Theorem lsslindf 21746

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 21724	. . . 4 ⊢ Rel LIndF
2	1	brrelex1i 5697	. . 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 5697	. . 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 484	. . . . . . . 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 2730	. . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊)
9	7, 8	ressbasss 17216	. . . . . . . 8 ⊢ (Base‘𝑋) ⊆ (Base‘𝑊)
10		fss 6707	. . . . . . . 8 ⊢ ((𝐹:dom 𝐹⟶(Base‘𝑋) ∧ (Base‘𝑋) ⊆ (Base‘𝑊)) → 𝐹:dom 𝐹⟶(Base‘𝑊))
11	6, 9, 10	sylancl 586	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑋)) → 𝐹:dom 𝐹⟶(Base‘𝑊))
12		ffn 6691	. . . . . . . . 9 ⊢ (𝐹:dom 𝐹⟶(Base‘𝑊) → 𝐹 Fn dom 𝐹)
13	12	adantl 481	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑊)) → 𝐹 Fn dom 𝐹)
14		simp3 1138	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → ran 𝐹 ⊆ 𝑆)
15		lsslindf.u	. . . . . . . . . . . . 13 ⊢ 𝑈 = (LSubSp‘𝑊)
16	8, 15	lssss 20849	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ 𝑈 → 𝑆 ⊆ (Base‘𝑊))
17	16	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → 𝑆 ⊆ (Base‘𝑊))
18	7, 8	ressbas2 17215	. . . . . . . . . . 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 3986	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → ran 𝐹 ⊆ (Base‘𝑋))
21	20	adantr 480	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑊)) → ran 𝐹 ⊆ (Base‘𝑋))
22		df-f 6518	. . . . . . . 8 ⊢ (𝐹:dom 𝐹⟶(Base‘𝑋) ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ (Base‘𝑋)))
23	13, 21, 22	sylanbrc 583	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑊)) → 𝐹:dom 𝐹⟶(Base‘𝑋))
24	11, 23	impbida 800	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹:dom 𝐹⟶(Base‘𝑋) ↔ 𝐹:dom 𝐹⟶(Base‘𝑊)))
25	24	adantr 480	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (𝐹:dom 𝐹⟶(Base‘𝑋) ↔ 𝐹:dom 𝐹⟶(Base‘𝑊)))
26		simpl2 1193	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → 𝑆 ∈ 𝑈)
27		eqid 2730	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
28	7, 27	resssca 17313	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝑈 → (Scalar‘𝑊) = (Scalar‘𝑋))
29	28	eqcomd 2736	. . . . . . . . . 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 6865	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
32	30	fveq2d 6865	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
33	32	sneqd 4604	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → {(0g‘(Scalar‘𝑋))} = {(0g‘(Scalar‘𝑊))})
34	31, 33	difeq12d 4093	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → ((Base‘(Scalar‘𝑋)) ∖ {(0g‘(Scalar‘𝑋))}) = ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))
35		eqid 2730	. . . . . . . . . . . . 13 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
36	7, 35	ressvsca 17314	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ 𝑈 → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑋))
37	36	eqcomd 2736	. . . . . . . . . . 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 7407	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (𝑘( ·𝑠 ‘𝑋)(𝐹‘𝑥)) = (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)))
40		simpl1 1192	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → 𝑊 ∈ LMod)
41		imassrn 6045	. . . . . . . . . . 11 ⊢ (𝐹 “ (dom 𝐹 ∖ {𝑥})) ⊆ ran 𝐹
42		simpl3 1194	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → ran 𝐹 ⊆ 𝑆)
43	41, 42	sstrid 3961	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (𝐹 “ (dom 𝐹 ∖ {𝑥})) ⊆ 𝑆)
44		eqid 2730	. . . . . . . . . . 11 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
45		eqid 2730	. . . . . . . . . . 11 ⊢ (LSpan‘𝑋) = (LSpan‘𝑋)
46	7, 44, 45, 15	lsslsp 20928	. . . . . . . . . 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 1373	. . . . . . . . 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 2823	. . . . . . . 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 318	. . . . . . 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 3321	. . . . . 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 3157	. . . . 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 632	. . . 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 7424	. . . . . 6 ⊢ 𝑋 ∈ V
54	53	a1i 11	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → 𝑋 ∈ V)
55		eqid 2730	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
56		eqid 2730	. . . . . 6 ⊢ ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑋)
57		eqid 2730	. . . . . 6 ⊢ (Scalar‘𝑋) = (Scalar‘𝑋)
58		eqid 2730	. . . . . 6 ⊢ (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋))
59		eqid 2730	. . . . . 6 ⊢ (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑋))
60	55, 56, 45, 57, 58, 59	islindf 21728	. . . . 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 580	. . . 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 2730	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
63		eqid 2730	. . . . . 6 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
64	8, 35, 44, 27, 62, 63	islindf 21728	. . . . 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 1186	. . . 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 311	. . 3 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊))
67	66	ex 412	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 ∈ V → (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊)))
68	3, 5, 67	pm5.21ndd 379	1 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   ∖ cdif 3914   ⊆ wss 3917  {csn 4592   class class class wbr 5110  dom cdm 5641  ran crn 5642   “ cima 5644   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Basecbs 17186   ↾_s cress 17207  Scalarcsca 17230   ·_𝑠 cvsca 17231  0_gc0g 17409  LModclmod 20773  LSubSpclss 20844  LSpanclspn 20884   LIndF clindf 21720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-sca 17243  df-vsca 17244  df-0g 17411  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-grp 18875  df-minusg 18876  df-sbg 18877  df-subg 19062  df-mgp 20057  df-ur 20098  df-ring 20151  df-lmod 20775  df-lss 20845  df-lsp 20885  df-lindf 21722

This theorem is referenced by:  lsslinds  21747