Theorem lmhmlnmsplit 44355

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 21359	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
2	1	3ad2ant1 1151	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LMod)
3		eqid 2763	. . . . . 6 ⊢ (LSubSp‘𝑆) = (LSubSp‘𝑆)
4		eqid 2763	. . . . . 6 ⊢ (𝑆 ↾s 𝑎) = (𝑆 ↾s 𝑎)
5	3, 4	reslmhm 21378	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝐹 ↾ 𝑎) ∈ ((𝑆 ↾s 𝑎) LMHom 𝑇))
6	5	3ad2antl1 1204	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝐹 ↾ 𝑎) ∈ ((𝑆 ↾s 𝑎) LMHom 𝑇))
7		cnvresima 6250	. . . . . . . 8 ⊢ (◡(𝐹 ↾ 𝑎) “ { 0 }) = ((◡𝐹 “ { 0 }) ∩ 𝑎)
8		lmhmfgsplit.k	. . . . . . . . . 10 ⊢ 𝐾 = (◡𝐹 “ { 0 })
9	8	eqcomi 2772	. . . . . . . . 9 ⊢ (◡𝐹 “ { 0 }) = 𝐾
10	9	ineq1i 4116	. . . . . . . 8 ⊢ ((◡𝐹 “ { 0 }) ∩ 𝑎) = (𝐾 ∩ 𝑎)
11		incom 4109	. . . . . . . 8 ⊢ (𝐾 ∩ 𝑎) = (𝑎 ∩ 𝐾)
12	7, 10, 11	3eqtri 2790	. . . . . . 7 ⊢ (◡(𝐹 ↾ 𝑎) “ { 0 }) = (𝑎 ∩ 𝐾)
13	12	oveq2i 7462	. . . . . 6 ⊢ ((𝑆 ↾s 𝑎) ↾s (◡(𝐹 ↾ 𝑎) “ { 0 })) = ((𝑆 ↾s 𝑎) ↾s (𝑎 ∩ 𝐾))
14		vex 3458	. . . . . . . 8 ⊢ 𝑎 ∈ V
15		inss1 4136	. . . . . . . 8 ⊢ (𝑎 ∩ 𝐾) ⊆ 𝑎
16		ressabs 17470	. . . . . . . 8 ⊢ ((𝑎 ∈ V ∧ (𝑎 ∩ 𝐾) ⊆ 𝑎) → ((𝑆 ↾s 𝑎) ↾s (𝑎 ∩ 𝐾)) = (𝑆 ↾s (𝑎 ∩ 𝐾)))
17	14, 15, 16	mp2an 705	. . . . . . 7 ⊢ ((𝑆 ↾s 𝑎) ↾s (𝑎 ∩ 𝐾)) = (𝑆 ↾s (𝑎 ∩ 𝐾))
18		lmhmfgsplit.u	. . . . . . . . 9 ⊢ 𝑈 = (𝑆 ↾s 𝐾)
19	18	oveq1i 7461	. . . . . . . 8 ⊢ (𝑈 ↾s (𝑎 ∩ 𝐾)) = ((𝑆 ↾s 𝐾) ↾s (𝑎 ∩ 𝐾))
20		simpl1 1210	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
21		cnvexg 7970	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ◡𝐹 ∈ V)
22		imaexg 7959	. . . . . . . . . . . 12 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ { 0 }) ∈ V)
23	21, 22	syl 18	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (◡𝐹 “ { 0 }) ∈ V)
24	8, 23	eqeltrid 2868	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐾 ∈ V)
25	20, 24	syl 18	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝐾 ∈ V)
26		inss2 4137	. . . . . . . . 9 ⊢ (𝑎 ∩ 𝐾) ⊆ 𝐾
27		ressabs 17470	. . . . . . . . 9 ⊢ ((𝐾 ∈ V ∧ (𝑎 ∩ 𝐾) ⊆ 𝐾) → ((𝑆 ↾s 𝐾) ↾s (𝑎 ∩ 𝐾)) = (𝑆 ↾s (𝑎 ∩ 𝐾)))
28	25, 26, 27	sylancl 598	. . . . . . . 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	eqtrid 2810	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑈 ↾s (𝑎 ∩ 𝐾)) = (𝑆 ↾s (𝑎 ∩ 𝐾)))
30	17, 29	eqtr4id 2817	. . . . . 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	eqtrid 2810	. . . . 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 1211	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑈 ∈ LNoeM)
33	2	adantr 486	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑆 ∈ LMod)
34		simpr 490	. . . . . . . 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 21377	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐾 ∈ (LSubSp‘𝑆))
37	20, 36	syl 18	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝐾 ∈ (LSubSp‘𝑆))
38	3	lssincl 21291	. . . . . . . 8 ⊢ ((𝑆 ∈ LMod ∧ 𝑎 ∈ (LSubSp‘𝑆) ∧ 𝐾 ∈ (LSubSp‘𝑆)) → (𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑆))
39	33, 34, 37, 38	syl3anc 1399	. . . . . . 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 2763	. . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
42	18, 3, 41	lsslss 21287	. . . . . . . 8 ⊢ ((𝑆 ∈ LMod ∧ 𝐾 ∈ (LSubSp‘𝑆)) → ((𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑈) ↔ ((𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑆) ∧ (𝑎 ∩ 𝐾) ⊆ 𝐾)))
43	33, 37, 42	syl2anc 596	. . . . . . 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 726	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑈))
45		eqid 2763	. . . . . . 7 ⊢ (𝑈 ↾s (𝑎 ∩ 𝐾)) = (𝑈 ↾s (𝑎 ∩ 𝐾))
46	41, 45	lnmlssfg 44348	. . . . . 6 ⊢ ((𝑈 ∈ LNoeM ∧ (𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑈)) → (𝑈 ↾s (𝑎 ∩ 𝐾)) ∈ LFinGen)
47	32, 44, 46	syl2anc 596	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑈 ↾s (𝑎 ∩ 𝐾)) ∈ LFinGen)
48	31, 47	eqeltrd 2864	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆 ↾s 𝑎) ↾s (◡(𝐹 ↾ 𝑎) “ { 0 })) ∈ LFinGen)
49		incom 4109	. . . . . . . . 9 ⊢ (ran 𝐹 ∩ ran (𝐹 ↾ 𝑎)) = (ran (𝐹 ↾ 𝑎) ∩ ran 𝐹)
50		resss 6008	. . . . . . . . . . 11 ⊢ (𝐹 ↾ 𝑎) ⊆ 𝐹
51		rnss 5933	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ 𝑎) ⊆ 𝐹 → ran (𝐹 ↾ 𝑎) ⊆ ran 𝐹)
52	50, 51	ax-mp 5	. . . . . . . . . 10 ⊢ ran (𝐹 ↾ 𝑎) ⊆ ran 𝐹
53		dfss2 4097	. . . . . . . . . 10 ⊢ (ran (𝐹 ↾ 𝑎) ⊆ ran 𝐹 ↔ (ran (𝐹 ↾ 𝑎) ∩ ran 𝐹) = ran (𝐹 ↾ 𝑎))
54	52, 53	mpbi 233	. . . . . . . . 9 ⊢ (ran (𝐹 ↾ 𝑎) ∩ ran 𝐹) = ran (𝐹 ↾ 𝑎)
55	49, 54	eqtr2i 2787	. . . . . . . 8 ⊢ ran (𝐹 ↾ 𝑎) = (ran 𝐹 ∩ ran (𝐹 ↾ 𝑎))
56	55	oveq2i 7462	. . . . . . 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 7461	. . . . . . . 8 ⊢ (𝑉 ↾s ran (𝐹 ↾ 𝑎)) = ((𝑇 ↾s ran 𝐹) ↾s ran (𝐹 ↾ 𝑎))
59		rnexg 7948	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ V)
60		resexg 6034	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹 ↾ 𝑎) ∈ V)
61		rnexg 7948	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝑎) ∈ V → ran (𝐹 ↾ 𝑎) ∈ V)
62	60, 61	syl 18	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ran (𝐹 ↾ 𝑎) ∈ V)
63		ressress 17469	. . . . . . . . 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 596	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑇 ↾s ran 𝐹) ↾s ran (𝐹 ↾ 𝑎)) = (𝑇 ↾s (ran 𝐹 ∩ ran (𝐹 ↾ 𝑎))))
65	58, 64	eqtrid 2810	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑉 ↾s ran (𝐹 ↾ 𝑎)) = (𝑇 ↾s (ran 𝐹 ∩ ran (𝐹 ↾ 𝑎))))
66	56, 65	eqtr4id 2817	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑇 ↾s ran (𝐹 ↾ 𝑎)) = (𝑉 ↾s ran (𝐹 ↾ 𝑎)))
67	20, 66	syl 18	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑇 ↾s ran (𝐹 ↾ 𝑎)) = (𝑉 ↾s ran (𝐹 ↾ 𝑎)))
68		simpl3 1212	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑉 ∈ LNoeM)
69		lmhmrnlss 21376	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝑎) ∈ ((𝑆 ↾s 𝑎) LMHom 𝑇) → ran (𝐹 ↾ 𝑎) ∈ (LSubSp‘𝑇))
70	6, 69	syl 18	. . . . . . 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 21358	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
73	20, 72	syl 18	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑇 ∈ LMod)
74		lmhmrnlss 21376	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ (LSubSp‘𝑇))
75	20, 74	syl 18	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ran 𝐹 ∈ (LSubSp‘𝑇))
76		eqid 2763	. . . . . . . . 9 ⊢ (LSubSp‘𝑇) = (LSubSp‘𝑇)
77		eqid 2763	. . . . . . . . 9 ⊢ (LSubSp‘𝑉) = (LSubSp‘𝑉)
78	57, 76, 77	lsslss 21287	. . . . . . . 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 596	. . . . . . 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 726	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ran (𝐹 ↾ 𝑎) ∈ (LSubSp‘𝑉))
81		eqid 2763	. . . . . . 7 ⊢ (𝑉 ↾s ran (𝐹 ↾ 𝑎)) = (𝑉 ↾s ran (𝐹 ↾ 𝑎))
82	77, 81	lnmlssfg 44348	. . . . . 6 ⊢ ((𝑉 ∈ LNoeM ∧ ran (𝐹 ↾ 𝑎) ∈ (LSubSp‘𝑉)) → (𝑉 ↾s ran (𝐹 ↾ 𝑎)) ∈ LFinGen)
83	68, 80, 82	syl2anc 596	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑉 ↾s ran (𝐹 ↾ 𝑎)) ∈ LFinGen)
84	67, 83	eqeltrd 2864	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑇 ↾s ran (𝐹 ↾ 𝑎)) ∈ LFinGen)
85		eqid 2763	. . . . 5 ⊢ (◡(𝐹 ↾ 𝑎) “ { 0 }) = (◡(𝐹 ↾ 𝑎) “ { 0 })
86		eqid 2763	. . . . 5 ⊢ ((𝑆 ↾s 𝑎) ↾s (◡(𝐹 ↾ 𝑎) “ { 0 })) = ((𝑆 ↾s 𝑎) ↾s (◡(𝐹 ↾ 𝑎) “ { 0 }))
87		eqid 2763	. . . . 5 ⊢ (𝑇 ↾s ran (𝐹 ↾ 𝑎)) = (𝑇 ↾s ran (𝐹 ↾ 𝑎))
88	35, 85, 86, 87	lmhmfgsplit 44354	. . . 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 1399	. . 3 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑆 ↾s 𝑎) ∈ LFinGen)
90	89	ralrimiva 3158	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → ∀𝑎 ∈ (LSubSp‘𝑆)(𝑆 ↾s 𝑎) ∈ LFinGen)
91	3	islnm 44345	. 2 ⊢ (𝑆 ∈ LNoeM ↔ (𝑆 ∈ LMod ∧ ∀𝑎 ∈ (LSubSp‘𝑆)(𝑆 ↾s 𝑎) ∈ LFinGen))
92	2, 90, 91	sylanbrc 595	1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LNoeM)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  {csn 4588  ^◡ccnv 5657  ran crn 5659   ↾ cres 5660   “ cima 5661  ‘cfv 6566  (class class class)co 7451   ↾_s cress 17452  0_gc0g 17660  LModclmod 21186  LSubSpclss 21257   LMHom clmhm 21345  LFinGenclfig 44335  LNoeMclnm 44343

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-sca 17489  df-vsca 17490  df-0g 17662  df-mgm 18866  df-sgrp 18958  df-mnd 18974  df-submnd 19029  df-grp 19197  df-minusg 19198  df-sbg 19199  df-subg 19383  df-ghm 19478  df-cntz 19582  df-lsm 19901  df-cmn 20047  df-abl 20048  df-mgp 20412  df-rng 20426  df-ur 20459  df-ring 20512  df-lmod 21188  df-lss 21258  df-lsp 21298  df-lmhm 21348  df-lfig 44336  df-lnm 44344

This theorem is used by:  pwslnmlem2  44361