Theorem lsslindf 21781

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 21759	. . . 4 ⊢ Rel LIndF
2	1	brrelex1i 5734	. . 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 5734	. . 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 483	. . . . . . . 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 2725	. . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊)
9	7, 8	ressbasss 17222	. . . . . . . 8 ⊢ (Base‘𝑋) ⊆ (Base‘𝑊)
10		fss 6739	. . . . . . . 8 ⊢ ((𝐹:dom 𝐹⟶(Base‘𝑋) ∧ (Base‘𝑋) ⊆ (Base‘𝑊)) → 𝐹:dom 𝐹⟶(Base‘𝑊))
11	6, 9, 10	sylancl 584	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑋)) → 𝐹:dom 𝐹⟶(Base‘𝑊))
12		ffn 6723	. . . . . . . . 9 ⊢ (𝐹:dom 𝐹⟶(Base‘𝑊) → 𝐹 Fn dom 𝐹)
13	12	adantl 480	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑊)) → 𝐹 Fn dom 𝐹)
14		simp3 1135	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → ran 𝐹 ⊆ 𝑆)
15		lsslindf.u	. . . . . . . . . . . . 13 ⊢ 𝑈 = (LSubSp‘𝑊)
16	8, 15	lssss 20832	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ 𝑈 → 𝑆 ⊆ (Base‘𝑊))
17	16	3ad2ant2 1131	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → 𝑆 ⊆ (Base‘𝑊))
18	7, 8	ressbas2 17221	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ (Base‘𝑊) → 𝑆 = (Base‘𝑋))
19	17, 18	syl 17	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → 𝑆 = (Base‘𝑋))
20	14, 19	sseqtrd 4017	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → ran 𝐹 ⊆ (Base‘𝑋))
21	20	adantr 479	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑊)) → ran 𝐹 ⊆ (Base‘𝑋))
22		df-f 6553	. . . . . . . 8 ⊢ (𝐹:dom 𝐹⟶(Base‘𝑋) ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ (Base‘𝑋)))
23	13, 21, 22	sylanbrc 581	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑊)) → 𝐹:dom 𝐹⟶(Base‘𝑋))
24	11, 23	impbida 799	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹:dom 𝐹⟶(Base‘𝑋) ↔ 𝐹:dom 𝐹⟶(Base‘𝑊)))
25	24	adantr 479	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (𝐹:dom 𝐹⟶(Base‘𝑋) ↔ 𝐹:dom 𝐹⟶(Base‘𝑊)))
26		simpl2 1189	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → 𝑆 ∈ 𝑈)
27		eqid 2725	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
28	7, 27	resssca 17327	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝑈 → (Scalar‘𝑊) = (Scalar‘𝑋))
29	28	eqcomd 2731	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝑈 → (Scalar‘𝑋) = (Scalar‘𝑊))
30	26, 29	syl 17	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (Scalar‘𝑋) = (Scalar‘𝑊))
31	30	fveq2d 6900	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
32	30	fveq2d 6900	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
33	32	sneqd 4642	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → {(0g‘(Scalar‘𝑋))} = {(0g‘(Scalar‘𝑊))})
34	31, 33	difeq12d 4119	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → ((Base‘(Scalar‘𝑋)) ∖ {(0g‘(Scalar‘𝑋))}) = ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))
35		eqid 2725	. . . . . . . . . . . . 13 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
36	7, 35	ressvsca 17328	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ 𝑈 → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑋))
37	36	eqcomd 2731	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝑈 → ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑊))
38	26, 37	syl 17	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑊))
39	38	oveqd 7436	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (𝑘( ·𝑠 ‘𝑋)(𝐹‘𝑥)) = (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)))
40		simpl1 1188	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → 𝑊 ∈ LMod)
41		imassrn 6075	. . . . . . . . . . 11 ⊢ (𝐹 “ (dom 𝐹 ∖ {𝑥})) ⊆ ran 𝐹
42		simpl3 1190	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → ran 𝐹 ⊆ 𝑆)
43	41, 42	sstrid 3988	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (𝐹 “ (dom 𝐹 ∖ {𝑥})) ⊆ 𝑆)
44		eqid 2725	. . . . . . . . . . 11 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
45		eqid 2725	. . . . . . . . . . 11 ⊢ (LSpan‘𝑋) = (LSpan‘𝑋)
46	7, 44, 45, 15	lsslsp 20911	. . . . . . . . . 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 1368	. . . . . . . . 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 2819	. . . . . . . 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 317	. . . . . . 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 3329	. . . . . 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 3167	. . . . 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 630	. . . 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 7453	. . . . . 6 ⊢ 𝑋 ∈ V
54	53	a1i 11	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → 𝑋 ∈ V)
55		eqid 2725	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
56		eqid 2725	. . . . . 6 ⊢ ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑋)
57		eqid 2725	. . . . . 6 ⊢ (Scalar‘𝑋) = (Scalar‘𝑋)
58		eqid 2725	. . . . . 6 ⊢ (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋))
59		eqid 2725	. . . . . 6 ⊢ (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑋))
60	55, 56, 45, 57, 58, 59	islindf 21763	. . . . 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 578	. . . 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 2725	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
63		eqid 2725	. . . . . 6 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
64	8, 35, 44, 27, 62, 63	islindf 21763	. . . . 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 1182	. . . 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 310	. . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊))
67	66	ex 411	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 ∈ V → (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊)))
68	3, 5, 67	pm5.21ndd 378	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3050  Vcvv 3461   ∖ cdif 3941   ⊆ wss 3944  {csn 4630   class class class wbr 5149  dom cdm 5678  ran crn 5679   “ cima 5681   Fn wfn 6544  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419  Basecbs 17183   ↾_s cress 17212  Scalarcsca 17239   ·_𝑠 cvsca 17240  0_gc0g 17424  LModclmod 20755  LSubSpclss 20827  LSpanclspn 20867   LIndF clindf 21755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-sca 17252  df-vsca 17253  df-0g 17426  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18901  df-minusg 18902  df-sbg 18903  df-subg 19086  df-mgp 20087  df-ur 20134  df-ring 20187  df-lmod 20757  df-lss 20828  df-lsp 20868  df-lindf 21757

This theorem is referenced by:  lsslinds  21782