Theorem lmhmlnmsplit 42284

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 20870	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
2	1	3ad2ant1 1130	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LMod)
3		eqid 2724	. . . . . 6 ⊢ (LSubSp‘𝑆) = (LSubSp‘𝑆)
4		eqid 2724	. . . . . 6 ⊢ (𝑆 ↾s 𝑎) = (𝑆 ↾s 𝑎)
5	3, 4	reslmhm 20889	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝐹 ↾ 𝑎) ∈ ((𝑆 ↾s 𝑎) LMHom 𝑇))
6	5	3ad2antl1 1182	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝐹 ↾ 𝑎) ∈ ((𝑆 ↾s 𝑎) LMHom 𝑇))
7		cnvresima 6219	. . . . . . . 8 ⊢ (◡(𝐹 ↾ 𝑎) “ { 0 }) = ((◡𝐹 “ { 0 }) ∩ 𝑎)
8		lmhmfgsplit.k	. . . . . . . . . 10 ⊢ 𝐾 = (◡𝐹 “ { 0 })
9	8	eqcomi 2733	. . . . . . . . 9 ⊢ (◡𝐹 “ { 0 }) = 𝐾
10	9	ineq1i 4200	. . . . . . . 8 ⊢ ((◡𝐹 “ { 0 }) ∩ 𝑎) = (𝐾 ∩ 𝑎)
11		incom 4193	. . . . . . . 8 ⊢ (𝐾 ∩ 𝑎) = (𝑎 ∩ 𝐾)
12	7, 10, 11	3eqtri 2756	. . . . . . 7 ⊢ (◡(𝐹 ↾ 𝑎) “ { 0 }) = (𝑎 ∩ 𝐾)
13	12	oveq2i 7412	. . . . . 6 ⊢ ((𝑆 ↾s 𝑎) ↾s (◡(𝐹 ↾ 𝑎) “ { 0 })) = ((𝑆 ↾s 𝑎) ↾s (𝑎 ∩ 𝐾))
14		vex 3470	. . . . . . . 8 ⊢ 𝑎 ∈ V
15		inss1 4220	. . . . . . . 8 ⊢ (𝑎 ∩ 𝐾) ⊆ 𝑎
16		ressabs 17192	. . . . . . . 8 ⊢ ((𝑎 ∈ V ∧ (𝑎 ∩ 𝐾) ⊆ 𝑎) → ((𝑆 ↾s 𝑎) ↾s (𝑎 ∩ 𝐾)) = (𝑆 ↾s (𝑎 ∩ 𝐾)))
17	14, 15, 16	mp2an 689	. . . . . . 7 ⊢ ((𝑆 ↾s 𝑎) ↾s (𝑎 ∩ 𝐾)) = (𝑆 ↾s (𝑎 ∩ 𝐾))
18		lmhmfgsplit.u	. . . . . . . . 9 ⊢ 𝑈 = (𝑆 ↾s 𝐾)
19	18	oveq1i 7411	. . . . . . . 8 ⊢ (𝑈 ↾s (𝑎 ∩ 𝐾)) = ((𝑆 ↾s 𝐾) ↾s (𝑎 ∩ 𝐾))
20		simpl1 1188	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
21		cnvexg 7908	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ◡𝐹 ∈ V)
22		imaexg 7899	. . . . . . . . . . . 12 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ { 0 }) ∈ V)
23	21, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (◡𝐹 “ { 0 }) ∈ V)
24	8, 23	eqeltrid 2829	. . . . . . . . . 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 4221	. . . . . . . . 9 ⊢ (𝑎 ∩ 𝐾) ⊆ 𝐾
27		ressabs 17192	. . . . . . . . 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	eqtrid 2776	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑈 ↾s (𝑎 ∩ 𝐾)) = (𝑆 ↾s (𝑎 ∩ 𝐾)))
30	17, 29	eqtr4id 2783	. . . . . 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 2776	. . . . 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 1189	. . . . . 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 20888	. . . . . . . . 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 20801	. . . . . . . 8 ⊢ ((𝑆 ∈ LMod ∧ 𝑎 ∈ (LSubSp‘𝑆) ∧ 𝐾 ∈ (LSubSp‘𝑆)) → (𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑆))
39	33, 34, 37, 38	syl3anc 1368	. . . . . . 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 2724	. . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
42	18, 3, 41	lsslss 20797	. . . . . . . 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 710	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑈))
45		eqid 2724	. . . . . . 7 ⊢ (𝑈 ↾s (𝑎 ∩ 𝐾)) = (𝑈 ↾s (𝑎 ∩ 𝐾))
46	41, 45	lnmlssfg 42277	. . . . . 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 2825	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆 ↾s 𝑎) ↾s (◡(𝐹 ↾ 𝑎) “ { 0 })) ∈ LFinGen)
49		incom 4193	. . . . . . . . 9 ⊢ (ran 𝐹 ∩ ran (𝐹 ↾ 𝑎)) = (ran (𝐹 ↾ 𝑎) ∩ ran 𝐹)
50		resss 5996	. . . . . . . . . . 11 ⊢ (𝐹 ↾ 𝑎) ⊆ 𝐹
51		rnss 5928	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ 𝑎) ⊆ 𝐹 → ran (𝐹 ↾ 𝑎) ⊆ ran 𝐹)
52	50, 51	ax-mp 5	. . . . . . . . . 10 ⊢ ran (𝐹 ↾ 𝑎) ⊆ ran 𝐹
53		df-ss 3957	. . . . . . . . . 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 2753	. . . . . . . 8 ⊢ ran (𝐹 ↾ 𝑎) = (ran 𝐹 ∩ ran (𝐹 ↾ 𝑎))
56	55	oveq2i 7412	. . . . . . 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 7411	. . . . . . . 8 ⊢ (𝑉 ↾s ran (𝐹 ↾ 𝑎)) = ((𝑇 ↾s ran 𝐹) ↾s ran (𝐹 ↾ 𝑎))
59		rnexg 7888	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ V)
60		resexg 6017	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹 ↾ 𝑎) ∈ V)
61		rnexg 7888	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝑎) ∈ V → ran (𝐹 ↾ 𝑎) ∈ V)
62	60, 61	syl 17	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ran (𝐹 ↾ 𝑎) ∈ V)
63		ressress 17191	. . . . . . . . 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	eqtrid 2776	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑉 ↾s ran (𝐹 ↾ 𝑎)) = (𝑇 ↾s (ran 𝐹 ∩ ran (𝐹 ↾ 𝑎))))
66	56, 65	eqtr4id 2783	. . . . . 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 1190	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑉 ∈ LNoeM)
69		lmhmrnlss 20887	. . . . . . . 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 20869	. . . . . . . . 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 20887	. . . . . . . . 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 2724	. . . . . . . . 9 ⊢ (LSubSp‘𝑇) = (LSubSp‘𝑇)
77		eqid 2724	. . . . . . . . 9 ⊢ (LSubSp‘𝑉) = (LSubSp‘𝑉)
78	57, 76, 77	lsslss 20797	. . . . . . . 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 710	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ran (𝐹 ↾ 𝑎) ∈ (LSubSp‘𝑉))
81		eqid 2724	. . . . . . 7 ⊢ (𝑉 ↾s ran (𝐹 ↾ 𝑎)) = (𝑉 ↾s ran (𝐹 ↾ 𝑎))
82	77, 81	lnmlssfg 42277	. . . . . 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 2825	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑇 ↾s ran (𝐹 ↾ 𝑎)) ∈ LFinGen)
85		eqid 2724	. . . . 5 ⊢ (◡(𝐹 ↾ 𝑎) “ { 0 }) = (◡(𝐹 ↾ 𝑎) “ { 0 })
86		eqid 2724	. . . . 5 ⊢ ((𝑆 ↾s 𝑎) ↾s (◡(𝐹 ↾ 𝑎) “ { 0 })) = ((𝑆 ↾s 𝑎) ↾s (◡(𝐹 ↾ 𝑎) “ { 0 }))
87		eqid 2724	. . . . 5 ⊢ (𝑇 ↾s ran (𝐹 ↾ 𝑎)) = (𝑇 ↾s ran (𝐹 ↾ 𝑎))
88	35, 85, 86, 87	lmhmfgsplit 42283	. . . 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 1368	. . 3 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑆 ↾s 𝑎) ∈ LFinGen)
90	89	ralrimiva 3138	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → ∀𝑎 ∈ (LSubSp‘𝑆)(𝑆 ↾s 𝑎) ∈ LFinGen)
91	3	islnm 42274	. 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 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3053  Vcvv 3466   ∩ cin 3939   ⊆ wss 3940  {csn 4620  ^◡ccnv 5665  ran crn 5667   ↾ cres 5668   “ cima 5669  ‘cfv 6533  (class class class)co 7401   ↾_s cress 17171  0_gc0g 17383  LModclmod 20695  LSubSpclss 20767   LMHom clmhm 20856  LFinGenclfig 42264  LNoeMclnm 42272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8698  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-sets 17095  df-slot 17113  df-ndx 17125  df-base 17143  df-ress 17172  df-plusg 17208  df-sca 17211  df-vsca 17212  df-0g 17385  df-mgm 18562  df-sgrp 18641  df-mnd 18657  df-submnd 18703  df-grp 18855  df-minusg 18856  df-sbg 18857  df-subg 19039  df-ghm 19128  df-cntz 19222  df-lsm 19545  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-lmod 20697  df-lss 20768  df-lsp 20808  df-lmhm 20859  df-lfig 42265  df-lnm 42273

This theorem is referenced by:  pwslnmlem2  42290