Theorem lsslindf 22116

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 22094	. . . 4 ⊢ Rel LIndF
2	1	brrelex1i 5707	. . 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 5707	. . 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 490	. . . . . . . 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 2761	. . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊)
9	7, 8	ressbasss 17397	. . . . . . . 8 ⊢ (Base‘𝑋) ⊆ (Base‘𝑊)
10		fss 6718	. . . . . . . 8 ⊢ ((𝐹:dom 𝐹⟶(Base‘𝑋) ∧ (Base‘𝑋) ⊆ (Base‘𝑊)) → 𝐹:dom 𝐹⟶(Base‘𝑊))
11	6, 9, 10	sylancl 598	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑋)) → 𝐹:dom 𝐹⟶(Base‘𝑊))
12		ffn 6701	. . . . . . . . 9 ⊢ (𝐹:dom 𝐹⟶(Base‘𝑊) → 𝐹 Fn dom 𝐹)
13	12	adantl 487	. . . . . . . 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 21191	. . . . . . . . . . . 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 17396	. . . . . . . . . . 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 3967	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → ran 𝐹 ⊆ (Base‘𝑋))
21	20	adantr 486	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑊)) → ran 𝐹 ⊆ (Base‘𝑋))
22		df-f 6535	. . . . . . . 8 ⊢ (𝐹:dom 𝐹⟶(Base‘𝑋) ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ (Base‘𝑋)))
23	13, 21, 22	sylanbrc 595	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹:dom 𝐹⟶(Base‘𝑊)) → 𝐹:dom 𝐹⟶(Base‘𝑋))
24	11, 23	impbida 813	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹:dom 𝐹⟶(Base‘𝑋) ↔ 𝐹:dom 𝐹⟶(Base‘𝑊)))
25	24	adantr 486	. . . . 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 2761	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
28	7, 27	resssca 17494	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝑈 → (Scalar‘𝑊) = (Scalar‘𝑋))
29	28	eqcomd 2767	. . . . . . . . . 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 6881	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
32	30	fveq2d 6881	. . . . . . . . 9 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
33	32	sneqd 4596	. . . . . . . 8 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → {(0g‘(Scalar‘𝑋))} = {(0g‘(Scalar‘𝑊))})
34	31, 33	difeq12d 4075	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → ((Base‘(Scalar‘𝑋)) ∖ {(0g‘(Scalar‘𝑋))}) = ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))
35		eqid 2761	. . . . . . . . . . . . 13 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
36	7, 35	ressvsca 17495	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ 𝑈 → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑋))
37	36	eqcomd 2767	. . . . . . . . . . 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 6065	. . . . . . . . . . 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 3942	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) ∧ 𝐹 ∈ V) → (𝐹 “ (dom 𝐹 ∖ {𝑥})) ⊆ 𝑆)
44		eqid 2761	. . . . . . . . . . 11 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
45		eqid 2761	. . . . . . . . . . 11 ⊢ (LSpan‘𝑋) = (LSpan‘𝑋)
46	7, 44, 45, 15	lsslsp 21270	. . . . . . . . . 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 2855	. . . . . . . 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 3335	. . . . . 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 3186	. . . . 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 644	. . . 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 2761	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
56		eqid 2761	. . . . . 6 ⊢ ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑋)
57		eqid 2761	. . . . . 6 ⊢ (Scalar‘𝑋) = (Scalar‘𝑋)
58		eqid 2761	. . . . . 6 ⊢ (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋))
59		eqid 2761	. . . . . 6 ⊢ (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑋))
60	55, 56, 45, 57, 58, 59	islindf 22098	. . . . 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 592	. . . 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 2761	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
63		eqid 2761	. . . . . 6 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
64	8, 35, 44, 27, 62, 63	islindf 22098	. . . . 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 418	. 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   ∖ cdif 3896   ⊆ wss 3899  {csn 4584   class class class wbr 5103  dom cdm 5651  ran crn 5652   “ cima 5654   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412  Basecbs 17367   ↾_s cress 17388  Scalarcsca 17411   ·_𝑠 cvsca 17412  0_gc0g 17590  LModclmod 21115  LSubSpclss 21186  LSpanclspn 21226   LIndF clindf 22090

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8701  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-3 12387  df-4 12388  df-5 12389  df-6 12390  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-ress 17389  df-plusg 17421  df-sca 17424  df-vsca 17425  df-0g 17592  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-grp 19127  df-minusg 19128  df-sbg 19129  df-subg 19313  df-mgp 20341  df-ur 20388  df-ring 20441  df-lmod 21117  df-lss 21187  df-lsp 21227  df-lindf 22092

This theorem is used by:  lsslinds  22117