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Theorem lindfrn 21758

Description: The range of an independent family is an independent set. (Contributed by Stefan O'Rear, 24-Feb-2015.)

**Assertion**
Ref	Expression
lindfrn	⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ran 𝐹 ∈ (LIndS‘𝑊))

Proof of Theorem lindfrn Dummy variables 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2731	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
2	1	lindff 21752	. . . 4 ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ LMod) → 𝐹:dom 𝐹⟶(Base‘𝑊))
3	2	ancoms 458	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹⟶(Base‘𝑊))
4	3	frnd 6659	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ran 𝐹 ⊆ (Base‘𝑊))
5		simpll 766	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → 𝑊 ∈ LMod)
6		imassrn 6019	. . . . . . . . 9 ⊢ (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ ran 𝐹
7	6, 4	sstrid 3941	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ (Base‘𝑊))
8	7	adantr 480	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ (Base‘𝑊))
9	3	ffund 6655	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → Fun 𝐹)
10		eldifsn 4735	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ran 𝐹 ∖ {(𝐹‘𝑦)}) ↔ (𝑥 ∈ ran 𝐹 ∧ 𝑥 ≠ (𝐹‘𝑦)))
11		funfn 6511	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
12		fvelrnb 6882	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑘 ∈ dom 𝐹(𝐹‘𝑘) = 𝑥))
13	11, 12	sylbi 217	. . . . . . . . . . . . 13 ⊢ (Fun 𝐹 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑘 ∈ dom 𝐹(𝐹‘𝑘) = 𝑥))
14	13	adantr 480	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑥 ∈ ran 𝐹 ↔ ∃𝑘 ∈ dom 𝐹(𝐹‘𝑘) = 𝑥))
15		difss 4083	. . . . . . . . . . . . . . . . . 18 ⊢ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹
16	15	jctr 524	. . . . . . . . . . . . . . . . 17 ⊢ (Fun 𝐹 → (Fun 𝐹 ∧ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹))
17	16	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦))) → (Fun 𝐹 ∧ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹))
18		simpl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦)) → 𝑘 ∈ dom 𝐹)
19		fveq2 6822	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑦 → (𝐹‘𝑘) = (𝐹‘𝑦))
20	19	necon3i 2960	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑘) ≠ (𝐹‘𝑦) → 𝑘 ≠ 𝑦)
21	20	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦)) → 𝑘 ≠ 𝑦)
22		eldifsn 4735	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (dom 𝐹 ∖ {𝑦}) ↔ (𝑘 ∈ dom 𝐹 ∧ 𝑘 ≠ 𝑦))
23	18, 21, 22	sylanbrc 583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦)) → 𝑘 ∈ (dom 𝐹 ∖ {𝑦}))
24	23	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦))) → 𝑘 ∈ (dom 𝐹 ∖ {𝑦}))
25		funfvima2 7165	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐹 ∧ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹) → (𝑘 ∈ (dom 𝐹 ∖ {𝑦}) → (𝐹‘𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
26	17, 24, 25	sylc 65	. . . . . . . . . . . . . . 15 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦))) → (𝐹‘𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))
27	26	expr 456	. . . . . . . . . . . . . 14 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹‘𝑘) ≠ (𝐹‘𝑦) → (𝐹‘𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
28		neeq1 2990	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) = 𝑥 → ((𝐹‘𝑘) ≠ (𝐹‘𝑦) ↔ 𝑥 ≠ (𝐹‘𝑦)))
29		eleq1 2819	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) = 𝑥 → ((𝐹‘𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})) ↔ 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
30	28, 29	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘) = 𝑥 → (((𝐹‘𝑘) ≠ (𝐹‘𝑦) → (𝐹‘𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))) ↔ (𝑥 ≠ (𝐹‘𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
31	27, 30	syl5ibcom 245	. . . . . . . . . . . . 13 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹‘𝑘) = 𝑥 → (𝑥 ≠ (𝐹‘𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
32	31	rexlimdva 3133	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (∃𝑘 ∈ dom 𝐹(𝐹‘𝑘) = 𝑥 → (𝑥 ≠ (𝐹‘𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
33	14, 32	sylbid 240	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑥 ∈ ran 𝐹 → (𝑥 ≠ (𝐹‘𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
34	33	impd 410	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → ((𝑥 ∈ ran 𝐹 ∧ 𝑥 ≠ (𝐹‘𝑦)) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
35	10, 34	biimtrid 242	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑥 ∈ (ran 𝐹 ∖ {(𝐹‘𝑦)}) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
36	35	ssrdv 3935	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (ran 𝐹 ∖ {(𝐹‘𝑦)}) ⊆ (𝐹 “ (dom 𝐹 ∖ {𝑦})))
37	9, 36	sylan 580	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → (ran 𝐹 ∖ {(𝐹‘𝑦)}) ⊆ (𝐹 “ (dom 𝐹 ∖ {𝑦})))
38		eqid 2731	. . . . . . . 8 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
39	1, 38	lspss 20917	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ (Base‘𝑊) ∧ (ran 𝐹 ∖ {(𝐹‘𝑦)}) ⊆ (𝐹 “ (dom 𝐹 ∖ {𝑦}))) → ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)})) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑦}))))
40	5, 8, 37, 39	syl3anc 1373	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)})) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑦}))))
41	40	adantrr 717	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)})) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑦}))))
42		simplr 768	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝐹 LIndF 𝑊)
43		simprl 770	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝑦 ∈ dom 𝐹)
44		eldifi 4078	. . . . . . 7 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
45	44	ad2antll 729	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
46		eldifsni 4739	. . . . . . 7 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) → 𝑘 ≠ (0_g‘(Scalar‘𝑊)))
47	46	ad2antll 729	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝑘 ≠ (0_g‘(Scalar‘𝑊)))
48		eqid 2731	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
49		eqid 2731	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
50		eqid 2731	. . . . . . 7 ⊢ (0_g‘(Scalar‘𝑊)) = (0_g‘(Scalar‘𝑊))
51		eqid 2731	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
52	48, 38, 49, 50, 51	lindfind 21753	. . . . . 6 ⊢ (((𝐹 LIndF 𝑊 ∧ 𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0_g‘(Scalar‘𝑊)))) → ¬ (𝑘( ·_𝑠 ‘𝑊)(𝐹‘𝑦)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑦}))))
53	42, 43, 45, 47, 52	syl22anc 838	. . . . 5 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → ¬ (𝑘( ·_𝑠 ‘𝑊)(𝐹‘𝑦)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑦}))))
54	41, 53	ssneldd 3932	. . . 4 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → ¬ (𝑘( ·_𝑠 ‘𝑊)(𝐹‘𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)})))
55	54	ralrimivva 3175	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ∀𝑦 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)(𝐹‘𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)})))
56	9	funfnd 6512	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹 Fn dom 𝐹)
57		oveq2 7354	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑦) → (𝑘( ·_𝑠 ‘𝑊)𝑥) = (𝑘( ·_𝑠 ‘𝑊)(𝐹‘𝑦)))
58		sneq 4583	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘𝑦) → {𝑥} = {(𝐹‘𝑦)})
59	58	difeq2d 4073	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑦) → (ran 𝐹 ∖ {𝑥}) = (ran 𝐹 ∖ {(𝐹‘𝑦)}))
60	59	fveq2d 6826	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑦) → ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) = ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)})))
61	57, 60	eleq12d 2825	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑦) → ((𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) ↔ (𝑘( ·_𝑠 ‘𝑊)(𝐹‘𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)}))))
62	61	notbid 318	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑦) → (¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) ↔ ¬ (𝑘( ·_𝑠 ‘𝑊)(𝐹‘𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)}))))
63	62	ralbidv 3155	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑦) → (∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)(𝐹‘𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)}))))
64	63	ralrn 7021	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (∀𝑥 ∈ ran 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) ↔ ∀𝑦 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)(𝐹‘𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)}))))
65	56, 64	syl 17	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → (∀𝑥 ∈ ran 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) ↔ ∀𝑦 ∈ dom 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)(𝐹‘𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)}))))
66	55, 65	mpbird 257	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ∀𝑥 ∈ ran 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})))
67	1, 48, 38, 49, 51, 50	islinds2 21750	. . 3 ⊢ (𝑊 ∈ LMod → (ran 𝐹 ∈ (LIndS‘𝑊) ↔ (ran 𝐹 ⊆ (Base‘𝑊) ∧ ∀𝑥 ∈ ran 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})))))
68	67	adantr 480	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → (ran 𝐹 ∈ (LIndS‘𝑊) ↔ (ran 𝐹 ⊆ (Base‘𝑊) ∧ ∀𝑥 ∈ ran 𝐹∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) ¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})))))
69	4, 66, 68	mpbir2and 713	1 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ran 𝐹 ∈ (LIndS‘𝑊))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 ∖ cdif 3894 ⊆ wss 3897 {csn 4573 class class class wbr 5089 dom cdm 5614 ran crn 5615 “ cima 5617 Fun wfun 6475 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 Basecbs 17120 Scalarcsca 17164 ·_𝑠 cvsca 17165 0_gc0g 17343 LModclmod 20793 LSpanclspn 20904 LIndF clindf 21741 LIndSclinds 21742

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-1cn 11064 ax-addcl 11066

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-om 7797 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-nn 12126 df-slot 17093 df-ndx 17105 df-base 17121 df-0g 17345 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-grp 18849 df-lmod 20795 df-lss 20865 df-lsp 20905 df-lindf 21743 df-linds 21744

This theorem is referenced by: islindf3 21763 lindsmm 21765 matunitlindflem2 37667

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