Theorem lmhmfgsplit 43531

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 1144	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑉 ∈ LFinGen)
2		lmhmlmod2 21022	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
3	2	3ad2ant1 1139	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑇 ∈ LMod)
4		lmhmrnlss 21040	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ (LSubSp‘𝑇))
5	4	3ad2ant1 1139	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → ran 𝐹 ∈ (LSubSp‘𝑇))
6		lmhmfgsplit.v	. . . . 5 ⊢ 𝑉 = (𝑇 ↾s ran 𝐹)
7		eqid 2739	. . . . 5 ⊢ (LSubSp‘𝑇) = (LSubSp‘𝑇)
8		eqid 2739	. . . . 5 ⊢ (LSpan‘𝑇) = (LSpan‘𝑇)
9	6, 7, 8	islssfg 43515	. . . 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 590	. . 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 233	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → ∃𝑎 ∈ 𝒫 ran 𝐹(𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))
12		simpl1 1198	. . . . 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 2739	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
14		eqid 2739	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
15	13, 14	lmhmf 21024	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
16		ffn 6655	. . . . 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 4536	. . . . 5 ⊢ (𝑎 ∈ 𝒫 ran 𝐹 → 𝑎 ⊆ ran 𝐹)
19	18	ad2antrl 734	. . . 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 786	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) ∧ (𝑎 ∈ 𝒫 ran 𝐹 ∧ (𝑎 ∈ Fin ∧ ((LSpan‘𝑇)‘𝑎) = ran 𝐹))) → 𝑎 ∈ Fin)
21		fipreima 9258	. . . 4 ⊢ ((𝐹 Fn (Base‘𝑆) ∧ 𝑎 ⊆ ran 𝐹 ∧ 𝑎 ∈ Fin) → ∃𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin)(𝐹 “ 𝑏) = 𝑎)
22	17, 19, 20, 21	syl3anc 1379	. . 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 2739	. . . . . . 7 ⊢ (LSubSp‘𝑆) = (LSubSp‘𝑆)
24		eqid 2739	. . . . . . 7 ⊢ (LSSum‘𝑆) = (LSSum‘𝑆)
25		lmhmfgsplit.z	. . . . . . 7 ⊢ 0 = (0g‘𝑇)
26		lmhmfgsplit.k	. . . . . . 7 ⊢ 𝐾 = (◡𝐹 “ { 0 })
27		simpll1 1219	. . . . . . 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 21023	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
29	28	3ad2ant1 1139	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑆 ∈ LMod)
30	29	ad2antrr 732	. . . . . . . 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 4165	. . . . . . . . . . 11 ⊢ (𝒫 (Base‘𝑆) ∩ Fin) ⊆ 𝒫 (Base‘𝑆)
32	31	sseli 3911	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) → 𝑏 ∈ 𝒫 (Base‘𝑆))
33		elpwi 4536	. . . . . . . . . 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 734	. . . . . . . 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 2739	. . . . . . . . 9 ⊢ (LSpan‘𝑆) = (LSpan‘𝑆)
37	13, 23, 36	lspcl 20966	. . . . . . . 8 ⊢ ((𝑆 ∈ LMod ∧ 𝑏 ⊆ (Base‘𝑆)) → ((LSpan‘𝑆)‘𝑏) ∈ (LSubSp‘𝑆))
38	30, 35, 37	syl2anc 590	. . . . . . 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 21039	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑏 ⊆ (Base‘𝑆)) → (𝐹 “ ((LSpan‘𝑆)‘𝑏)) = ((LSpan‘𝑇)‘(𝐹 “ 𝑏)))
40	27, 35, 39	syl2anc 590	. . . . . . . 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 6827	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑏) = 𝑎 → ((LSpan‘𝑇)‘(𝐹 “ 𝑏)) = ((LSpan‘𝑇)‘𝑎))
42	41	ad2antll 735	. . . . . . . 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 1250	. . . . . . . . 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 1124	. . . . . . . 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 2778	. . . . . . 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 43528	. . . . . 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 7372	. . . . 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 17205	. . . . . . 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 732	. . . . 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 2775	. . . 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 2739	. . . . 5 ⊢ (𝑆 ↾s ((LSpan‘𝑆)‘𝑏)) = (𝑆 ↾s ((LSpan‘𝑆)‘𝑏))
54		eqid 2739	. . . . 5 ⊢ (𝑆 ↾s (𝐾(LSSum‘𝑆)((LSpan‘𝑆)‘𝑏))) = (𝑆 ↾s (𝐾(LSSum‘𝑆)((LSpan‘𝑆)‘𝑏)))
55	26, 25, 23	lmhmkerlss 21041	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐾 ∈ (LSubSp‘𝑆))
56	55	3ad2ant1 1139	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝐾 ∈ (LSubSp‘𝑆))
57	56	ad2antrr 732	. . . . 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 1220	. . . . 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 4166	. . . . . . . 8 ⊢ (𝒫 (Base‘𝑆) ∩ Fin) ⊆ Fin
60	59	sseli 3911	. . . . . . 7 ⊢ (𝑏 ∈ (𝒫 (Base‘𝑆) ∩ Fin) → 𝑏 ∈ Fin)
61	60	ad2antrl 734	. . . . . 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 43517	. . . . . 6 ⊢ ((𝑆 ∈ LMod ∧ 𝑏 ⊆ (Base‘𝑆) ∧ 𝑏 ∈ Fin) → (𝑆 ↾s ((LSpan‘𝑆)‘𝑏)) ∈ LFinGen)
63	30, 35, 61, 62	syl3anc 1379	. . . . 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 43519	. . . 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 2839	. . 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 3146	. 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 3146	1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LFinGen ∧ 𝑉 ∈ LFinGen) → 𝑆 ∈ LFinGen)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∃wrex 3063   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4529  {csn 4555  ^◡ccnv 5617  ran crn 5619   “ cima 5621   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  Fincfn 8883  Basecbs 17170   ↾_s cress 17191  0_gc0g 17393  LSSumclsm 19600  LModclmod 20850  LSubSpclss 20921  LSpanclspn 20961   LMHom clmhm 21009  LFinGenclfig 43512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-sca 17227  df-vsca 17228  df-0g 17395  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-submnd 18743  df-grp 18903  df-minusg 18904  df-sbg 18905  df-subg 19090  df-ghm 19179  df-cntz 19283  df-lsm 19602  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-lmod 20852  df-lss 20922  df-lsp 20962  df-lmhm 21012  df-lfig 43513

This theorem is referenced by:  lmhmlnmsplit  43532