Theorem lkrlss 39118

Description: The kernel of a linear functional is a subspace. (nlelshi 32046 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 2736	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2736	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
3		eqid 2736	. . . 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 39113	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∈ (Base‘𝑊) ∣ (𝐺‘𝑥) = (0g‘(Scalar‘𝑊))})
7		ssrab2 4060	. . 3 ⊢ {𝑥 ∈ (Base‘𝑊) ∣ (𝐺‘𝑥) = (0g‘(Scalar‘𝑊))} ⊆ (Base‘𝑊)
8	6, 7	eqsstrdi 4008	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) ⊆ (Base‘𝑊))
9		eqid 2736	. . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊)
10	1, 9	lmod0vcl 20853	. . . . 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 39088	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐺‘(0g‘𝑊)) = (0g‘(Scalar‘𝑊)))
13	1, 2, 3, 4, 5	ellkr 39112	. . . 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 4322	. 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 39115	. . . . . . . 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 2736	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
23		eqid 2736	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
24	1, 2, 22, 23	lmodvscl 20840	. . . . . . 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 39115	. . . . . . 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 2736	. . . . . . 7 ⊢ (+g‘𝑊) = (+g‘𝑊)
30	1, 29	lmodvacl 20837	. . . . . 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 2736	. . . . . . . 8 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
33		eqid 2736	. . . . . . . 8 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
34	1, 29, 2, 22, 23, 32, 33, 4	lfli 39084	. . . . . . 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 39116	. . . . . . . . . 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 7426	. . . . . . . 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 20830	. . . . . . . . . 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 20259	. . . . . . . . 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 2771	. . . . . . 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 39116	. . . . . . . 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 7428	. . . . . 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 20831	. . . . . . . 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 18953	. . . . . . 7 ⊢ ((Scalar‘𝑊) ∈ Grp → (0g‘(Scalar‘𝑊)) ∈ (Base‘(Scalar‘𝑊)))
50	23, 32, 3	grplid 18955	. . . . . . 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 2775	. . . . 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 39112	. . . . . 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 3188	. . 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 3133	. 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 20896	. 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 2933  ∀wral 3052  {crab 3420   ⊆ wss 3931  ∅c0 4313  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  +_gcplusg 17276  ._rcmulr 17277  Scalarcsca 17279   ·_𝑠 cvsca 17280  0_gc0g 17458  Grpcgrp 18921  Ringcrg 20198  LModclmod 20822  LSubSpclss 20893  LFnlclfn 39080  LKerclk 39108

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-plusg 17289  df-0g 17460  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-grp 18924  df-minusg 18925  df-sbg 18926  df-cmn 19768  df-abl 19769  df-mgp 20106  df-rng 20118  df-ur 20147  df-ring 20200  df-lmod 20824  df-lss 20894  df-lfl 39081  df-lkr 39109

This theorem is referenced by:  lkrssv  39119  lkrlsp  39125  lkrlsp3  39127  lkrshp  39128  lclkrlem2f  41536  lclkrlem2n  41544  lclkrlem2v  41552  lcfrlem25  41591  lcfrlem35  41601