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Theorem lmhmfgsplit 38154
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 1133 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑉 ∈ LFinGen)
2 lmhmlmod2 19230 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
323ad2ant1 1128 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑇 ∈ LMod)
4 lmhmrnlss 19248 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ (LSubSp‘𝑇))
543ad2ant1 1128 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → ran 𝐹 ∈ (LSubSp‘𝑇))
6 lmhmfgsplit.v . . . . 5 𝑉 = (𝑇s ran 𝐹)
7 eqid 2756 . . . . 5 (LSubSp‘𝑇) = (LSubSp‘𝑇)
8 eqid 2756 . . . . 5 (LSpan‘𝑇) = (LSpan‘𝑇)
96, 7, 8islssfg 38138 . . . 4 ((𝑇 ∈ LMod ∧ ran 𝐹 ∈ (LSubSp‘𝑇)) → (𝑉 ∈ LFinGen ↔ ∃𝑎 ∈ 𝒫 ran 𝐹(𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹)))
103, 5, 9syl2anc 696 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → (𝑉 ∈ LFinGen ↔ ∃𝑎 ∈ 𝒫 ran 𝐹(𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹)))
111, 10mpbid 222 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → ∃𝑎 ∈ 𝒫 ran 𝐹(𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))
12 simpl1 1228 . . . . 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 2756 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
14 eqid 2756 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
1513, 14lmhmf 19232 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
16 ffn 6202 . . . . 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 4308 . . . . 5 (𝑎 ∈ 𝒫 ran 𝐹𝑎 ⊆ ran 𝐹)
1918ad2antrl 766 . . . 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 823 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) → 𝑎 ∈ Fin)
21 fipreima 8433 . . . 4 ((𝐹 Fn (Base‘𝑆) ∧ 𝑎 ⊆ ran 𝐹𝑎 ∈ Fin) → ∃𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin)(𝐹𝑏) = 𝑎)
2217, 19, 20, 21syl3anc 1477 . . 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 2756 . . . . . . 7 (LSubSp‘𝑆) = (LSubSp‘𝑆)
24 eqid 2756 . . . . . . 7 (LSSum‘𝑆) = (LSSum‘𝑆)
25 lmhmfgsplit.z . . . . . . 7 0 = (0g𝑇)
26 lmhmfgsplit.k . . . . . . 7 𝐾 = (𝐹 “ { 0 })
27 simpll1 1255 . . . . . . 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 19231 . . . . . . . . . 10 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
29283ad2ant1 1128 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑆 ∈ LMod)
3029ad2antrr 764 . . . . . . . 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 3972 . . . . . . . . . . 11 (𝒫 (Base‘𝑆) ∩ Fin) ⊆ 𝒫 (Base‘𝑆)
3231sseli 3736 . . . . . . . . . 10 (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) → 𝑏 ∈ 𝒫 (Base‘𝑆))
33 elpwi 4308 . . . . . . . . . 10 (𝑏 ∈ 𝒫 (Base‘𝑆) → 𝑏 ⊆ (Base‘𝑆))
3432, 33syl 17 . . . . . . . . 9 (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) → 𝑏 ⊆ (Base‘𝑆))
3534ad2antrl 766 . . . . . . . 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 2756 . . . . . . . . 9 (LSpan‘𝑆) = (LSpan‘𝑆)
3713, 23, 36lspcl 19174 . . . . . . . 8 ((𝑆 ∈ LMod ∧ 𝑏 ⊆ (Base‘𝑆)) → ((LSpan‘𝑆)‘𝑏) ∈ (LSubSp‘𝑆))
3830, 35, 37syl2anc 696 . . . . . . 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 19247 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑏 ⊆ (Base‘𝑆)) → (𝐹 “ ((LSpan‘𝑆)‘𝑏)) = ((LSpan‘𝑇)‘(𝐹𝑏)))
4027, 35, 39syl2anc 696 . . . . . . . 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 6348 . . . . . . . . 9 ((𝐹𝑏) = 𝑎 → ((LSpan‘𝑇)‘(𝐹𝑏)) = ((LSpan‘𝑇)‘𝑎))
4241ad2antll 767 . . . . . . . 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 1310 . . . . . . . . 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 1112 . . . . . . . 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 2794 . . . . . . 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 38151 . . . . . 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 6825 . . . . 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 16133 . . . . . . 7 (𝑆 ∈ LMod → (𝑆s (Base‘𝑆)) = 𝑆)
4929, 48syl 17 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → (𝑆s (Base‘𝑆)) = 𝑆)
5049ad2antrr 764 . . . . 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 2791 . . . 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 2756 . . . . 5 (𝑆s ((LSpan‘𝑆)‘𝑏)) = (𝑆s ((LSpan‘𝑆)‘𝑏))
54 eqid 2756 . . . . 5 (𝑆s (𝐾(LSSum‘𝑆)((LSpan‘𝑆)‘𝑏))) = (𝑆s (𝐾(LSSum‘𝑆)((LSpan‘𝑆)‘𝑏)))
5526, 25, 23lmhmkerlss 19249 . . . . . . 7 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐾 ∈ (LSubSp‘𝑆))
56553ad2ant1 1128 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝐾 ∈ (LSubSp‘𝑆))
5756ad2antrr 764 . . . . 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 1257 . . . . 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 3973 . . . . . . . 8 (𝒫 (Base‘𝑆) ∩ Fin) ⊆ Fin
6059sseli 3736 . . . . . . 7 (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) → 𝑏 ∈ Fin)
6160ad2antrl 766 . . . . . 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 38140 . . . . . 6 ((𝑆 ∈ LMod ∧ 𝑏 ⊆ (Base‘𝑆) ∧ 𝑏 ∈ Fin) → (𝑆s ((LSpan‘𝑆)‘𝑏)) ∈ LFinGen)
6330, 35, 61, 62syl3anc 1477 . . . . 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 38142 . . . 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 2835 . . 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 3169 . 2 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) → 𝑆 ∈ LFinGen)
6711, 66rexlimddv 3169 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑆 ∈ LFinGen)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1628  wcel 2135  wrex 3047  cin 3710  wss 3711  𝒫 cpw 4298  {csn 4317  ccnv 5261  ran crn 5263  cima 5265   Fn wfn 6040  wf 6041  cfv 6045  (class class class)co 6809  Fincfn 8117  Basecbs 16055  s cress 16056  0gc0g 16298  LSSumclsm 18245  LModclmod 19061  LSubSpclss 19130  LSpanclspn 19169   LMHom clmhm 19217  LFinGenclfig 38135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110  ax-cnex 10180  ax-resscn 10181  ax-1cn 10182  ax-icn 10183  ax-addcl 10184  ax-addrcl 10185  ax-mulcl 10186  ax-mulrcl 10187  ax-mulcom 10188  ax-addass 10189  ax-mulass 10190  ax-distr 10191  ax-i2m1 10192  ax-1ne0 10193  ax-1rid 10194  ax-rnegex 10195  ax-rrecex 10196  ax-cnre 10197  ax-pre-lttri 10198  ax-pre-lttrn 10199  ax-pre-ltadd 10200  ax-pre-mulgt0 10201
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-nel 3032  df-ral 3051  df-rex 3052  df-reu 3053  df-rmo 3054  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-int 4624  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-riota 6770  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-om 7227  df-1st 7329  df-2nd 7330  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-1o 7725  df-oadd 7729  df-er 7907  df-en 8118  df-dom 8119  df-sdom 8120  df-fin 8121  df-pnf 10264  df-mnf 10265  df-xr 10266  df-ltxr 10267  df-le 10268  df-sub 10456  df-neg 10457  df-nn 11209  df-2 11267  df-3 11268  df-4 11269  df-5 11270  df-6 11271  df-ndx 16058  df-slot 16059  df-base 16061  df-sets 16062  df-ress 16063  df-plusg 16152  df-sca 16155  df-vsca 16156  df-0g 16300  df-mgm 17439  df-sgrp 17481  df-mnd 17492  df-submnd 17533  df-grp 17622  df-minusg 17623  df-sbg 17624  df-subg 17788  df-ghm 17855  df-cntz 17946  df-lsm 18247  df-cmn 18391  df-abl 18392  df-mgp 18686  df-ur 18698  df-ring 18745  df-lmod 19063  df-lss 19131  df-lsp 19170  df-lmhm 19220  df-lfig 38136
This theorem is referenced by:  lmhmlnmsplit  38155
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