Theorem lmhmfgsplit 36674

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 = (0g‘𝑇)
lmhmfgsplit.k	⊢ 𝐾 = (◡𝐹 “ { 0 })
lmhmfgsplit.u	⊢ 𝑈 = (𝑆 ↾s 𝐾)
lmhmfgsplit.v	⊢ 𝑉 = (𝑇 ↾s 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.

Step	Hyp	Ref	Expression
1		simp3 1056	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑉 ∈ LFinGen)
2		lmhmlmod2 18853	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
3	2	3ad2ant1 1075	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑇 ∈ LMod)
4		lmhmrnlss 18871	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ (LSubSp‘𝑇))
5	4	3ad2ant1 1075	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → ran 𝐹 ∈ (LSubSp‘𝑇))
6		lmhmfgsplit.v	. . . . 5 ⊢ 𝑉 = (𝑇 ↾s ran 𝐹)
7		eqid 2610	. . . . 5 ⊢ (LSubSp‘𝑇) = (LSubSp‘𝑇)
8		eqid 2610	. . . . 5 ⊢ (LSpan‘𝑇) = (LSpan‘𝑇)
9	6, 7, 8	islssfg 36658	. . . 4 ⊢ ((𝑇 ∈ LMod ∧ ran 𝐹 ∈ (LSubSp‘𝑇)) → (𝑉 ∈ LFinGen ↔ ∃𝑎 ∈ 𝒫 ran 𝐹(𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹)))
10	3, 5, 9	syl2anc 691	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → (𝑉 ∈ LFinGen ↔ ∃𝑎 ∈ 𝒫 ran 𝐹(𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹)))
11	1, 10	mpbid 221	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → ∃𝑎 ∈ 𝒫 ran 𝐹(𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))
12		simpl1 1057	. . . . 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 2610	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
14		eqid 2610	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
15	13, 14	lmhmf 18855	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
16		ffn 5958	. . . . 5 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
17	12, 15, 16	3syl 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 4117	. . . . 5 ⊢ (𝑎 ∈ 𝒫 ran 𝐹 → 𝑎 ⊆ ran 𝐹)
19	18	ad2antrl 760	. . . 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 800	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) → 𝑎 ∈ Fin)
21		fipreima 8155	. . . 4 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ 𝑎 ⊆ ran 𝐹 ∧ 𝑎 ∈ Fin) → ∃𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin)(𝐹 “ 𝑏) = 𝑎)
22	17, 19, 20, 21	syl3anc 1318	. . 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)(𝐹 “ 𝑏) = 𝑎)
23		eqid 2610	. . . . . . 7 ⊢ (LSubSp‘𝑆) = (LSubSp‘𝑆)
24		eqid 2610	. . . . . . 7 ⊢ (LSSum‘𝑆) = (LSSum‘𝑆)
25		lmhmfgsplit.z	. . . . . . 7 ⊢ 0 = (0g‘𝑇)
26		lmhmfgsplit.k	. . . . . . 7 ⊢ 𝐾 = (◡𝐹 “ { 0 })
27		simpll1 1093	. . . . . . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
28		lmhmlmod1 18854	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
29	28	3ad2ant1 1075	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑆 ∈ LMod)
30	29	ad2antrr 758	. . . . . . . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → 𝑆 ∈ LMod)
31		inss1 3795	. . . . . . . . . . 11 ⊢ (𝒫 (Base‘𝑆) ∩ Fin) ⊆ 𝒫 (Base‘𝑆)
32	31	sseli 3564	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) → 𝑏 ∈ 𝒫 (Base‘𝑆))
33		elpwi 4117	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝒫 (Base‘𝑆) → 𝑏 ⊆ (Base‘𝑆))
34	32, 33	syl 17	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) → 𝑏 ⊆ (Base‘𝑆))
35	34	ad2antrl 760	. . . . . . . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → 𝑏 ⊆ (Base‘𝑆))
36		eqid 2610	. . . . . . . . 9 ⊢ (LSpan‘𝑆) = (LSpan‘𝑆)
37	13, 23, 36	lspcl 18797	. . . . . . . 8 ⊢ ((𝑆 ∈ LMod ∧ 𝑏 ⊆ (Base‘𝑆)) → ((LSpan‘𝑆)‘𝑏) ∈ (LSubSp‘𝑆))
38	30, 35, 37	syl2anc 691	. . . . . . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → ((LSpan‘𝑆)‘𝑏) ∈ (LSubSp‘𝑆))
39	13, 36, 8	lmhmlsp 18870	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑏 ⊆ (Base‘𝑆)) → (𝐹 “ ((LSpan‘𝑆)‘𝑏)) = ((LSpan‘𝑇)‘(𝐹 “ 𝑏)))
40	27, 35, 39	syl2anc 691	. . . . . . . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → (𝐹 “ ((LSpan‘𝑆)‘𝑏)) = ((LSpan‘𝑇)‘(𝐹 “ 𝑏)))
41		fveq2 6103	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑏) = 𝑎 → ((LSpan‘𝑇)‘(𝐹 “ 𝑏)) = ((LSpan‘𝑇)‘𝑎))
42	41	ad2antll 761	. . . . . . . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → ((LSpan‘𝑇)‘(𝐹 “ 𝑏)) = ((LSpan‘𝑇)‘𝑎))
43		simp2rr 1124	. . . . . . . . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → ((LSpan‘𝑇)‘𝑎) = ran 𝐹)
44	43	3expa 1257	. . . . . . . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → ((LSpan‘𝑇)‘𝑎) = ran 𝐹)
45	40, 42, 44	3eqtrd 2648	. . . . . . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → (𝐹 “ ((LSpan‘𝑆)‘𝑏)) = ran 𝐹)
46	23, 24, 25, 26, 13, 27, 38, 45	kercvrlsm 36671	. . . . . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → (𝐾(LSSum‘𝑆)((LSpan‘𝑆)‘𝑏)) = (Base‘𝑆))
47	46	oveq2d 6565	. . . . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → (𝑆 ↾s (𝐾(LSSum‘𝑆)((LSpan‘𝑆)‘𝑏))) = (𝑆 ↾s (Base‘𝑆)))
48	13	ressid 15762	. . . . . . 7 ⊢ (𝑆 ∈ LMod → (𝑆 ↾s (Base‘𝑆)) = 𝑆)
49	29, 48	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → (𝑆 ↾s (Base‘𝑆)) = 𝑆)
50	49	ad2antrr 758	. . . . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → (𝑆 ↾s (Base‘𝑆)) = 𝑆)
51	47, 50	eqtr2d 2645	. . . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → 𝑆 = (𝑆 ↾s (𝐾(LSSum‘𝑆)((LSpan‘𝑆)‘𝑏))))
52		lmhmfgsplit.u	. . . . 5 ⊢ 𝑈 = (𝑆 ↾s 𝐾)
53		eqid 2610	. . . . 5 ⊢ (𝑆 ↾s ((LSpan‘𝑆)‘𝑏)) = (𝑆 ↾s ((LSpan‘𝑆)‘𝑏))
54		eqid 2610	. . . . 5 ⊢ (𝑆 ↾s (𝐾(LSSum‘𝑆)((LSpan‘𝑆)‘𝑏))) = (𝑆 ↾s (𝐾(LSSum‘𝑆)((LSpan‘𝑆)‘𝑏)))
55	26, 25, 23	lmhmkerlss 18872	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐾 ∈ (LSubSp‘𝑆))
56	55	3ad2ant1 1075	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝐾 ∈ (LSubSp‘𝑆))
57	56	ad2antrr 758	. . . . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → 𝐾 ∈ (LSubSp‘𝑆))
58		simpll2 1094	. . . . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → 𝑈 ∈ LFinGen)
59		inss2 3796	. . . . . . . 8 ⊢ (𝒫 (Base‘𝑆) ∩ Fin) ⊆ Fin
60	59	sseli 3564	. . . . . . 7 ⊢ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) → 𝑏 ∈ Fin)
61	60	ad2antrl 760	. . . . . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → 𝑏 ∈ Fin)
62	36, 13, 53	islssfgi 36660	. . . . . 6 ⊢ ((𝑆 ∈ LMod ∧ 𝑏 ⊆ (Base‘𝑆) ∧ 𝑏 ∈ Fin) → (𝑆 ↾s ((LSpan‘𝑆)‘𝑏)) ∈ LFinGen)
63	30, 35, 61, 62	syl3anc 1318	. . . . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → (𝑆 ↾s ((LSpan‘𝑆)‘𝑏)) ∈ LFinGen)
64	23, 24, 52, 53, 54, 30, 57, 38, 58, 63	lsmfgcl 36662	. . . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → (𝑆 ↾s (𝐾(LSSum‘𝑆)((LSpan‘𝑆)‘𝑏))) ∈ LFinGen)
65	51, 64	eqeltrd 2688	. . 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) ∧ (𝐹 “ 𝑏) = 𝑎)) → 𝑆 ∈ LFinGen)
66	22, 65	rexlimddv 3017	. 2 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) → 𝑆 ∈ LFinGen)
67	11, 66	rexlimddv 3017	1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑆 ∈ LFinGen)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ^◡ccnv 5037  ran crn 5039   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  Basecbs 15695   ↾_s cress 15696  0_gc0g 15923  LSSumclsm 17872  LModclmod 18686  LSubSpclss 18753  LSpanclspn 18792   LMHom clmhm 18840  LFinGenclfig 36655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-sca 15784  df-vsca 15785  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-ghm 17481  df-cntz 17573  df-lsm 17874  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-lmod 18688  df-lss 18754  df-lsp 18793  df-lmhm 18843  df-lfig 36656

This theorem is referenced by:  lmhmlnmsplit  36675