Theorem lsslindf 21797

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 21775	. . . 4 ⊢ Rel LIndF
2	1	brrelex1i 5688	. . 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 5688	. . 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 2737	. . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊)
9	7, 8	ressbasss 17178	. . . . . . . 8 ⊢ (Base‘𝑋) ⊆ (Base‘𝑊)
10		fss 6686	. . . . . . . 8 ⊢ ((𝐹:dom 𝐹⟶(Base‘𝑋) ∧ (Base‘𝑋) ⊆ (Base‘𝑊)) → 𝐹:dom 𝐹⟶(Base‘𝑊))
11	6, 9, 10	sylancl 587	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑋)) → 𝐹:dom 𝐹⟶(Base‘𝑊))
12		ffn 6670	. . . . . . . . 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 1139	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → ran 𝐹 ⊆ 𝑆)
15		lsslindf.u	. . . . . . . . . . . . 13 ⊢ 𝑈 = (LSubSp‘𝑊)
16	8, 15	lssss 20899	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ 𝑈 → 𝑆 ⊆ (Base‘𝑊))
17	16	3ad2ant2 1135	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → 𝑆 ⊆ (Base‘𝑊))
18	7, 8	ressbas2 17177	. . . . . . . . . . 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 3972	. . . . . . . . 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 6504	. . . . . . . 8 ⊢ (𝐹:dom 𝐹⟶(Base‘𝑋) ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ (Base‘𝑋)))
23	13, 21, 22	sylanbrc 584	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑊)) → 𝐹:dom 𝐹⟶(Base‘𝑋))
24	11, 23	impbida 801	. . . . . 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 1194	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → 𝑆 ∈ 𝑈)
27		eqid 2737	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
28	7, 27	resssca 17275	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝑈 → (Scalar‘𝑊) = (Scalar‘𝑋))
29	28	eqcomd 2743	. . . . . . . . . 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 6846	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
32	30	fveq2d 6846	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
33	32	sneqd 4594	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → {(0g‘(Scalar‘𝑋))} = {(0g‘(Scalar‘𝑊))})
34	31, 33	difeq12d 4081	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → ((Base‘(Scalar‘𝑋)) ∖ {(0g‘(Scalar‘𝑋))}) = ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))
35		eqid 2737	. . . . . . . . . . . . 13 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
36	7, 35	ressvsca 17276	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ 𝑈 → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑋))
37	36	eqcomd 2743	. . . . . . . . . . 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 7385	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (𝑘( ·𝑠 ‘𝑋)(𝐹‘𝑥)) = (𝑘( ·𝑠 ‘𝑊)(𝐹‘𝑥)))
40		simpl1 1193	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → 𝑊 ∈ LMod)
41		imassrn 6038	. . . . . . . . . . 11 ⊢ (𝐹 “ (dom 𝐹 ∖ {𝑥})) ⊆ ran 𝐹
42		simpl3 1195	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → ran 𝐹 ⊆ 𝑆)
43	41, 42	sstrid 3947	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (𝐹 “ (dom 𝐹 ∖ {𝑥})) ⊆ 𝑆)
44		eqid 2737	. . . . . . . . . . 11 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
45		eqid 2737	. . . . . . . . . . 11 ⊢ (LSpan‘𝑋) = (LSpan‘𝑋)
46	7, 44, 45, 15	lsslsp 20978	. . . . . . . . . 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 1374	. . . . . . . . 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 2831	. . . . . . . 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 3318	. . . . . 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 3161	. . . . 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 633	. . . 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 7402	. . . . . 6 ⊢ 𝑋 ∈ V
54	53	a1i 11	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → 𝑋 ∈ V)
55		eqid 2737	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
56		eqid 2737	. . . . . 6 ⊢ ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑋)
57		eqid 2737	. . . . . 6 ⊢ (Scalar‘𝑋) = (Scalar‘𝑋)
58		eqid 2737	. . . . . 6 ⊢ (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋))
59		eqid 2737	. . . . . 6 ⊢ (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑋))
60	55, 56, 45, 57, 58, 59	islindf 21779	. . . . 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 581	. . . 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 2737	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
63		eqid 2737	. . . . . 6 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
64	8, 35, 44, 27, 62, 63	islindf 21779	. . . . 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 1187	. . . 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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3442   ∖ cdif 3900   ⊆ wss 3903  {csn 4582   class class class wbr 5100  dom cdm 5632  ran crn 5633   “ cima 5635   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  Basecbs 17148   ↾_s cress 17169  Scalarcsca 17192   ·_𝑠 cvsca 17193  0_gc0g 17371  LModclmod 20823  LSubSpclss 20894  LSpanclspn 20934   LIndF clindf 21771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-sca 17205  df-vsca 17206  df-0g 17373  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-grp 18878  df-minusg 18879  df-sbg 18880  df-subg 19065  df-mgp 20088  df-ur 20129  df-ring 20182  df-lmod 20825  df-lss 20895  df-lsp 20935  df-lindf 21773

This theorem is referenced by:  lsslinds  21798