Theorem lkrlss 36846

Description: The kernel of a linear functional is a subspace. (nlelshi 30141 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 2737	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2737	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
3		eqid 2737	. . . 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 36841	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) = {𝑥 ∈ (Base‘𝑊) ∣ (𝐺‘𝑥) = (0g‘(Scalar‘𝑊))})
7		ssrab2 3993	. . 3 ⊢ {𝑥 ∈ (Base‘𝑊) ∣ (𝐺‘𝑥) = (0g‘(Scalar‘𝑊))} ⊆ (Base‘𝑊)
8	6, 7	eqsstrdi 3955	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) ⊆ (Base‘𝑊))
9		eqid 2737	. . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊)
10	1, 9	lmod0vcl 19928	. . . . 5 ⊢ (𝑊 ∈ LMod → (0g‘𝑊) ∈ (Base‘𝑊))
11	10	adantr 484	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (0g‘𝑊) ∈ (Base‘𝑊))
12	2, 3, 9, 4	lfl0 36816	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐺‘(0g‘𝑊)) = (0g‘(Scalar‘𝑊)))
13	1, 2, 3, 4, 5	ellkr 36840	. . . 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 4250	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) ≠ ∅)
16		simplll 775	. . . . . 6 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝑊 ∈ LMod)
17		simplr 769	. . . . . . 7 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝑟 ∈ (Base‘(Scalar‘𝑊)))
18		simpllr 776	. . . . . . . 8 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝐺 ∈ 𝐹)
19		simprl 771	. . . . . . . 8 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝑥 ∈ (𝐾‘𝐺))
20	1, 4, 5	lkrcl 36843	. . . . . . . 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 2737	. . . . . . . 8 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
23		eqid 2737	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
24	1, 2, 22, 23	lmodvscl 19916	. . . . . . 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 773	. . . . . . 7 ⊢ ((((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) ∧ (𝑥 ∈ (𝐾‘𝐺) ∧ 𝑦 ∈ (𝐾‘𝐺))) → 𝑦 ∈ (𝐾‘𝐺))
27	1, 4, 5	lkrcl 36843	. . . . . . 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 2737	. . . . . . 7 ⊢ (+g‘𝑊) = (+g‘𝑊)
30	1, 29	lmodvacl 19913	. . . . . 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 2737	. . . . . . . 8 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
33		eqid 2737	. . . . . . . 8 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
34	1, 29, 2, 22, 23, 32, 33, 4	lfli 36812	. . . . . . 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 36844	. . . . . . . . . 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 7229	. . . . . . . 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 19907	. . . . . . . . . 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 19606	. . . . . . . . 9 ⊢ (((Scalar‘𝑊) ∈ Ring ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) → (𝑟(.r‘(Scalar‘𝑊))(0g‘(Scalar‘𝑊))) = (0g‘(Scalar‘𝑊)))
42	40, 17, 41	syl2anc 587	. . . . . . . 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 2777	. . . . . . 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 36844	. . . . . . . 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 7231	. . . . . 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 19908	. . . . . . . 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 18395	. . . . . . 7 ⊢ ((Scalar‘𝑊) ∈ Grp → (0g‘(Scalar‘𝑊)) ∈ (Base‘(Scalar‘𝑊)))
50	23, 32, 3	grplid 18397	. . . . . . 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 588	. . . . . 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 2781	. . . . 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 36840	. . . . . 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 3112	. . 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 3105	. 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 19971	. 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 1345	1 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐾‘𝐺) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∀wral 3061  {crab 3065   ⊆ wss 3866  ∅c0 4237  ‘cfv 6380  (class class class)co 7213  Basecbs 16760  +_gcplusg 16802  ._rcmulr 16803  Scalarcsca 16805   ·_𝑠 cvsca 16806  0_gc0g 16944  Grpcgrp 18365  Ringcrg 19562  LModclmod 19899  LSubSpclss 19968  LFnlclfn 36808  LKerclk 36836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-plusg 16815  df-0g 16946  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-grp 18368  df-minusg 18369  df-sbg 18370  df-mgp 19505  df-ur 19517  df-ring 19564  df-lmod 19901  df-lss 19969  df-lfl 36809  df-lkr 36837

This theorem is referenced by:  lkrssv  36847  lkrlsp  36853  lkrlsp3  36855  lkrshp  36856  lclkrlem2f  39263  lclkrlem2n  39271  lclkrlem2v  39279  lcfrlem25  39318  lcfrlem35  39328