lindfrn - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lindfrn		Structured version Visualization version GIF version

Theorem lindfrn 20957

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 2819	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
2	1	lindff 20951	. . . 4 ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ LMod) → 𝐹:dom 𝐹⟶(Base‘𝑊))
3	2	ancoms 461	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹⟶(Base‘𝑊))
4	3	frnd 6514	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ran 𝐹 ⊆ (Base‘𝑊))
5		simpll 765	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → 𝑊 ∈ LMod)
6		imassrn 5933	. . . . . . . . 9 ⊢ (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ ran 𝐹
7	6, 4	sstrid 3976	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ (Base‘𝑊))
8	7	adantr 483	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ (Base‘𝑊))
9	3	ffund 6511	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → Fun 𝐹)
10		eldifsn 4711	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ran 𝐹 ∖ {(𝐹‘𝑦)}) ↔ (𝑥 ∈ ran 𝐹 ∧ 𝑥 ≠ (𝐹‘𝑦)))
11		funfn 6378	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
12		fvelrnb 6719	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑘 ∈ dom 𝐹(𝐹‘𝑘) = 𝑥))
13	11, 12	sylbi 219	. . . . . . . . . . . . 13 ⊢ (Fun 𝐹 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑘 ∈ dom 𝐹(𝐹‘𝑘) = 𝑥))
14	13	adantr 483	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑥 ∈ ran 𝐹 ↔ ∃𝑘 ∈ dom 𝐹(𝐹‘𝑘) = 𝑥))
15		difss 4106	. . . . . . . . . . . . . . . . . 18 ⊢ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹
16	15	jctr 527	. . . . . . . . . . . . . . . . 17 ⊢ (Fun 𝐹 → (Fun 𝐹 ∧ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹))
17	16	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦))) → (Fun 𝐹 ∧ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹))
18		simpl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦)) → 𝑘 ∈ dom 𝐹)
19		fveq2 6663	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑦 → (𝐹‘𝑘) = (𝐹‘𝑦))
20	19	necon3i 3046	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑘) ≠ (𝐹‘𝑦) → 𝑘 ≠ 𝑦)
21	20	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦)) → 𝑘 ≠ 𝑦)
22		eldifsn 4711	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (dom 𝐹 ∖ {𝑦}) ↔ (𝑘 ∈ dom 𝐹 ∧ 𝑘 ≠ 𝑦))
23	18, 21, 22	sylanbrc 585	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦)) → 𝑘 ∈ (dom 𝐹 ∖ {𝑦}))
24	23	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦))) → 𝑘 ∈ (dom 𝐹 ∖ {𝑦}))
25		funfvima2 6985	. . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . 14 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹‘𝑘) ≠ (𝐹‘𝑦) → (𝐹‘𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
28		neeq1 3076	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) = 𝑥 → ((𝐹‘𝑘) ≠ (𝐹‘𝑦) ↔ 𝑥 ≠ (𝐹‘𝑦)))
29		eleq1 2898	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) = 𝑥 → ((𝐹‘𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})) ↔ 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
30	28, 29	imbi12d 347	. . . . . . . . . . . . . 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 247	. . . . . . . . . . . . 13 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹‘𝑘) = 𝑥 → (𝑥 ≠ (𝐹‘𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
32	31	rexlimdva 3282	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (∃𝑘 ∈ dom 𝐹(𝐹‘𝑘) = 𝑥 → (𝑥 ≠ (𝐹‘𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
33	14, 32	sylbid 242	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑥 ∈ ran 𝐹 → (𝑥 ≠ (𝐹‘𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
34	33	impd 413	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → ((𝑥 ∈ ran 𝐹 ∧ 𝑥 ≠ (𝐹‘𝑦)) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
35	10, 34	syl5bi 244	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑥 ∈ (ran 𝐹 ∖ {(𝐹‘𝑦)}) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
36	35	ssrdv 3971	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (ran 𝐹 ∖ {(𝐹‘𝑦)}) ⊆ (𝐹 “ (dom 𝐹 ∖ {𝑦})))
37	9, 36	sylan 582	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → (ran 𝐹 ∖ {(𝐹‘𝑦)}) ⊆ (𝐹 “ (dom 𝐹 ∖ {𝑦})))
38		eqid 2819	. . . . . . . 8 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
39	1, 38	lspss 19748	. . . . . . 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 1366	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)})) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑦}))))
41	40	adantrr 715	. . . . 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 767	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝐹 LIndF 𝑊)
43		simprl 769	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝑦 ∈ dom 𝐹)
44		eldifi 4101	. . . . . . 7 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
45	44	ad2antll 727	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
46		eldifsni 4714	. . . . . . 7 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) → 𝑘 ≠ (0_g‘(Scalar‘𝑊)))
47	46	ad2antll 727	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝑘 ≠ (0_g‘(Scalar‘𝑊)))
48		eqid 2819	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
49		eqid 2819	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
50		eqid 2819	. . . . . . 7 ⊢ (0_g‘(Scalar‘𝑊)) = (0_g‘(Scalar‘𝑊))
51		eqid 2819	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
52	48, 38, 49, 50, 51	lindfind 20952	. . . . . 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 836	. . . . 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 3968	. . . 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 3189	. . 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 6379	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹 Fn dom 𝐹)
57		oveq2 7156	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑦) → (𝑘( ·_𝑠 ‘𝑊)𝑥) = (𝑘( ·_𝑠 ‘𝑊)(𝐹‘𝑦)))
58		sneq 4569	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘𝑦) → {𝑥} = {(𝐹‘𝑦)})
59	58	difeq2d 4097	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑦) → (ran 𝐹 ∖ {𝑥}) = (ran 𝐹 ∖ {(𝐹‘𝑦)}))
60	59	fveq2d 6667	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑦) → ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) = ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)})))
61	57, 60	eleq12d 2905	. . . . . . 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 320	. . . . . 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 3195	. . . . 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 6847	. . . 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 259	. 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 20949	. . 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 483	. 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 711	1 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ran 𝐹 ∈ (LIndS‘𝑊))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ∀wral 3136 ∃wrex 3137 ∖ cdif 3931 ⊆ wss 3934 {csn 4559 class class class wbr 5057 dom cdm 5548 ran crn 5549 “ cima 5551 Fun wfun 6342 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 Scalarcsca 16560 ·_𝑠 cvsca 16561 0_gc0g 16705 LModclmod 19626 LSpanclspn 19735 LIndF clindf 20940 LIndSclinds 20941

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-slot 16479 df-base 16481 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-lmod 19628 df-lss 19696 df-lsp 19736 df-lindf 20942 df-linds 20943

This theorem is referenced by: islindf3 20962 lindsmm 20964 matunitlindflem2 34881

W3C validator