Theorem lmhmlnmsplit 43076

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 20940	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
2	1	3ad2ant1 1133	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LMod)
3		eqid 2729	. . . . . 6 ⊢ (LSubSp‘𝑆) = (LSubSp‘𝑆)
4		eqid 2729	. . . . . 6 ⊢ (𝑆 ↾s 𝑎) = (𝑆 ↾s 𝑎)
5	3, 4	reslmhm 20959	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝐹 ↾ 𝑎) ∈ ((𝑆 ↾s 𝑎) LMHom 𝑇))
6	5	3ad2antl1 1186	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝐹 ↾ 𝑎) ∈ ((𝑆 ↾s 𝑎) LMHom 𝑇))
7		cnvresima 6203	. . . . . . . 8 ⊢ (◡(𝐹 ↾ 𝑎) “ { 0 }) = ((◡𝐹 “ { 0 }) ∩ 𝑎)
8		lmhmfgsplit.k	. . . . . . . . . 10 ⊢ 𝐾 = (◡𝐹 “ { 0 })
9	8	eqcomi 2738	. . . . . . . . 9 ⊢ (◡𝐹 “ { 0 }) = 𝐾
10	9	ineq1i 4179	. . . . . . . 8 ⊢ ((◡𝐹 “ { 0 }) ∩ 𝑎) = (𝐾 ∩ 𝑎)
11		incom 4172	. . . . . . . 8 ⊢ (𝐾 ∩ 𝑎) = (𝑎 ∩ 𝐾)
12	7, 10, 11	3eqtri 2756	. . . . . . 7 ⊢ (◡(𝐹 ↾ 𝑎) “ { 0 }) = (𝑎 ∩ 𝐾)
13	12	oveq2i 7398	. . . . . 6 ⊢ ((𝑆 ↾s 𝑎) ↾s (◡(𝐹 ↾ 𝑎) “ { 0 })) = ((𝑆 ↾s 𝑎) ↾s (𝑎 ∩ 𝐾))
14		vex 3451	. . . . . . . 8 ⊢ 𝑎 ∈ V
15		inss1 4200	. . . . . . . 8 ⊢ (𝑎 ∩ 𝐾) ⊆ 𝑎
16		ressabs 17218	. . . . . . . 8 ⊢ ((𝑎 ∈ V ∧ (𝑎 ∩ 𝐾) ⊆ 𝑎) → ((𝑆 ↾s 𝑎) ↾s (𝑎 ∩ 𝐾)) = (𝑆 ↾s (𝑎 ∩ 𝐾)))
17	14, 15, 16	mp2an 692	. . . . . . 7 ⊢ ((𝑆 ↾s 𝑎) ↾s (𝑎 ∩ 𝐾)) = (𝑆 ↾s (𝑎 ∩ 𝐾))
18		lmhmfgsplit.u	. . . . . . . . 9 ⊢ 𝑈 = (𝑆 ↾s 𝐾)
19	18	oveq1i 7397	. . . . . . . 8 ⊢ (𝑈 ↾s (𝑎 ∩ 𝐾)) = ((𝑆 ↾s 𝐾) ↾s (𝑎 ∩ 𝐾))
20		simpl1 1192	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
21		cnvexg 7900	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ◡𝐹 ∈ V)
22		imaexg 7889	. . . . . . . . . . . 12 ⊢ (◡𝐹 ∈ V → (◡𝐹 “ { 0 }) ∈ V)
23	21, 22	syl 17	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (◡𝐹 “ { 0 }) ∈ V)
24	8, 23	eqeltrid 2832	. . . . . . . . . 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 4201	. . . . . . . . 9 ⊢ (𝑎 ∩ 𝐾) ⊆ 𝐾
27		ressabs 17218	. . . . . . . . 9 ⊢ ((𝐾 ∈ V ∧ (𝑎 ∩ 𝐾) ⊆ 𝐾) → ((𝑆 ↾s 𝐾) ↾s (𝑎 ∩ 𝐾)) = (𝑆 ↾s (𝑎 ∩ 𝐾)))
28	25, 26, 27	sylancl 586	. . . . . . . 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 1193	. . . . . 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 20958	. . . . . . . . 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 20871	. . . . . . . 8 ⊢ ((𝑆 ∈ LMod ∧ 𝑎 ∈ (LSubSp‘𝑆) ∧ 𝐾 ∈ (LSubSp‘𝑆)) → (𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑆))
39	33, 34, 37, 38	syl3anc 1373	. . . . . . 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 2729	. . . . . . . . 9 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
42	18, 3, 41	lsslss 20867	. . . . . . . 8 ⊢ ((𝑆 ∈ LMod ∧ 𝐾 ∈ (LSubSp‘𝑆)) → ((𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑈) ↔ ((𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑆) ∧ (𝑎 ∩ 𝐾) ⊆ 𝐾)))
43	33, 37, 42	syl2anc 584	. . . . . . 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 713	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑈))
45		eqid 2729	. . . . . . 7 ⊢ (𝑈 ↾s (𝑎 ∩ 𝐾)) = (𝑈 ↾s (𝑎 ∩ 𝐾))
46	41, 45	lnmlssfg 43069	. . . . . 6 ⊢ ((𝑈 ∈ LNoeM ∧ (𝑎 ∩ 𝐾) ∈ (LSubSp‘𝑈)) → (𝑈 ↾s (𝑎 ∩ 𝐾)) ∈ LFinGen)
47	32, 44, 46	syl2anc 584	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑈 ↾s (𝑎 ∩ 𝐾)) ∈ LFinGen)
48	31, 47	eqeltrd 2828	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ((𝑆 ↾s 𝑎) ↾s (◡(𝐹 ↾ 𝑎) “ { 0 })) ∈ LFinGen)
49		incom 4172	. . . . . . . . 9 ⊢ (ran 𝐹 ∩ ran (𝐹 ↾ 𝑎)) = (ran (𝐹 ↾ 𝑎) ∩ ran 𝐹)
50		resss 5972	. . . . . . . . . . 11 ⊢ (𝐹 ↾ 𝑎) ⊆ 𝐹
51		rnss 5903	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ 𝑎) ⊆ 𝐹 → ran (𝐹 ↾ 𝑎) ⊆ ran 𝐹)
52	50, 51	ax-mp 5	. . . . . . . . . 10 ⊢ ran (𝐹 ↾ 𝑎) ⊆ ran 𝐹
53		dfss2 3932	. . . . . . . . . 10 ⊢ (ran (𝐹 ↾ 𝑎) ⊆ ran 𝐹 ↔ (ran (𝐹 ↾ 𝑎) ∩ ran 𝐹) = ran (𝐹 ↾ 𝑎))
54	52, 53	mpbi 230	. . . . . . . . 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 7398	. . . . . . 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 7397	. . . . . . . 8 ⊢ (𝑉 ↾s ran (𝐹 ↾ 𝑎)) = ((𝑇 ↾s ran 𝐹) ↾s ran (𝐹 ↾ 𝑎))
59		rnexg 7878	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ V)
60		resexg 5998	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹 ↾ 𝑎) ∈ V)
61		rnexg 7878	. . . . . . . . . 10 ⊢ ((𝐹 ↾ 𝑎) ∈ V → ran (𝐹 ↾ 𝑎) ∈ V)
62	60, 61	syl 17	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ran (𝐹 ↾ 𝑎) ∈ V)
63		ressress 17217	. . . . . . . . 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 584	. . . . . . . 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 1194	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → 𝑉 ∈ LNoeM)
69		lmhmrnlss 20957	. . . . . . . 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 20939	. . . . . . . . 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 20957	. . . . . . . . 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 2729	. . . . . . . . 9 ⊢ (LSubSp‘𝑇) = (LSubSp‘𝑇)
77		eqid 2729	. . . . . . . . 9 ⊢ (LSubSp‘𝑉) = (LSubSp‘𝑉)
78	57, 76, 77	lsslss 20867	. . . . . . . 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 584	. . . . . . 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 713	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → ran (𝐹 ↾ 𝑎) ∈ (LSubSp‘𝑉))
81		eqid 2729	. . . . . . 7 ⊢ (𝑉 ↾s ran (𝐹 ↾ 𝑎)) = (𝑉 ↾s ran (𝐹 ↾ 𝑎))
82	77, 81	lnmlssfg 43069	. . . . . 6 ⊢ ((𝑉 ∈ LNoeM ∧ ran (𝐹 ↾ 𝑎) ∈ (LSubSp‘𝑉)) → (𝑉 ↾s ran (𝐹 ↾ 𝑎)) ∈ LFinGen)
83	68, 80, 82	syl2anc 584	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑉 ↾s ran (𝐹 ↾ 𝑎)) ∈ LFinGen)
84	67, 83	eqeltrd 2828	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑇 ↾s ran (𝐹 ↾ 𝑎)) ∈ LFinGen)
85		eqid 2729	. . . . 5 ⊢ (◡(𝐹 ↾ 𝑎) “ { 0 }) = (◡(𝐹 ↾ 𝑎) “ { 0 })
86		eqid 2729	. . . . 5 ⊢ ((𝑆 ↾s 𝑎) ↾s (◡(𝐹 ↾ 𝑎) “ { 0 })) = ((𝑆 ↾s 𝑎) ↾s (◡(𝐹 ↾ 𝑎) “ { 0 }))
87		eqid 2729	. . . . 5 ⊢ (𝑇 ↾s ran (𝐹 ↾ 𝑎)) = (𝑇 ↾s ran (𝐹 ↾ 𝑎))
88	35, 85, 86, 87	lmhmfgsplit 43075	. . . 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 1373	. . 3 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) ∧ 𝑎 ∈ (LSubSp‘𝑆)) → (𝑆 ↾s 𝑎) ∈ LFinGen)
90	89	ralrimiva 3125	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → ∀𝑎 ∈ (LSubSp‘𝑆)(𝑆 ↾s 𝑎) ∈ LFinGen)
91	3	islnm 43066	. 2 ⊢ (𝑆 ∈ LNoeM ↔ (𝑆 ∈ LMod ∧ ∀𝑎 ∈ (LSubSp‘𝑆)(𝑆 ↾s 𝑎) ∈ LFinGen))
92	2, 90, 91	sylanbrc 583	1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ LNoeM ∧ 𝑉 ∈ LNoeM) → 𝑆 ∈ LNoeM)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914  {csn 4589  ^◡ccnv 5637  ran crn 5639   ↾ cres 5640   “ cima 5641  ‘cfv 6511  (class class class)co 7387   ↾_s cress 17200  0_gc0g 17402  LModclmod 20766  LSubSpclss 20837   LMHom clmhm 20926  LFinGenclfig 43056  LNoeMclnm 43064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-sca 17236  df-vsca 17237  df-0g 17404  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-submnd 18711  df-grp 18868  df-minusg 18869  df-sbg 18870  df-subg 19055  df-ghm 19145  df-cntz 19249  df-lsm 19566  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-ring 20144  df-lmod 20768  df-lss 20838  df-lsp 20878  df-lmhm 20929  df-lfig 43057  df-lnm 43065

This theorem is referenced by:  pwslnmlem2  43082