Theorem lkrlss 36230

Description: The kernel of a linear functional is a subspace. (nlelshi 29836 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 2821	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2821	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
3		eqid 2821	. . . 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 36225	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∈ (Base‘𝑊) ∣ (𝐺‘𝑥) = (0g‘(Scalar‘𝑊))})
7		ssrab2 4055	. . 3 ⊢ {𝑥 ∈ (Base‘𝑊) ∣ (𝐺‘𝑥) = (0g‘(Scalar‘𝑊))} ⊆ (Base‘𝑊)
8	6, 7	eqsstrdi 4020	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) ⊆ (Base‘𝑊))
9		eqid 2821	. . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊)
10	1, 9	lmod0vcl 19662	. . . . 5 ⊢ (𝑊 ∈ LMod → (0g‘𝑊) ∈ (Base‘𝑊))
11	10	adantr 483	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (0g‘𝑊) ∈ (Base‘𝑊))
12	2, 3, 9, 4	lfl0 36200	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐺‘(0g‘𝑊)) = (0g‘(Scalar‘𝑊)))
13	1, 2, 3, 4, 5	ellkr 36224	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((0g‘𝑊) ∈ (𝐾‘𝐺) ↔ ((0g‘𝑊) ∈ (Base‘𝑊) ∧ (𝐺‘(0g‘𝑊)) = (0g‘(Scalar‘𝑊)))))
14	11, 12, 13	mpbir2and 711	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (0g‘𝑊) ∈ (𝐾‘𝐺))
15	14	ne0d 4300	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) ≠ ∅)
16		simplll 773	. . . . . 6 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝑊 ∈ LMod)
17		simplr 767	. . . . . . 7 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝑟 ∈ (Base‘(Scalar‘𝑊)))
18		simpllr 774	. . . . . . . 8 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝐺 ∈ 𝐹)
19		simprl 769	. . . . . . . 8 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝑥 ∈ (𝐾‘𝐺))
20	1, 4, 5	lkrcl 36227	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑥 ∈ (𝐾‘𝐺)) → 𝑥 ∈ (Base‘𝑊))
21	16, 18, 19, 20	syl3anc 1367	. . . . . . 7 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝑥 ∈ (Base‘𝑊))
22		eqid 2821	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
23		eqid 2821	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
24	1, 2, 22, 23	lmodvscl 19650	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑟( ·𝑠 ‘𝑊)𝑥) ∈ (Base‘𝑊))
25	16, 17, 21, 24	syl3anc 1367	. . . . . 6 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → (𝑟( ·𝑠 ‘𝑊)𝑥) ∈ (Base‘𝑊))
26		simprr 771	. . . . . . 7 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝑦 ∈ (𝐾‘𝐺))
27	1, 4, 5	lkrcl 36227	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑦 ∈ (𝐾‘𝐺)) → 𝑦 ∈ (Base‘𝑊))
28	16, 18, 26, 27	syl3anc 1367	. . . . . 6 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝑦 ∈ (Base‘𝑊))
29		eqid 2821	. . . . . . 7 ⊢ (+g‘𝑊) = (+g‘𝑊)
30	1, 29	lmodvacl 19647	. . . . . 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 1367	. . . . 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 2821	. . . . . . . 8 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
33		eqid 2821	. . . . . . . 8 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
34	1, 29, 2, 22, 23, 32, 33, 4	lfli 36196	. . . . . . 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 1378	. . . . . 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 36228	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑥 ∈ (𝐾‘𝐺)) → (𝐺‘𝑥) = (0g‘(Scalar‘𝑊)))
37	16, 18, 19, 36	syl3anc 1367	. . . . . . . . 9 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → (𝐺‘𝑥) = (0g‘(Scalar‘𝑊)))
38	37	oveq2d 7171	. . . . . . . 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 19641	. . . . . . . . . 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 19337	. . . . . . . . 9 ⊢ (((Scalar‘𝑊) ∈ Ring ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) → (𝑟(.r‘(Scalar‘𝑊))(0g‘(Scalar‘𝑊))) = (0g‘(Scalar‘𝑊)))
42	40, 17, 41	syl2anc 586	. . . . . . . 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 2856	. . . . . . 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 36228	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑦 ∈ (𝐾‘𝐺)) → (𝐺‘𝑦) = (0g‘(Scalar‘𝑊)))
45	16, 18, 26, 44	syl3anc 1367	. . . . . . 7 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → (𝐺‘𝑦) = (0g‘(Scalar‘𝑊)))
46	43, 45	oveq12d 7173	. . . . . 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 19642	. . . . . . . 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 18130	. . . . . . 7 ⊢ ((Scalar‘𝑊) ∈ Grp → (0g‘(Scalar‘𝑊)) ∈ (Base‘(Scalar‘𝑊)))
50	23, 32, 3	grplid 18132	. . . . . . 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 587	. . . . . 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 2860	. . . . 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 36224	. . . . . 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 724	. . . . 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 711	. . . 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 3191	. . 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 3182	. 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 19705	. 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 1339	1 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  {crab 3142   ⊆ wss 3935  ∅c0 4290  ‘cfv 6354  (class class class)co 7155  Basecbs 16482  +_gcplusg 16564  ._rcmulr 16565  Scalarcsca 16567   ·_𝑠 cvsca 16568  0_gc0g 16712  Grpcgrp 18102  Ringcrg 19296  LModclmod 19633  LSubSpclss 19702  LFnlclfn 36192  LKerclk 36220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-er 8288  df-map 8407  df-en 8509  df-dom 8510  df-sdom 8511  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-2 11699  df-ndx 16485  df-slot 16486  df-base 16488  df-sets 16489  df-plusg 16577  df-0g 16714  df-mgm 17851  df-sgrp 17900  df-mnd 17911  df-grp 18105  df-minusg 18106  df-sbg 18107  df-mgp 19239  df-ur 19251  df-ring 19298  df-lmod 19635  df-lss 19703  df-lfl 36193  df-lkr 36221

This theorem is referenced by:  lkrssv  36231  lkrlsp  36237  lkrlsp3  36239  lkrshp  36240  lclkrlem2f  38647  lclkrlem2n  38655  lclkrlem2v  38663  lcfrlem25  38702  lcfrlem35  38712