Theorem lmhmlnmsplit 40828

Description: If the kernel and range of a homomorphism of left modules are Noetherian, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 12-Jun-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
lmhmlnmsplit	⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LNoeM)

Proof of Theorem lmhmlnmsplit

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmhmlmod1 20210	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
2	1	3ad2ant1 1131	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LMod)
3		eqid 2738	. . . . . 6 ⊢ (LSubSp‘𝑆) = (LSubSp‘𝑆)
4		eqid 2738	. . . . . 6 ⊢ (𝑆 ↾s 𝑎) = (𝑆 ↾s 𝑎)
5	3, 4	reslmhm 20229	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝐹 ↾ 𝑎) ∈ ((𝑆 ↾s 𝑎) LMHom 𝑇))
6	5	3ad2antl1 1183	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝐹 ↾ 𝑎) ∈ ((𝑆 ↾s 𝑎) LMHom 𝑇))
7		cnvresima 6122	. . . . . . . 8 ⊢ (◡(𝐹 ↾ 𝑎) “ { 0 }) = ((◡𝐹 “ { 0 }) ∩ 𝑎)
8		lmhmfgsplit.k	. . . . . . . . . 10 ⊢ 𝐾 = (◡𝐹 “ { 0 })
9	8	eqcomi 2747	. . . . . . . . 9 ⊢ (◡𝐹 “ { 0 }) = 𝐾
10	9	ineq1i 4139	. . . . . . . 8 ⊢ ((◡𝐹 “ { 0 }) ∩ 𝑎) = (𝐾 ∩ 𝑎)
11		incom 4131	. . . . . . . 8 ⊢ (𝐾 ∩ 𝑎) = (𝑎 ∩ 𝐾)
12	7, 10, 11	3eqtri 2770	. . . . . . 7 ⊢ (◡(𝐹 ↾ 𝑎) “ { 0 }) = (𝑎 ∩ 𝐾)
13	12	oveq2i 7266	. . . . . 6 ⊢ ((𝑆 ↾s 𝑎) ↾s (◡(𝐹 ↾ 𝑎) “ { 0 })) = ((𝑆 ↾s 𝑎) ↾s (𝑎 ∩ 𝐾))
14		vex 3426	. . . . . . . 8 ⊢ 𝑎 ∈ V
15		inss1 4159	. . . . . . . 8 ⊢ (𝑎 ∩ 𝐾) ⊆ 𝑎
16		ressabs 16885	. . . . . . . 8 ⊢ ((𝑎 ∈ V ∧ (𝑎 ∩ 𝐾) ⊆ 𝑎) → ((𝑆 ↾s 𝑎) ↾s (𝑎 ∩ 𝐾)) = (𝑆 ↾s (𝑎 ∩ 𝐾)))
17	14, 15, 16	mp2an 688	. . . . . . 7 ⊢ ((𝑆 ↾s 𝑎) ↾s (𝑎 ∩ 𝐾)) = (𝑆 ↾s (𝑎 ∩ 𝐾))
18		lmhmfgsplit.u	. . . . . . . . 9 ⊢ 𝑈 = (𝑆 ↾s 𝐾)
19	18	oveq1i 7265	. . . . . . . 8 ⊢ (𝑈 ↾s (𝑎 ∩ 𝐾)) = ((𝑆 ↾s 𝐾) ↾s (𝑎 ∩ 𝐾))
20		simpl1 1189	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
21		cnvexg 7745	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ◡𝐹 ∈ V)
22		imaexg 7736	. . . . . . . . . . . 12 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ { 0 }) ∈ V)
23	21, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (◡𝐹 “ { 0 }) ∈ V)
24	8, 23	eqeltrid 2843	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐾 ∈ V)
25	20, 24	syl 17	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝐾 ∈ V)
26		inss2 4160	. . . . . . . . 9 ⊢ (𝑎 ∩ 𝐾) ⊆ 𝐾
27		ressabs 16885	. . . . . . . . 9 ⊢ ((𝐾 ∈ V ∧ (𝑎 ∩ 𝐾) ⊆ 𝐾) → ((𝑆 ↾s 𝐾) ↾s (𝑎 ∩ 𝐾)) = (𝑆 ↾s (𝑎 ∩ 𝐾)))
28	25, 26, 27	sylancl 585	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆 ↾s 𝐾) ↾s (𝑎 ∩ 𝐾)) = (𝑆 ↾s (𝑎 ∩ 𝐾)))
29	19, 28	syl5eq 2791	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑈 ↾s (𝑎 ∩ 𝐾)) = (𝑆 ↾s (𝑎 ∩ 𝐾)))
30	17, 29	eqtr4id 2798	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆 ↾s 𝑎) ↾s (𝑎 ∩ 𝐾)) = (𝑈 ↾s (𝑎 ∩ 𝐾)))
31	13, 30	syl5eq 2791	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆 ↾s 𝑎) ↾s (◡(𝐹 ↾ 𝑎) “ { 0 })) = (𝑈 ↾s (𝑎 ∩ 𝐾)))
32		simpl2 1190	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑈 ∈ LNoeM)
33	2	adantr 480	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑆 ∈ LMod)
34		simpr 484	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑎 ∈ (LSubSp‘𝑆))
35		lmhmfgsplit.z	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑇)
36	8, 35, 3	lmhmkerlss 20228	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐾 ∈ (LSubSp‘𝑆))
37	20, 36	syl 17	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝐾 ∈ (LSubSp‘𝑆))
38	3	lssincl 20142	. . . . . . . 8 ⊢ ((𝑆 ∈ LMod ∧ 𝑎 ∈ (LSubSp‘𝑆) ∧ 𝐾 ∈ (LSubSp‘𝑆)) → (𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑆))
39	33, 34, 37, 38	syl3anc 1369	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑆))
40	26	a1i 11	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑎 ∩ 𝐾) ⊆ 𝐾)
41		eqid 2738	. . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
42	18, 3, 41	lsslss 20138	. . . . . . . 8 ⊢ ((𝑆 ∈ LMod ∧ 𝐾 ∈ (LSubSp‘𝑆)) → ((𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑈) ↔ ((𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑆) ∧ (𝑎 ∩ 𝐾) ⊆ 𝐾)))
43	33, 37, 42	syl2anc 583	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑈) ↔ ((𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑆) ∧ (𝑎 ∩ 𝐾) ⊆ 𝐾)))
44	39, 40, 43	mpbir2and 709	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑈))
45		eqid 2738	. . . . . . 7 ⊢ (𝑈 ↾s (𝑎 ∩ 𝐾)) = (𝑈 ↾s (𝑎 ∩ 𝐾))
46	41, 45	lnmlssfg 40821	. . . . . 6 ⊢ ((𝑈 ∈ LNoeM ∧ (𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑈)) → (𝑈 ↾s (𝑎 ∩ 𝐾)) ∈ LFinGen)
47	32, 44, 46	syl2anc 583	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑈 ↾s (𝑎 ∩ 𝐾)) ∈ LFinGen)
48	31, 47	eqeltrd 2839	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆 ↾s 𝑎) ↾s (◡(𝐹 ↾ 𝑎) “ { 0 })) ∈ LFinGen)
49		incom 4131	. . . . . . . . 9 ⊢ (ran 𝐹 ∩ ran (𝐹 ↾ 𝑎)) = (ran (𝐹 ↾ 𝑎) ∩ ran 𝐹)
50		resss 5905	. . . . . . . . . . 11 ⊢ (𝐹 ↾ 𝑎) ⊆ 𝐹
51		rnss 5837	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ 𝑎) ⊆ 𝐹 → ran (𝐹 ↾ 𝑎) ⊆ ran 𝐹)
52	50, 51	ax-mp 5	. . . . . . . . . 10 ⊢ ran (𝐹 ↾ 𝑎) ⊆ ran 𝐹
53		df-ss 3900	. . . . . . . . . 10 ⊢ (ran (𝐹 ↾ 𝑎) ⊆ ran 𝐹 ↔ (ran (𝐹 ↾ 𝑎) ∩ ran 𝐹) = ran (𝐹 ↾ 𝑎))
54	52, 53	mpbi 229	. . . . . . . . 9 ⊢ (ran (𝐹 ↾ 𝑎) ∩ ran 𝐹) = ran (𝐹 ↾ 𝑎)
55	49, 54	eqtr2i 2767	. . . . . . . 8 ⊢ ran (𝐹 ↾ 𝑎) = (ran 𝐹 ∩ ran (𝐹 ↾ 𝑎))
56	55	oveq2i 7266	. . . . . . 7 ⊢ (𝑇 ↾s ran (𝐹 ↾ 𝑎)) = (𝑇 ↾s (ran 𝐹 ∩ ran (𝐹 ↾ 𝑎)))
57		lmhmfgsplit.v	. . . . . . . . 9 ⊢ 𝑉 = (𝑇 ↾s ran 𝐹)
58	57	oveq1i 7265	. . . . . . . 8 ⊢ (𝑉 ↾s ran (𝐹 ↾ 𝑎)) = ((𝑇 ↾s ran 𝐹) ↾s ran (𝐹 ↾ 𝑎))
59		rnexg 7725	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ V)
60		resexg 5926	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹 ↾ 𝑎) ∈ V)
61		rnexg 7725	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝑎) ∈ V → ran (𝐹 ↾ 𝑎) ∈ V)
62	60, 61	syl 17	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ran (𝐹 ↾ 𝑎) ∈ V)
63		ressress 16884	. . . . . . . . 9 ⊢ ((ran 𝐹 ∈ V ∧ ran (𝐹 ↾ 𝑎) ∈ V) → ((𝑇 ↾s ran 𝐹) ↾s ran (𝐹 ↾ 𝑎)) = (𝑇 ↾s (ran 𝐹 ∩ ran (𝐹 ↾ 𝑎))))
64	59, 62, 63	syl2anc 583	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑇 ↾s ran 𝐹) ↾s ran (𝐹 ↾ 𝑎)) = (𝑇 ↾s (ran 𝐹 ∩ ran (𝐹 ↾ 𝑎))))
65	58, 64	syl5eq 2791	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑉 ↾s ran (𝐹 ↾ 𝑎)) = (𝑇 ↾s (ran 𝐹 ∩ ran (𝐹 ↾ 𝑎))))
66	56, 65	eqtr4id 2798	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑇 ↾s ran (𝐹 ↾ 𝑎)) = (𝑉 ↾s ran (𝐹 ↾ 𝑎)))
67	20, 66	syl 17	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑇 ↾s ran (𝐹 ↾ 𝑎)) = (𝑉 ↾s ran (𝐹 ↾ 𝑎)))
68		simpl3 1191	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑉 ∈ LNoeM)
69		lmhmrnlss 20227	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝑎) ∈ ((𝑆 ↾s 𝑎) LMHom 𝑇) → ran (𝐹 ↾ 𝑎) ∈ (LSubSp‘𝑇))
70	6, 69	syl 17	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ran (𝐹 ↾ 𝑎) ∈ (LSubSp‘𝑇))
71	52	a1i 11	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ran (𝐹 ↾ 𝑎) ⊆ ran 𝐹)
72		lmhmlmod2 20209	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
73	20, 72	syl 17	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑇 ∈ LMod)
74		lmhmrnlss 20227	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ (LSubSp‘𝑇))
75	20, 74	syl 17	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ran 𝐹 ∈ (LSubSp‘𝑇))
76		eqid 2738	. . . . . . . . 9 ⊢ (LSubSp‘𝑇) = (LSubSp‘𝑇)
77		eqid 2738	. . . . . . . . 9 ⊢ (LSubSp‘𝑉) = (LSubSp‘𝑉)
78	57, 76, 77	lsslss 20138	. . . . . . . 8 ⊢ ((𝑇 ∈ LMod ∧ ran 𝐹 ∈ (LSubSp‘𝑇)) → (ran (𝐹 ↾ 𝑎) ∈ (LSubSp‘𝑉) ↔ (ran (𝐹 ↾ 𝑎) ∈ (LSubSp‘𝑇) ∧ ran (𝐹 ↾ 𝑎) ⊆ ran 𝐹)))
79	73, 75, 78	syl2anc 583	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (ran (𝐹 ↾ 𝑎) ∈ (LSubSp‘𝑉) ↔ (ran (𝐹 ↾ 𝑎) ∈ (LSubSp‘𝑇) ∧ ran (𝐹 ↾ 𝑎) ⊆ ran 𝐹)))
80	70, 71, 79	mpbir2and 709	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ran (𝐹 ↾ 𝑎) ∈ (LSubSp‘𝑉))
81		eqid 2738	. . . . . . 7 ⊢ (𝑉 ↾s ran (𝐹 ↾ 𝑎)) = (𝑉 ↾s ran (𝐹 ↾ 𝑎))
82	77, 81	lnmlssfg 40821	. . . . . 6 ⊢ ((𝑉 ∈ LNoeM ∧ ran (𝐹 ↾ 𝑎) ∈ (LSubSp‘𝑉)) → (𝑉 ↾s ran (𝐹 ↾ 𝑎)) ∈ LFinGen)
83	68, 80, 82	syl2anc 583	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑉 ↾s ran (𝐹 ↾ 𝑎)) ∈ LFinGen)
84	67, 83	eqeltrd 2839	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑇 ↾s ran (𝐹 ↾ 𝑎)) ∈ LFinGen)
85		eqid 2738	. . . . 5 ⊢ (◡(𝐹 ↾ 𝑎) “ { 0 }) = (◡(𝐹 ↾ 𝑎) “ { 0 })
86		eqid 2738	. . . . 5 ⊢ ((𝑆 ↾s 𝑎) ↾s (◡(𝐹 ↾ 𝑎) “ { 0 })) = ((𝑆 ↾s 𝑎) ↾s (◡(𝐹 ↾ 𝑎) “ { 0 }))
87		eqid 2738	. . . . 5 ⊢ (𝑇 ↾s ran (𝐹 ↾ 𝑎)) = (𝑇 ↾s ran (𝐹 ↾ 𝑎))
88	35, 85, 86, 87	lmhmfgsplit 40827	. . . 4 ⊢ (((𝐹 ↾ 𝑎) ∈ ((𝑆 ↾s 𝑎) LMHom 𝑇) ∧ ((𝑆 ↾s 𝑎) ↾s (◡(𝐹 ↾ 𝑎) “ { 0 })) ∈ LFinGen ∧ (𝑇 ↾s ran (𝐹 ↾ 𝑎)) ∈ LFinGen) → (𝑆 ↾s 𝑎) ∈ LFinGen)
89	6, 48, 84, 88	syl3anc 1369	. . 3 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑆 ↾s 𝑎) ∈ LFinGen)
90	89	ralrimiva 3107	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → ∀𝑎 ∈ (LSubSp‘𝑆)(𝑆 ↾s 𝑎) ∈ LFinGen)
91	3	islnm 40818	. 2 ⊢ (𝑆 ∈ LNoeM ↔ (𝑆 ∈ LMod ∧ ∀𝑎 ∈ (LSubSp‘𝑆)(𝑆 ↾s 𝑎) ∈ LFinGen))
92	2, 90, 91	sylanbrc 582	1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LNoeM)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  {csn 4558  ^◡ccnv 5579  ran crn 5581   ↾ cres 5582   “ cima 5583  ‘cfv 6418  (class class class)co 7255   ↾_s cress 16867  0_gc0g 17067  LModclmod 20038  LSubSpclss 20108   LMHom clmhm 20196  LFinGenclfig 40808  LNoeMclnm 40816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-sca 16904  df-vsca 16905  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-subg 18667  df-ghm 18747  df-cntz 18838  df-lsm 19156  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-lmod 20040  df-lss 20109  df-lsp 20149  df-lmhm 20199  df-lfig 40809  df-lnm 40817

This theorem is referenced by:  pwslnmlem2  40834