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Theorem lindfrn 21873

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 2762	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
2	1	lindff 21867	. . . 4 ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ LMod) → 𝐹:dom 𝐹⟶(Base‘𝑊))
3	2	ancoms 462	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹⟶(Base‘𝑊))
4	3	frnd 6700	. 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ran 𝐹 ⊆ (Base‘𝑊))
5		simpll 776	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → 𝑊 ∈ LMod)
6		imassrn 6060	. . . . . . . . 9 ⊢ (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ ran 𝐹
7	6, 4	sstrid 3947	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ (Base‘𝑊))
8	7	adantr 484	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ (Base‘𝑊))
9	3	ffund 6696	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → Fun 𝐹)
10		eldifsn 4746	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ran 𝐹 ∖ {(𝐹‘𝑦)}) ↔ (𝑥 ∈ ran 𝐹 ∧ 𝑥 ≠ (𝐹‘𝑦)))
11		funfn 6551	. . . . . . . . . . . . . 14 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
12		fvelrnb 6927	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑥 ∈ ran 𝐹 ↔ ∃𝑘 ∈ dom 𝐹(𝐹‘𝑘) = 𝑥))
15		difss 4089	. . . . . . . . . . . . . . . . . 18 ⊢ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹
16	15	jctr 532	. . . . . . . . . . . . . . . . 17 ⊢ (Fun 𝐹 → (Fun 𝐹 ∧ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹))
17	16	ad2antrr 736	. . . . . . . . . . . . . . . 16 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦))) → (Fun 𝐹 ∧ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹))
18		simpl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦)) → 𝑘 ∈ dom 𝐹)
19		fveq2 6867	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑦 → (𝐹‘𝑘) = (𝐹‘𝑦))
20	19	necon3i 2989	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑘) ≠ (𝐹‘𝑦) → 𝑘 ≠ 𝑦)
21	20	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦)) → 𝑘 ≠ 𝑦)
22		eldifsn 4746	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (dom 𝐹 ∖ {𝑦}) ↔ (𝑘 ∈ dom 𝐹 ∧ 𝑘 ≠ 𝑦))
23	18, 21, 22	sylanbrc 592	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦)) → 𝑘 ∈ (dom 𝐹 ∖ {𝑦}))
24	23	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ≠ (𝐹‘𝑦))) → 𝑘 ∈ (dom 𝐹 ∖ {𝑦}))
25		funfvima2 7215	. . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . 14 ⊢ (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹‘𝑘) ≠ (𝐹‘𝑦) → (𝐹‘𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
28		neeq1 3019	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) = 𝑥 → ((𝐹‘𝑘) ≠ (𝐹‘𝑦) ↔ 𝑥 ≠ (𝐹‘𝑦)))
29		eleq1 2850	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) = 𝑥 → ((𝐹‘𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})) ↔ 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
30	28, 29	imbi12d 346	. . . . . . . . . . . . . 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 3163	. . . . . . . . . . . 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 414	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → ((𝑥 ∈ ran 𝐹 ∧ 𝑥 ≠ (𝐹‘𝑦)) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
35	10, 34	biimtrid 244	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑥 ∈ (ran 𝐹 ∖ {(𝐹‘𝑦)}) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
36	35	ssrdv 3942	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (ran 𝐹 ∖ {(𝐹‘𝑦)}) ⊆ (𝐹 “ (dom 𝐹 ∖ {𝑦})))
37	9, 36	sylan 589	. . . . . . 7 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → (ran 𝐹 ∖ {(𝐹‘𝑦)}) ⊆ (𝐹 “ (dom 𝐹 ∖ {𝑦})))
38		eqid 2762	. . . . . . . 8 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
39	1, 38	lspss 21051	. . . . . . 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 1390	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)})) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑦}))))
41	40	adantrr 727	. . . . 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 778	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝐹 LIndF 𝑊)
43		simprl 780	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝑦 ∈ dom 𝐹)
44		eldifi 4084	. . . . . . 7 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
45	44	ad2antll 739	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
46		eldifsni 4750	. . . . . . 7 ⊢ (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}) → 𝑘 ≠ (0_g‘(Scalar‘𝑊)))
47	46	ad2antll 739	. . . . . 6 ⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹 ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0_g‘(Scalar‘𝑊))}))) → 𝑘 ≠ (0_g‘(Scalar‘𝑊)))
48		eqid 2762	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
49		eqid 2762	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
50		eqid 2762	. . . . . . 7 ⊢ (0_g‘(Scalar‘𝑊)) = (0_g‘(Scalar‘𝑊))
51		eqid 2762	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
52	48, 38, 49, 50, 51	lindfind 21868	. . . . . 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 849	. . . . 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 3939	. . . 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 3205	. . 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 6552	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹 Fn dom 𝐹)
57		oveq2 7404	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑦) → (𝑘( ·_𝑠 ‘𝑊)𝑥) = (𝑘( ·_𝑠 ‘𝑊)(𝐹‘𝑦)))
58		sneq 4592	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘𝑦) → {𝑥} = {(𝐹‘𝑦)})
59	58	difeq2d 4080	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑦) → (ran 𝐹 ∖ {𝑥}) = (ran 𝐹 ∖ {(𝐹‘𝑦)}))
60	59	fveq2d 6871	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑦) → ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) = ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹‘𝑦)})))
61	57, 60	eleq12d 2856	. . . . . . 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 3185	. . . . 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 7069	. . . 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 21865	. . 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 484	. 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 723	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 399 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 ∀wral 3076 ∃wrex 3086 ∖ cdif 3901 ⊆ wss 3904 {csn 4582 class class class wbr 5100 dom cdm 5647 ran crn 5648 “ cima 5650 Fun wfun 6515 Fn wfn 6516 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 Basecbs 17245 Scalarcsca 17289 ·_𝑠 cvsca 17290 0_gc0g 17468 LModclmod 20927 LSpanclspn 21038 LIndF clindf 21856 LIndSclinds 21857

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-1cn 11131 ax-addcl 11133

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-om 7847 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-nn 12211 df-slot 17218 df-ndx 17230 df-base 17246 df-0g 17470 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-grp 18978 df-lmod 20929 df-lss 20999 df-lsp 21039 df-lindf 21858 df-linds 21859

This theorem is referenced by: islindf3 21878 lindsmm 21880 matunitlindflem2 38116

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