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Theorem lmhmfgsplit 41174
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 1137 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑉 ∈ LFinGen)
2 lmhmlmod2 20400 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
323ad2ant1 1132 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑇 ∈ LMod)
4 lmhmrnlss 20418 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ (LSubSp‘𝑇))
543ad2ant1 1132 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → ran 𝐹 ∈ (LSubSp‘𝑇))
6 lmhmfgsplit.v . . . . 5 𝑉 = (𝑇s ran 𝐹)
7 eqid 2736 . . . . 5 (LSubSp‘𝑇) = (LSubSp‘𝑇)
8 eqid 2736 . . . . 5 (LSpan‘𝑇) = (LSpan‘𝑇)
96, 7, 8islssfg 41158 . . . 4 ((𝑇 ∈ LMod ∧ ran 𝐹 ∈ (LSubSp‘𝑇)) → (𝑉 ∈ LFinGen ↔ ∃𝑎 ∈ 𝒫 ran 𝐹(𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹)))
103, 5, 9syl2anc 584 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → (𝑉 ∈ LFinGen ↔ ∃𝑎 ∈ 𝒫 ran 𝐹(𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹)))
111, 10mpbid 231 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → ∃𝑎 ∈ 𝒫 ran 𝐹(𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))
12 simpl1 1190 . . . . 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 2736 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
14 eqid 2736 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
1513, 14lmhmf 20402 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
16 ffn 6651 . . . . 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 4554 . . . . 5 (𝑎 ∈ 𝒫 ran 𝐹𝑎 ⊆ ran 𝐹)
1918ad2antrl 725 . . . 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 778 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) → 𝑎 ∈ Fin)
21 fipreima 9223 . . . 4 ((𝐹 Fn (Base‘𝑆) ∧ 𝑎 ⊆ ran 𝐹𝑎 ∈ Fin) → ∃𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin)(𝐹𝑏) = 𝑎)
2217, 19, 20, 21syl3anc 1370 . . 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 2736 . . . . . . 7 (LSubSp‘𝑆) = (LSubSp‘𝑆)
24 eqid 2736 . . . . . . 7 (LSSum‘𝑆) = (LSSum‘𝑆)
25 lmhmfgsplit.z . . . . . . 7 0 = (0g𝑇)
26 lmhmfgsplit.k . . . . . . 7 𝐾 = (𝐹 “ { 0 })
27 simpll1 1211 . . . . . . 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 20401 . . . . . . . . . 10 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
29283ad2ant1 1132 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑆 ∈ LMod)
3029ad2antrr 723 . . . . . . . 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 4175 . . . . . . . . . . 11 (𝒫 (Base‘𝑆) ∩ Fin) ⊆ 𝒫 (Base‘𝑆)
3231sseli 3928 . . . . . . . . . 10 (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) → 𝑏 ∈ 𝒫 (Base‘𝑆))
33 elpwi 4554 . . . . . . . . . 10 (𝑏 ∈ 𝒫 (Base‘𝑆) → 𝑏 ⊆ (Base‘𝑆))
3432, 33syl 17 . . . . . . . . 9 (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) → 𝑏 ⊆ (Base‘𝑆))
3534ad2antrl 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)) → 𝑏 ⊆ (Base‘𝑆))
36 eqid 2736 . . . . . . . . 9 (LSpan‘𝑆) = (LSpan‘𝑆)
3713, 23, 36lspcl 20344 . . . . . . . 8 ((𝑆 ∈ LMod ∧ 𝑏 ⊆ (Base‘𝑆)) → ((LSpan‘𝑆)‘𝑏) ∈ (LSubSp‘𝑆))
3830, 35, 37syl2anc 584 . . . . . . 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 20417 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑏 ⊆ (Base‘𝑆)) → (𝐹 “ ((LSpan‘𝑆)‘𝑏)) = ((LSpan‘𝑇)‘(𝐹𝑏)))
4027, 35, 39syl2anc 584 . . . . . . . 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 6825 . . . . . . . . 9 ((𝐹𝑏) = 𝑎 → ((LSpan‘𝑇)‘(𝐹𝑏)) = ((LSpan‘𝑇)‘𝑎))
4241ad2antll 726 . . . . . . . 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 1242 . . . . . . . . 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 1117 . . . . . . . 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 2780 . . . . . . 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 41171 . . . . . 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 7353 . . . . 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 17051 . . . . . . 7 (𝑆 ∈ LMod → (𝑆s (Base‘𝑆)) = 𝑆)
4929, 48syl 17 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → (𝑆s (Base‘𝑆)) = 𝑆)
5049ad2antrr 723 . . . . 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 2777 . . . 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 2736 . . . . 5 (𝑆s ((LSpan‘𝑆)‘𝑏)) = (𝑆s ((LSpan‘𝑆)‘𝑏))
54 eqid 2736 . . . . 5 (𝑆s (𝐾(LSSum‘𝑆)((LSpan‘𝑆)‘𝑏))) = (𝑆s (𝐾(LSSum‘𝑆)((LSpan‘𝑆)‘𝑏)))
5526, 25, 23lmhmkerlss 20419 . . . . . . 7 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐾 ∈ (LSubSp‘𝑆))
56553ad2ant1 1132 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝐾 ∈ (LSubSp‘𝑆))
5756ad2antrr 723 . . . . 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 1212 . . . . 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 4176 . . . . . . . 8 (𝒫 (Base‘𝑆) ∩ Fin) ⊆ Fin
6059sseli 3928 . . . . . . 7 (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) → 𝑏 ∈ Fin)
6160ad2antrl 725 . . . . . 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 41160 . . . . . 6 ((𝑆 ∈ LMod ∧ 𝑏 ⊆ (Base‘𝑆) ∧ 𝑏 ∈ Fin) → (𝑆s ((LSpan‘𝑆)‘𝑏)) ∈ LFinGen)
6330, 35, 61, 62syl3anc 1370 . . . . 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 41162 . . . 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 2837 . . 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 3154 . 2 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) → 𝑆 ∈ LFinGen)
6711, 66rexlimddv 3154 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑆 ∈ LFinGen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  wrex 3070  cin 3897  wss 3898  𝒫 cpw 4547  {csn 4573  ccnv 5619  ran crn 5621  cima 5623   Fn wfn 6474  wf 6475  cfv 6479  (class class class)co 7337  Fincfn 8804  Basecbs 17009  s cress 17038  0gc0g 17247  LSSumclsm 19335  LModclmod 20229  LSubSpclss 20299  LSpanclspn 20339   LMHom clmhm 20387  LFinGenclfig 41155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-1st 7899  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-1o 8367  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-fin 8808  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-nn 12075  df-2 12137  df-3 12138  df-4 12139  df-5 12140  df-6 12141  df-sets 16962  df-slot 16980  df-ndx 16992  df-base 17010  df-ress 17039  df-plusg 17072  df-sca 17075  df-vsca 17076  df-0g 17249  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-submnd 18528  df-grp 18676  df-minusg 18677  df-sbg 18678  df-subg 18848  df-ghm 18928  df-cntz 19019  df-lsm 19337  df-cmn 19483  df-abl 19484  df-mgp 19816  df-ur 19833  df-ring 19880  df-lmod 20231  df-lss 20300  df-lsp 20340  df-lmhm 20390  df-lfig 41156
This theorem is referenced by:  lmhmlnmsplit  41175
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