 5 𝑉 = (𝑇s ran 𝐹)
7 eqid 2740 . . . . 5 (LSubSp‘𝑇) = (LSubSp‘𝑇)
8 eqid 2740 . . . . 5 (LSpan‘𝑇) = (LSpan‘𝑇)
96, 7, 8islssfg 43027 . . . 4 ((𝑇 ∈ LMod ∧ ran 𝐹 ∈ (LSubSp‘𝑇)) → (𝑉 ∈ LFinGen ↔ ∃𝑎 ∈ 𝒫 ran 𝐹(𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹)))
103, 5, 9syl2anc 583 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → (𝑉 ∈ LFinGen ↔ ∃𝑎 ∈ 𝒫 ran 𝐹(𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹)))
111, 10mpbid 232 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → ∃𝑎 ∈ 𝒫 ran 𝐹(𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))
12 simpl1 1191 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
13 eqid 2740 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
14 eqid 2740 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
1513, 14lmhmf 21056 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
16 ffn 6747 . . . . 5 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
1712, 15, 163syl 18 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) → 𝐹 Fn (Base‘𝑆))
18 elpwi 4629 . . . . 5 (𝑎 ∈ 𝒫 ran 𝐹𝑎 ⊆ ran 𝐹)
1918ad2antrl 727 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) → 𝑎 ⊆ ran 𝐹)
20 simprrl 780 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) → 𝑎 ∈ Fin)
21 fipreima 9428 . . . 4 ((𝐹 Fn (Base‘𝑆) ∧ 𝑎 ⊆ ran 𝐹𝑎 ∈ Fin) → ∃𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin)(𝐹𝑏) = 𝑎)
2217, 19, 20, 21syl3anc 1371 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) → ∃𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin)(𝐹𝑏) = 𝑎)
23 eqid 2740 . . . . . . 7 (LSubSp‘𝑆) = (LSubSp‘𝑆)
24 eqid 2740 . . . . . . 7 (LSSum‘𝑆) = (LSSum‘𝑆)
25 lmhmfgsplit.z . . . . . . 7 0 = (0g𝑇)
26 lmhmfgsplit.k . . . . . . 7 𝐾 = (𝐹 “ { 0 })
27 simpll1 1212 . . . . . . 7 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
28 lmhmlmod1 21055 . . . . . . . . . 10 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
29283ad2ant1 1133 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑆 ∈ LMod)
3029ad2antrr 725 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → 𝑆 ∈ LMod)
31 inss1 4258 . . . . . . . . . . 11 (𝒫 (Base‘𝑆) ∩ Fin) ⊆ 𝒫 (Base‘𝑆)
3231sseli 4004 . . . . . . . . . 10 (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) → 𝑏 ∈ 𝒫 (Base‘𝑆))
33 elpwi 4629 . . . . . . . . . 10 (𝑏 ∈ 𝒫 (Base‘𝑆) → 𝑏 ⊆ (Base‘𝑆))
3432, 33syl 17 . . . . . . . . 9 (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) → 𝑏 ⊆ (Base‘𝑆))
3534ad2antrl 727 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → 𝑏 ⊆ (Base‘𝑆))
36 eqid 2740 . . . . . . . . 9 (LSpan‘𝑆) = (LSpan‘𝑆)
3713, 23, 36lspcl 20997 . . . . . . . 8 ((𝑆 ∈ LMod ∧ 𝑏 ⊆ (Base‘𝑆)) → ((LSpan‘𝑆)‘𝑏) ∈ (LSubSp‘𝑆))
3830, 35, 37syl2anc 583 . . . . . . 7 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → ((LSpan‘𝑆)‘𝑏) ∈ (LSubSp‘𝑆))
3913, 36, 8lmhmlsp 21071 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑏 ⊆ (Base‘𝑆)) → (𝐹 “ ((LSpan‘𝑆)‘𝑏)) = ((LSpan‘𝑇)‘(𝐹𝑏)))
4027, 35, 39syl2anc 583 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → (𝐹 “ ((LSpan‘𝑆)‘𝑏)) = ((LSpan‘𝑇)‘(𝐹𝑏)))
41 fveq2 6920 . . . . . . . . 9 ((𝐹𝑏) = 𝑎 → ((LSpan‘𝑇)‘(𝐹𝑏)) = ((LSpan‘𝑇)‘𝑎))
4241ad2antll 728 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → ((LSpan‘𝑇)‘(𝐹𝑏)) = ((LSpan‘𝑇)‘𝑎))
43 simp2rr 1243 . . . . . . . . 9 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹)) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → ((LSpan‘𝑇)‘𝑎) = ran 𝐹)
44433expa 1118 . . . . . . . 8 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → ((LSpan‘𝑇)‘𝑎) = ran 𝐹)
4540, 42, 443eqtrd 2784 . . . . . . 7 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → (𝐹 “ ((LSpan‘𝑆)‘𝑏)) = ran 𝐹)
4623, 24, 25, 26, 13, 27, 38, 45kercvrlsm 43040 . . . . . 6 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → (𝐾(LSSum‘𝑆)((LSpan‘𝑆)‘𝑏)) = (Base‘𝑆))
4746oveq2d 7464 . . . . 5 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → (𝑆s (𝐾(LSSum‘𝑆)((LSpan‘𝑆)‘𝑏))) = (𝑆s (Base‘𝑆)))
4813ressid 17303 . . . . . . 7 (𝑆 ∈ LMod → (𝑆s (Base‘𝑆)) = 𝑆)
4929, 48syl 17 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → (𝑆s (Base‘𝑆)) = 𝑆)
5049ad2antrr 725 . . . . 5 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → (𝑆s (Base‘𝑆)) = 𝑆)
5147, 50eqtr2d 2781 . . . 4 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → 𝑆 = (𝑆s (𝐾(LSSum‘𝑆)((LSpan‘𝑆)‘𝑏))))
52 lmhmfgsplit.u . . . . 5 𝑈 = (𝑆s 𝐾)
53 eqid 2740 . . . . 5 (𝑆s ((LSpan‘𝑆)‘𝑏)) = (𝑆s ((LSpan‘𝑆)‘𝑏))
54 eqid 2740 . . . . 5 (𝑆s (𝐾(LSSum‘𝑆)((LSpan‘𝑆)‘𝑏))) = (𝑆s (𝐾(LSSum‘𝑆)((LSpan‘𝑆)‘𝑏)))
5526, 25, 23lmhmkerlss 21073 . . . . . . 7 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐾 ∈ (LSubSp‘𝑆))
56553ad2ant1 1133 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝐾 ∈ (LSubSp‘𝑆))
5756ad2antrr 725 . . . . 5 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → 𝐾 ∈ (LSubSp‘𝑆))
58 simpll2 1213 . . . . 5 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → 𝑈 ∈ LFinGen)
59 inss2 4259 . . . . . . . 8 (𝒫 (Base‘𝑆) ∩ Fin) ⊆ Fin
6059sseli 4004 . . . . . . 7 (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) → 𝑏 ∈ Fin)
6160ad2antrl 727 . . . . . 6 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → 𝑏 ∈ Fin)
6236, 13, 53islssfgi 43029 . . . . . 6 ((𝑆 ∈ LMod ∧ 𝑏 ⊆ (Base‘𝑆) ∧ 𝑏 ∈ Fin) → (𝑆s ((LSpan‘𝑆)‘𝑏)) ∈ LFinGen)
6330, 35, 61, 62syl3anc 1371 . . . . 5 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → (𝑆s ((LSpan‘𝑆)‘𝑏)) ∈ LFinGen)
6423, 24, 52, 53, 54, 30, 57, 38, 58, 63lsmfgcl 43031 . . . 4 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → (𝑆s (𝐾(LSSum‘𝑆)((LSpan‘𝑆)‘𝑏))) ∈ LFinGen)
6551, 64eqeltrd 2844 . . 3 ((((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) ∧ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) ∧ (𝐹𝑏) = 𝑎)) → 𝑆 ∈ LFinGen)
6622, 65rexlimddv 3167 . 2 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) → 𝑆 ∈ LFinGen)
6711, 66rexlimddv 3167 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑆 ∈ LFinGen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1537  wcel 2108  wrex 3076  cin 3975  wss 3976  𝒫 cpw 4622  {csn 4648  ccnv 5699  ran crn 5701  cima 5703   Fn wfn 6568  wf 6569  cfv 6573  (class class class)co 7448  Fincfn 9003  Basecbs 17258  s cress 17287  0gc0g 17499  LSSumclsm 19676  LModclmod 20880  LSubSpclss 20952  LSpanclspn 20992   LMHom clmhm 21041  LFinGenclfig 43024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-sca 17327  df-vsca 17328  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-subg 19163  df-ghm 19253  df-cntz 19357  df-lsm 19678  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-lmod 20882  df-lss 20953  df-lsp 20993  df-lmhm 21044  df-lfig 43025
This theorem is referenced by:  lmhmlnmsplit  43044
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