Theorem lsslindf 21961

Description: Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)

Hypotheses

Ref	Expression
lsslindf.u	⊢ 𝑈 = (LSubSp‘𝑊)
lsslindf.x	⊢ 𝑋 = (𝑊 ↾s 𝑆)

Assertion

Ref	Expression
lsslindf	⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊))

Proof of Theorem lsslindf

Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rellindf 21939	. . . 4 ⊢ Rel LIndF
2	1	brrelex1i 5719	. . 3 ⊢ (𝐹 LIndF 𝑋 → 𝐹 ∈ V)
3	2	a1i 11	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 LIndF 𝑋 → 𝐹 ∈ V))
4	1	brrelex1i 5719	. . 3 ⊢ (𝐹 LIndF 𝑊 → 𝐹 ∈ V)
5	4	a1i 11	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 LIndF 𝑊 → 𝐹 ∈ V))
6		simpr 489	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑋)) → 𝐹:dom 𝐹⟶(Base‘𝑋))
7		lsslindf.x	. . . . . . . . 9 ⊢ 𝑋 = (𝑊 ↾s 𝑆)
8		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊)
9	7, 8	ressbasss 17300	. . . . . . . 8 ⊢ (Base‘𝑋) ⊆ (Base‘𝑊)
10		fss 6724	. . . . . . . 8 ⊢ ((𝐹:dom 𝐹⟶(Base‘𝑋) ∧ (Base‘𝑋) ⊆ (Base‘𝑊)) → 𝐹:dom 𝐹⟶(Base‘𝑊))
11	6, 9, 10	sylancl 597	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑋)) → 𝐹:dom 𝐹⟶(Base‘𝑊))
12		ffn 6707	. . . . . . . . 9 ⊢ (𝐹:dom 𝐹⟶(Base‘𝑊) → 𝐹 Fn dom 𝐹)
13	12	adantl 486	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑊)) → 𝐹 Fn dom 𝐹)
14		simp3 1156	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → ran 𝐹 ⊆ 𝑆)
15		lsslindf.u	. . . . . . . . . . . . 13 ⊢ 𝑈 = (LSubSp‘𝑊)
16	8, 15	lssss 21038	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ 𝑈 → 𝑆 ⊆ (Base‘𝑊))
17	16	3ad2ant2 1152	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → 𝑆 ⊆ (Base‘𝑊))
18	7, 8	ressbas2 17299	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ (Base‘𝑊) → 𝑆 = (Base‘𝑋))
19	17, 18	syl 18	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → 𝑆 = (Base‘𝑋))
20	14, 19	sseqtrd 3974	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → ran 𝐹 ⊆ (Base‘𝑋))
21	20	adantr 485	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑊)) → ran 𝐹 ⊆ (Base‘𝑋))
22		df-f 6542	. . . . . . . 8 ⊢ (𝐹:dom 𝐹⟶(Base‘𝑋) ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ (Base‘𝑋)))
23	13, 21, 22	sylanbrc 594	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑊)) → 𝐹:dom 𝐹⟶(Base‘𝑋))
24	11, 23	impbida 812	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹:dom 𝐹⟶(Base‘𝑋) ↔ 𝐹:dom 𝐹⟶(Base‘𝑊)))
25	24	adantr 485	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (𝐹:dom 𝐹⟶(Base‘𝑋) ↔ 𝐹:dom 𝐹⟶(Base‘𝑊)))
26		simpl2 1211	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → 𝑆 ∈ 𝑈)
27		eqid 2763	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
28	7, 27	resssca 17397	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝑈 → (Scalar‘𝑊) = (Scalar‘𝑋))
29	28	eqcomd 2769	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝑈 → (Scalar‘𝑋) = (Scalar‘𝑊))
30	26, 29	syl 18	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (Scalar‘𝑋) = (Scalar‘𝑊))
31	30	fveq2d 6887	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
32	30	fveq2d 6887	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
33	32	sneqd 4602	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → {(0g‘(Scalar‘𝑋))} = {(0g‘(Scalar‘𝑊))})
34	31, 33	difeq12d 4083	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → ((Base‘(Scalar‘𝑋)) ∖ {(0g‘(Scalar‘𝑋))}) = ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))
35		eqid 2763	. . . . . . . . . . . . 13 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
36	7, 35	ressvsca 17398	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ 𝑈 → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑋))
37	36	eqcomd 2769	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝑈 → ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑊))
38	26, 37	syl 18	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑊))
39	38	oveqd 7429	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (𝑘( ·𝑠 ‘𝑋)(𝐹‘𝑥)) = (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)))
40		simpl1 1210	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → 𝑊 ∈ LMod)
41		imassrn 6075	. . . . . . . . . . 11 ⊢ (𝐹 “ (dom 𝐹 ∖ {𝑥})) ⊆ ran 𝐹
42		simpl3 1212	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → ran 𝐹 ⊆ 𝑆)
43	41, 42	sstrid 3949	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (𝐹 “ (dom 𝐹 ∖ {𝑥})) ⊆ 𝑆)
44		eqid 2763	. . . . . . . . . . 11 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
45		eqid 2763	. . . . . . . . . . 11 ⊢ (LSpan‘𝑋) = (LSpan‘𝑋)
46	7, 44, 45, 15	lsslsp 21117	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ (𝐹 “ (dom 𝐹 ∖ {𝑥})) ⊆ 𝑆) → ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) = ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))
47	40, 26, 43, 46	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) = ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))
48	39, 47	eleq12d 2857	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → ((𝑘( ·𝑠 ‘𝑋)(𝐹‘𝑥)) ∈ ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
49	48	notbid 321	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (¬ (𝑘( ·𝑠 ‘𝑋)(𝐹‘𝑥)) ∈ ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ ¬ (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
50	34, 49	raleqbidv 3338	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (∀𝑘 ∈ ((Base‘(Scalar‘𝑋)) ∖ {(0g‘(Scalar‘𝑋))}) ¬ (𝑘( ·𝑠 ‘𝑋)(𝐹‘𝑥)) ∈ ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
51	50	ralbidv 3188	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (∀𝑥 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑋)) ∖ {(0g‘(Scalar‘𝑋))}) ¬ (𝑘( ·𝑠 ‘𝑋)(𝐹‘𝑥)) ∈ ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))
52	25, 51	anbi12d 643	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → ((𝐹:dom 𝐹⟶(Base‘𝑋) ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑋)) ∖ {(0g‘(Scalar‘𝑋))}) ¬ (𝑘( ·𝑠 ‘𝑋)(𝐹‘𝑥)) ∈ ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))) ↔ (𝐹:dom 𝐹⟶(Base‘𝑊) ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
53	7	ovexi 7446	. . . . . 6 ⊢ 𝑋 ∈ V
54	53	a1i 11	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → 𝑋 ∈ V)
55		eqid 2763	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
56		eqid 2763	. . . . . 6 ⊢ ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑋)
57		eqid 2763	. . . . . 6 ⊢ (Scalar‘𝑋) = (Scalar‘𝑋)
58		eqid 2763	. . . . . 6 ⊢ (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋))
59		eqid 2763	. . . . . 6 ⊢ (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑋))
60	55, 56, 45, 57, 58, 59	islindf 21943	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑋 ↔ (𝐹:dom 𝐹⟶(Base‘𝑋) ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑋)) ∖ {(0g‘(Scalar‘𝑋))}) ¬ (𝑘( ·𝑠 ‘𝑋)(𝐹‘𝑥)) ∈ ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
61	54, 60	sylan 591	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑋 ↔ (𝐹:dom 𝐹⟶(Base‘𝑋) ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑋)) ∖ {(0g‘(Scalar‘𝑋))}) ¬ (𝑘( ·𝑠 ‘𝑋)(𝐹‘𝑥)) ∈ ((LSpan‘𝑋)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
62		eqid 2763	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
63		eqid 2763	. . . . . 6 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
64	8, 35, 44, 27, 62, 63	islindf 21943	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶(Base‘𝑊) ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
65	64	3ad2antl1 1204	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶(Base‘𝑊) ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
66	52, 61, 65	3bitr4d 314	. . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊))
67	66	ex 417	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 ∈ V → (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊)))
68	3, 5, 67	pm5.21ndd 382	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ∖ cdif 3903   ⊆ wss 3906  {csn 4590   class class class wbr 5110  dom cdm 5663  ran crn 5664   “ cima 5666   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412  Basecbs 17270   ↾_s cress 17291  Scalarcsca 17314   ·_𝑠 cvsca 17315  0_gc0g 17493  LModclmod 20962  LSubSpclss 21033  LSpanclspn 21073   LIndF clindf 21935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-3 12305  df-4 12306  df-5 12307  df-6 12308  df-sets 17225  df-slot 17243  df-ndx 17255  df-base 17271  df-ress 17292  df-plusg 17324  df-sca 17327  df-vsca 17328  df-0g 17495  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-grp 19004  df-minusg 19005  df-sbg 19006  df-subg 19190  df-mgp 20218  df-ur 20265  df-ring 20318  df-lmod 20964  df-lss 21034  df-lsp 21074  df-lindf 21937

This theorem is referenced by:  lsslinds  21962