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Theorem lindfrn 20160

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 2622	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
2	1	lindff 20154	. . . 4 ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ LMod) → 𝐹:dom 𝐹⟶(Base‘𝑊))
3	2	ancoms 469	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹⟶(Base‘𝑊))
4		frn 6053	. . 3 ⊢ (𝐹:dom 𝐹⟶(Base‘𝑊) → ran 𝐹 ⊆ (Base‘𝑊))
5	3, 4	syl 17	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ran 𝐹 ⊆ (Base‘𝑊))
6		simpll 790	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → 𝑊 ∈ LMod)
7		imassrn 5477	. . . . . . . . 9 ⊢ (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ ran 𝐹
8	7, 5	syl5ss 3614	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ (Base‘𝑊))
9	8	adantr 481	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ (Base‘𝑊))
10		ffun 6048	. . . . . . . . 9 ⊢ (𝐹:dom 𝐹⟶(Base‘𝑊) → Fun 𝐹)
11	3, 10	syl 17	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → Fun 𝐹)
12		eldifsn 4317	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ran 𝐹 ∖ {(𝐹‘𝑦)}) ↔ (𝑥 ∈ ran 𝐹 ∧ 𝑥 ≠ (𝐹‘𝑦)))
13		funfn 5918	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
14		fvelrnb 6243	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑘 ∈ dom 𝐹(𝐹‘𝑘) = 𝑥))
15	13, 14	sylbi 207	. . . . . . . . . . . . 13 ⊢ (Fun 𝐹 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑘 ∈ dom 𝐹(𝐹‘𝑘) = 𝑥))
16	15	adantr 481	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑥 ∈ ran 𝐹 ↔ ∃𝑘 ∈ dom 𝐹(𝐹‘𝑘) = 𝑥))
17		difss 3737	. . . . . . . . . . . . . . . . . 18 ⊢ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹
18	17	jctr 565	. . . . . . . . . . . . . . . . 17 ⊢ (Fun 𝐹 → (Fun 𝐹 ∧ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹))
19	18	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦))) → (Fun 𝐹 ∧ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹))
20		simpl 473	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦)) → 𝑘 ∈ dom 𝐹)
21		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑦 → (𝐹‘𝑘) = (𝐹‘𝑦))
22	21	necon3i 2826	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑘) ≠ (𝐹‘𝑦) → 𝑘 ≠ 𝑦)
23	22	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦)) → 𝑘 ≠ 𝑦)
24		eldifsn 4317	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (dom 𝐹 ∖ {𝑦}) ↔ (𝑘 ∈ dom 𝐹 ∧ 𝑘 ≠ 𝑦))
25	20, 23, 24	sylanbrc 698	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦)) → 𝑘 ∈ (dom 𝐹 ∖ {𝑦}))
26	25	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦))) → 𝑘 ∈ (dom 𝐹 ∖ {𝑦}))
27		funfvima2 6493	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐹 ∧ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹) → (𝑘 ∈ (dom 𝐹 ∖ {𝑦}) → (𝐹‘𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
28	19, 26, 27	sylc 65	. . . . . . . . . . . . . . 15 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦))) → (𝐹‘𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))
29	28	expr 643	. . . . . . . . . . . . . 14 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹‘𝑘) ≠ (𝐹‘𝑦) → (𝐹‘𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
30		neeq1 2856	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) = 𝑥 → ((𝐹‘𝑘) ≠ (𝐹‘𝑦) ↔ 𝑥 ≠ (𝐹‘𝑦)))
31		eleq1 2689	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) = 𝑥 → ((𝐹‘𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})) ↔ 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
32	30, 31	imbi12d 334	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑘) = 𝑥 → (((𝐹‘𝑘) ≠ (𝐹‘𝑦) → (𝐹‘𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))) ↔ (𝑥 ≠ (𝐹‘𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
33	29, 32	syl5ibcom 235	. . . . . . . . . . . . 13 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹‘𝑘) = 𝑥 → (𝑥 ≠ (𝐹‘𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
34	33	rexlimdva 3031	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (∃𝑘 ∈ dom 𝐹(𝐹‘𝑘) = 𝑥 → (𝑥 ≠ (𝐹‘𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
35	16, 34	sylbid 230	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑥 ∈ ran 𝐹 → (𝑥 ≠ (𝐹‘𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
36	35	impd 447	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → ((𝑥 ∈ ran 𝐹 ∧ 𝑥 ≠ (𝐹‘𝑦)) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
37	12, 36	syl5bi 232	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑥 ∈ (ran 𝐹 ∖ {(𝐹‘𝑦)}) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
38	37	ssrdv 3609	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (ran 𝐹 ∖ {(𝐹‘𝑦)}) ⊆ (𝐹 “ (dom 𝐹 ∖ {𝑦})))
39	11, 38	sylan 488	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → (ran 𝐹 ∖ {(𝐹‘𝑦)}) ⊆ (𝐹 “ (dom 𝐹 ∖ {𝑦})))
40		eqid 2622	. . . . . . . 8 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
41	1, 40	lspss 18984	. . . . . . 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}))))
42	6, 9, 39, 41	syl3anc 1326	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)})) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑦}))))
43	42	adantrr 753	. . . . 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}))))
44		simplr 792	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝐹 LIndF 𝑊)
45		simprl 794	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝑦 ∈ dom 𝐹)
46		eldifi 3732	. . . . . . 7 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
47	46	ad2antll 765	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
48		eldifsni 4320	. . . . . . 7 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) → 𝑘 ≠ (0_g‘(Scalar‘𝑊)))
49	48	ad2antll 765	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝑘 ≠ (0_g‘(Scalar‘𝑊)))
50		eqid 2622	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
51		eqid 2622	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
52		eqid 2622	. . . . . . 7 ⊢ (0_g‘(Scalar‘𝑊)) = (0_g‘(Scalar‘𝑊))
53		eqid 2622	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
54	50, 40, 51, 52, 53	lindfind 20155	. . . . . 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}))))
55	44, 45, 47, 49, 54	syl22anc 1327	. . . . 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}))))
56	43, 55	ssneldd 3606	. . . 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)})))
57	56	ralrimivva 2971	. . 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)})))
58	11, 13	sylib 208	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹 Fn dom 𝐹)
59		oveq2 6658	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑦) → (𝑘( ·_𝑠 ‘𝑊)𝑥) = (𝑘( ·_𝑠 ‘𝑊)(𝐹‘𝑦)))
60		sneq 4187	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘𝑦) → {𝑥} = {(𝐹‘𝑦)})
61	60	difeq2d 3728	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑦) → (ran 𝐹 ∖ {𝑥}) = (ran 𝐹 ∖ {(𝐹‘𝑦)}))
62	61	fveq2d 6195	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑦) → ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) = ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)})))
63	59, 62	eleq12d 2695	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑦) → ((𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) ↔ (𝑘( ·_𝑠 ‘𝑊)(𝐹‘𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)}))))
64	63	notbid 308	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑦) → (¬ (𝑘( ·_𝑠 ‘𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) ↔ ¬ (𝑘( ·_𝑠 ‘𝑊)(𝐹‘𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)}))))
65	64	ralbidv 2986	. . . . 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)}))))
66	65	ralrn 6362	. . . 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)}))))
67	58, 66	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)}))))
68	57, 67	mpbird 247	. 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})))
69	1, 50, 40, 51, 53, 52	islinds2 20152	. . 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})))))
70	69	adantr 481	. 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})))))
71	5, 68, 70	mpbir2and 957	1 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ran 𝐹 ∈ (LIndS‘𝑊))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ∖ cdif 3571 ⊆ wss 3574 {csn 4177 class class class wbr 4653 dom cdm 5114 ran crn 5115 “ cima 5117 Fun wfun 5882 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 Scalarcsca 15944 ·_𝑠 cvsca 15945 0_gc0g 16100 LModclmod 18863 LSpanclspn 18971 LIndF clindf 20143 LIndSclinds 20144

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-slot 15861 df-base 15863 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lindf 20145 df-linds 20146

This theorem is referenced by: islindf3 20165 lindsmm 20167 matunitlindflem2 33406

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