islss3 - Metamath Proof Explorer

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Theorem islss3 18780

Description: A linear subspace of a module is a subset which is a module in its own right. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)

**Hypotheses**
Ref	Expression
islss3.x	⊢ 𝑋 = (𝑊 ↾_s 𝑈)
islss3.v	⊢ 𝑉 = (Base‘𝑊)
islss3.s	⊢ 𝑆 = (LSubSp‘𝑊)

**Assertion**
Ref	Expression
islss3	⊢ (𝑊 ∈ LMod → (𝑈 ∈ 𝑆 ↔ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)))

Proof of Theorem islss3 Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		islss3.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑊)
2		islss3.s	. . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊)
3	1, 2	lssss 18758	. . . 4 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉)
4	3	adantl 481	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 ⊆ 𝑉)
5		islss3.x	. . . . . . 7 ⊢ 𝑋 = (𝑊 ↾_s 𝑈)
6	5, 1	ressbas2 15758	. . . . . 6 ⊢ (𝑈 ⊆ 𝑉 → 𝑈 = (Base‘𝑋))
7	6	adantl 481	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → 𝑈 = (Base‘𝑋))
8	3, 7	sylan2 490	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 = (Base‘𝑋))
9		eqid 2610	. . . . . 6 ⊢ (+_g‘𝑊) = (+_g‘𝑊)
10	5, 9	ressplusg 15818	. . . . 5 ⊢ (𝑈 ∈ 𝑆 → (+_g‘𝑊) = (+_g‘𝑋))
11	10	adantl 481	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (+_g‘𝑊) = (+_g‘𝑋))
12		eqid 2610	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
13	5, 12	resssca 15854	. . . . 5 ⊢ (𝑈 ∈ 𝑆 → (Scalar‘𝑊) = (Scalar‘𝑋))
14	13	adantl 481	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑊) = (Scalar‘𝑋))
15		eqid 2610	. . . . . 6 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
16	5, 15	ressvsca 15855	. . . . 5 ⊢ (𝑈 ∈ 𝑆 → ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑋))
17	16	adantl 481	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑋))
18		eqidd 2611	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)))
19		eqidd 2611	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (+_g‘(Scalar‘𝑊)) = (+_g‘(Scalar‘𝑊)))
20		eqidd 2611	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (._r‘(Scalar‘𝑊)) = (._r‘(Scalar‘𝑊)))
21		eqidd 2611	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (1_r‘(Scalar‘𝑊)) = (1_r‘(Scalar‘𝑊)))
22	12	lmodring 18694	. . . . 5 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring)
23	22	adantr 480	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑊) ∈ Ring)
24	2	lsssubg 18778	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (SubGrp‘𝑊))
25	5	subggrp 17420	. . . . 5 ⊢ (𝑈 ∈ (SubGrp‘𝑊) → 𝑋 ∈ Grp)
26	24, 25	syl 17	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ Grp)
27		eqid 2610	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
28	12, 15, 27, 2	lssvscl 18776	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑈)) → (𝑥( ·_𝑠 ‘𝑊)𝑎) ∈ 𝑈)
29	28	3impb 1252	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑈) → (𝑥( ·_𝑠 ‘𝑊)𝑎) ∈ 𝑈)
30		simpll 786	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑊 ∈ LMod)
31		simpr1 1060	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
32	3	ad2antlr 759	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑈 ⊆ 𝑉)
33		simpr2 1061	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑎 ∈ 𝑈)
34	32, 33	sseldd 3569	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑎 ∈ 𝑉)
35		simpr3 1062	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑏 ∈ 𝑈)
36	32, 35	sseldd 3569	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → 𝑏 ∈ 𝑉)
37	1, 9, 12, 15, 27	lmodvsdi 18709	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑥( ·_𝑠 ‘𝑊)(𝑎(+_g‘𝑊)𝑏)) = ((𝑥( ·_𝑠 ‘𝑊)𝑎)(+_g‘𝑊)(𝑥( ·_𝑠 ‘𝑊)𝑏)))
38	30, 31, 34, 36, 37	syl13anc 1320	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → (𝑥( ·_𝑠 ‘𝑊)(𝑎(+_g‘𝑊)𝑏)) = ((𝑥( ·_𝑠 ‘𝑊)𝑎)(+_g‘𝑊)(𝑥( ·_𝑠 ‘𝑊)𝑏)))
39		simpll 786	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑏 ∈ 𝑈)) → 𝑊 ∈ LMod)
40		simpr1 1060	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑏 ∈ 𝑈)) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
41		simpr2 1061	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑏 ∈ 𝑈)) → 𝑎 ∈ (Base‘(Scalar‘𝑊)))
42	3	ad2antlr 759	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑏 ∈ 𝑈)) → 𝑈 ⊆ 𝑉)
43		simpr3 1062	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑏 ∈ 𝑈)) → 𝑏 ∈ 𝑈)
44	42, 43	sseldd 3569	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑏 ∈ 𝑈)) → 𝑏 ∈ 𝑉)
45		eqid 2610	. . . . . 6 ⊢ (+_g‘(Scalar‘𝑊)) = (+_g‘(Scalar‘𝑊))
46	1, 9, 12, 15, 27, 45	lmodvsdir 18710	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑏 ∈ 𝑉)) → ((𝑥(+_g‘(Scalar‘𝑊))𝑎)( ·_𝑠 ‘𝑊)𝑏) = ((𝑥( ·_𝑠 ‘𝑊)𝑏)(+_g‘𝑊)(𝑎( ·_𝑠 ‘𝑊)𝑏)))
47	39, 40, 41, 44, 46	syl13anc 1320	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑏 ∈ 𝑈)) → ((𝑥(+_g‘(Scalar‘𝑊))𝑎)( ·_𝑠 ‘𝑊)𝑏) = ((𝑥( ·_𝑠 ‘𝑊)𝑏)(+_g‘𝑊)(𝑎( ·_𝑠 ‘𝑊)𝑏)))
48		eqid 2610	. . . . . 6 ⊢ (._r‘(Scalar‘𝑊)) = (._r‘(Scalar‘𝑊))
49	1, 12, 15, 27, 48	lmodvsass 18711	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑏 ∈ 𝑉)) → ((𝑥(._r‘(Scalar‘𝑊))𝑎)( ·_𝑠 ‘𝑊)𝑏) = (𝑥( ·_𝑠 ‘𝑊)(𝑎( ·_𝑠 ‘𝑊)𝑏)))
50	39, 40, 41, 44, 49	syl13anc 1320	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑏 ∈ 𝑈)) → ((𝑥(._r‘(Scalar‘𝑊))𝑎)( ·_𝑠 ‘𝑊)𝑏) = (𝑥( ·_𝑠 ‘𝑊)(𝑎( ·_𝑠 ‘𝑊)𝑏)))
51	4	sselda 3568	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑉)
52		eqid 2610	. . . . . . 7 ⊢ (1_r‘(Scalar‘𝑊)) = (1_r‘(Scalar‘𝑊))
53	1, 12, 15, 52	lmodvs1 18714	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝑉) → ((1_r‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑥) = 𝑥)
54	53	adantlr 747	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ 𝑉) → ((1_r‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑥) = 𝑥)
55	51, 54	syldan 486	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ 𝑈) → ((1_r‘(Scalar‘𝑊))( ·_𝑠 ‘𝑊)𝑥) = 𝑥)
56	8, 11, 14, 17, 18, 19, 20, 21, 23, 26, 29, 38, 47, 50, 55	islmodd 18692	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ LMod)
57	4, 56	jca 553	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod))
58		simprl 790	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) → 𝑈 ⊆ 𝑉)
59	58, 6	syl 17	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) → 𝑈 = (Base‘𝑋))
60		fvex 6113	. . . . . . 7 ⊢ (Base‘𝑋) ∈ V
61	59, 60	syl6eqel 2696	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) → 𝑈 ∈ V)
62	5, 12	resssca 15854	. . . . . 6 ⊢ (𝑈 ∈ V → (Scalar‘𝑊) = (Scalar‘𝑋))
63	61, 62	syl 17	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) → (Scalar‘𝑊) = (Scalar‘𝑋))
64	63	eqcomd 2616	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) → (Scalar‘𝑋) = (Scalar‘𝑊))
65		eqidd 2611	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋)))
66	1	a1i 11	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) → 𝑉 = (Base‘𝑊))
67	5, 9	ressplusg 15818	. . . . . 6 ⊢ (𝑈 ∈ V → (+_g‘𝑊) = (+_g‘𝑋))
68	61, 67	syl 17	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) → (+_g‘𝑊) = (+_g‘𝑋))
69	68	eqcomd 2616	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) → (+_g‘𝑋) = (+_g‘𝑊))
70	5, 15	ressvsca 15855	. . . . . 6 ⊢ (𝑈 ∈ V → ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑋))
71	61, 70	syl 17	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) → ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑋))
72	71	eqcomd 2616	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) → ( ·_𝑠 ‘𝑋) = ( ·_𝑠 ‘𝑊))
73	2	a1i 11	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) → 𝑆 = (LSubSp‘𝑊))
74	59, 58	eqsstr3d 3603	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) → (Base‘𝑋) ⊆ 𝑉)
75		lmodgrp 18693	. . . . . 6 ⊢ (𝑋 ∈ LMod → 𝑋 ∈ Grp)
76	75	ad2antll 761	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) → 𝑋 ∈ Grp)
77		eqid 2610	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
78	77	grpbn0 17274	. . . . 5 ⊢ (𝑋 ∈ Grp → (Base‘𝑋) ≠ ∅)
79	76, 78	syl 17	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) → (Base‘𝑋) ≠ ∅)
80		eqid 2610	. . . . . . 7 ⊢ (LSubSp‘𝑋) = (LSubSp‘𝑋)
81	77, 80	lss1 18760	. . . . . 6 ⊢ (𝑋 ∈ LMod → (Base‘𝑋) ∈ (LSubSp‘𝑋))
82	81	ad2antll 761	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) → (Base‘𝑋) ∈ (LSubSp‘𝑋))
83		eqid 2610	. . . . . 6 ⊢ (Scalar‘𝑋) = (Scalar‘𝑋)
84		eqid 2610	. . . . . 6 ⊢ (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋))
85		eqid 2610	. . . . . 6 ⊢ (+_g‘𝑋) = (+_g‘𝑋)
86		eqid 2610	. . . . . 6 ⊢ ( ·_𝑠 ‘𝑋) = ( ·_𝑠 ‘𝑋)
87	83, 84, 85, 86, 80	lsscl 18764	. . . . 5 ⊢ (((Base‘𝑋) ∈ (LSubSp‘𝑋) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑋)) ∧ 𝑎 ∈ (Base‘𝑋) ∧ 𝑏 ∈ (Base‘𝑋))) → ((𝑥( ·_𝑠 ‘𝑋)𝑎)(+_g‘𝑋)𝑏) ∈ (Base‘𝑋))
88	82, 87	sylan 487	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑋)) ∧ 𝑎 ∈ (Base‘𝑋) ∧ 𝑏 ∈ (Base‘𝑋))) → ((𝑥( ·_𝑠 ‘𝑋)𝑎)(+_g‘𝑋)𝑏) ∈ (Base‘𝑋))
89	64, 65, 66, 69, 72, 73, 74, 79, 88	islssd 18757	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) → (Base‘𝑋) ∈ 𝑆)
90	59, 89	eqeltrd 2688	. 2 ⊢ ((𝑊 ∈ LMod ∧ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)) → 𝑈 ∈ 𝑆)
91	57, 90	impbida 873	1 ⊢ (𝑊 ∈ LMod → (𝑈 ∈ 𝑆 ↔ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 ↾_s cress 15696 +_gcplusg 15768 ._rcmulr 15769 Scalarcsca 15771 ·_𝑠 cvsca 15772 Grpcgrp 17245 SubGrpcsubg 17411 1_rcur 18324 Ringcrg 18370 LModclmod 18686 LSubSpclss 18753

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-sca 15784 df-vsca 15785 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-mgp 18313 df-ur 18325 df-ring 18372 df-lmod 18688 df-lss 18754

This theorem is referenced by: lsslmod 18781 lsslss 18782 issubassa 19145

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