Theorem lkrlss 39074

Description: The kernel of a linear functional is a subspace. (nlelshi 32004 analog.) (Contributed by NM, 16-Apr-2014.)

Hypotheses

Ref	Expression
lkrlss.f	⊢ 𝐹 = (LFnl‘𝑊)
lkrlss.k	⊢ 𝐾 = (LKer‘𝑊)
lkrlss.s	⊢ 𝑆 = (LSubSp‘𝑊)

Assertion

Ref	Expression
lkrlss	⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) ∈ 𝑆)

Proof of Theorem lkrlss

Dummy variables 𝑥 𝑟 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2729	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
3		eqid 2729	. . . 4 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
4		lkrlss.f	. . . 4 ⊢ 𝐹 = (LFnl‘𝑊)
5		lkrlss.k	. . . 4 ⊢ 𝐾 = (LKer‘𝑊)
6	1, 2, 3, 4, 5	lkrval2 39069	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∈ (Base‘𝑊) ∣ (𝐺‘𝑥) = (0g‘(Scalar‘𝑊))})
7		ssrab2 4031	. . 3 ⊢ {𝑥 ∈ (Base‘𝑊) ∣ (𝐺‘𝑥) = (0g‘(Scalar‘𝑊))} ⊆ (Base‘𝑊)
8	6, 7	eqsstrdi 3980	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) ⊆ (Base‘𝑊))
9		eqid 2729	. . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊)
10	1, 9	lmod0vcl 20794	. . . . 5 ⊢ (𝑊 ∈ LMod → (0g‘𝑊) ∈ (Base‘𝑊))
11	10	adantr 480	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (0g‘𝑊) ∈ (Base‘𝑊))
12	2, 3, 9, 4	lfl0 39044	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐺‘(0g‘𝑊)) = (0g‘(Scalar‘𝑊)))
13	1, 2, 3, 4, 5	ellkr 39068	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((0g‘𝑊) ∈ (𝐾‘𝐺) ↔ ((0g‘𝑊) ∈ (Base‘𝑊) ∧ (𝐺‘(0g‘𝑊)) = (0g‘(Scalar‘𝑊)))))
14	11, 12, 13	mpbir2and 713	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (0g‘𝑊) ∈ (𝐾‘𝐺))
15	14	ne0d 4293	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) ≠ ∅)
16		simplll 774	. . . . . 6 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝑊 ∈ LMod)
17		simplr 768	. . . . . . 7 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝑟 ∈ (Base‘(Scalar‘𝑊)))
18		simpllr 775	. . . . . . . 8 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝐺 ∈ 𝐹)
19		simprl 770	. . . . . . . 8 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝑥 ∈ (𝐾‘𝐺))
20	1, 4, 5	lkrcl 39071	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑥 ∈ (𝐾‘𝐺)) → 𝑥 ∈ (Base‘𝑊))
21	16, 18, 19, 20	syl3anc 1373	. . . . . . 7 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝑥 ∈ (Base‘𝑊))
22		eqid 2729	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
23		eqid 2729	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
24	1, 2, 22, 23	lmodvscl 20781	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑟( ·𝑠 ‘𝑊)𝑥) ∈ (Base‘𝑊))
25	16, 17, 21, 24	syl3anc 1373	. . . . . 6 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → (𝑟( ·𝑠 ‘𝑊)𝑥) ∈ (Base‘𝑊))
26		simprr 772	. . . . . . 7 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝑦 ∈ (𝐾‘𝐺))
27	1, 4, 5	lkrcl 39071	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑦 ∈ (𝐾‘𝐺)) → 𝑦 ∈ (Base‘𝑊))
28	16, 18, 26, 27	syl3anc 1373	. . . . . 6 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝑦 ∈ (Base‘𝑊))
29		eqid 2729	. . . . . . 7 ⊢ (+g‘𝑊) = (+g‘𝑊)
30	1, 29	lmodvacl 20778	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑟( ·𝑠 ‘𝑊)𝑥) ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → ((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦) ∈ (Base‘𝑊))
31	16, 25, 28, 30	syl3anc 1373	. . . . 5 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → ((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦) ∈ (Base‘𝑊))
32		eqid 2729	. . . . . . . 8 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
33		eqid 2729	. . . . . . . 8 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
34	1, 29, 2, 22, 23, 32, 33, 4	lfli 39040	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝐺‘((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺‘𝑥))(+g‘(Scalar‘𝑊))(𝐺‘𝑦)))
35	16, 18, 17, 21, 28, 34	syl113anc 1384	. . . . . 6 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → (𝐺‘((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = ((𝑟(.r‘(Scalar‘𝑊))(𝐺‘𝑥))(+g‘(Scalar‘𝑊))(𝐺‘𝑦)))
36	2, 3, 4, 5	lkrf0 39072	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑥 ∈ (𝐾‘𝐺)) → (𝐺‘𝑥) = (0g‘(Scalar‘𝑊)))
37	16, 18, 19, 36	syl3anc 1373	. . . . . . . . 9 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → (𝐺‘𝑥) = (0g‘(Scalar‘𝑊)))
38	37	oveq2d 7365	. . . . . . . 8 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → (𝑟(.r‘(Scalar‘𝑊))(𝐺‘𝑥)) = (𝑟(.r‘(Scalar‘𝑊))(0g‘(Scalar‘𝑊))))
39	2	lmodring 20771	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring)
40	16, 39	syl 17	. . . . . . . . 9 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → (Scalar‘𝑊) ∈ Ring)
41	23, 33, 3	ringrz 20179	. . . . . . . . 9 ⊢ (((Scalar‘𝑊) ∈ Ring ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) → (𝑟(.r‘(Scalar‘𝑊))(0g‘(Scalar‘𝑊))) = (0g‘(Scalar‘𝑊)))
42	40, 17, 41	syl2anc 584	. . . . . . . 8 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → (𝑟(.r‘(Scalar‘𝑊))(0g‘(Scalar‘𝑊))) = (0g‘(Scalar‘𝑊)))
43	38, 42	eqtrd 2764	. . . . . . 7 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → (𝑟(.r‘(Scalar‘𝑊))(𝐺‘𝑥)) = (0g‘(Scalar‘𝑊)))
44	2, 3, 4, 5	lkrf0 39072	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑦 ∈ (𝐾‘𝐺)) → (𝐺‘𝑦) = (0g‘(Scalar‘𝑊)))
45	16, 18, 26, 44	syl3anc 1373	. . . . . . 7 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → (𝐺‘𝑦) = (0g‘(Scalar‘𝑊)))
46	43, 45	oveq12d 7367	. . . . . 6 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → ((𝑟(.r‘(Scalar‘𝑊))(𝐺‘𝑥))(+g‘(Scalar‘𝑊))(𝐺‘𝑦)) = ((0g‘(Scalar‘𝑊))(+g‘(Scalar‘𝑊))(0g‘(Scalar‘𝑊))))
47	2	lmodfgrp 20772	. . . . . . . 8 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Grp)
48	16, 47	syl 17	. . . . . . 7 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → (Scalar‘𝑊) ∈ Grp)
49	23, 3	grpidcl 18844	. . . . . . 7 ⊢ ((Scalar‘𝑊) ∈ Grp → (0g‘(Scalar‘𝑊)) ∈ (Base‘(Scalar‘𝑊)))
50	23, 32, 3	grplid 18846	. . . . . . 7 ⊢ (((Scalar‘𝑊) ∈ Grp ∧ (0g‘(Scalar‘𝑊)) ∈ (Base‘(Scalar‘𝑊))) → ((0g‘(Scalar‘𝑊))(+g‘(Scalar‘𝑊))(0g‘(Scalar‘𝑊))) = (0g‘(Scalar‘𝑊)))
51	48, 49, 50	syl2anc2 585	. . . . . 6 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → ((0g‘(Scalar‘𝑊))(+g‘(Scalar‘𝑊))(0g‘(Scalar‘𝑊))) = (0g‘(Scalar‘𝑊)))
52	35, 46, 51	3eqtrd 2768	. . . . 5 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → (𝐺‘((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = (0g‘(Scalar‘𝑊)))
53	1, 2, 3, 4, 5	ellkr 39068	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦) ∈ (𝐾‘𝐺) ↔ (((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦) ∈ (Base‘𝑊) ∧ (𝐺‘((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = (0g‘(Scalar‘𝑊)))))
54	53	ad2antrr 726	. . . . 5 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → (((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦) ∈ (𝐾‘𝐺) ↔ (((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦) ∈ (Base‘𝑊) ∧ (𝐺‘((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)) = (0g‘(Scalar‘𝑊)))))
55	31, 52, 54	mpbir2and 713	. . . 4 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → ((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦) ∈ (𝐾‘𝐺))
56	55	ralrimivva 3172	. . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) → ∀𝑥 ∈ (𝐾‘𝐺)∀𝑦 ∈ (𝐾‘𝐺)((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦) ∈ (𝐾‘𝐺))
57	56	ralrimiva 3121	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (𝐾‘𝐺)∀𝑦 ∈ (𝐾‘𝐺)((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦) ∈ (𝐾‘𝐺))
58		lkrlss.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑊)
59	2, 23, 1, 29, 22, 58	islss 20837	. 2 ⊢ ((𝐾‘𝐺) ∈ 𝑆 ↔ ((𝐾‘𝐺) ⊆ (Base‘𝑊) ∧ (𝐾‘𝐺) ≠ ∅ ∧ ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑥 ∈ (𝐾‘𝐺)∀𝑦 ∈ (𝐾‘𝐺)((𝑟( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦) ∈ (𝐾‘𝐺)))
60	8, 15, 57, 59	syl3anbrc 1344	1 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  {crab 3394   ⊆ wss 3903  ∅c0 4284  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  +_gcplusg 17161  ._rcmulr 17162  Scalarcsca 17164   ·_𝑠 cvsca 17165  0_gc0g 17343  Grpcgrp 18812  Ringcrg 20118  LModclmod 20763  LSubSpclss 20834  LFnlclfn 39036  LKerclk 39064

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

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 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-plusg 17174  df-0g 17345  df-mgm 18514  df-sgrp 18593  df-mnd 18609  df-grp 18815  df-minusg 18816  df-sbg 18817  df-cmn 19661  df-abl 19662  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-lmod 20765  df-lss 20835  df-lfl 39037  df-lkr 39065

This theorem is referenced by:  lkrssv  39075  lkrlsp  39081  lkrlsp3  39083  lkrshp  39084  lclkrlem2f  41491  lclkrlem2n  41499  lclkrlem2v  41507  lcfrlem25  41546  lcfrlem35  41556