Theorem islbs 19076

Description: The predicate "𝐵 is a basis for the left module or vector space 𝑊". A subset of the base set is a basis if zero is not in the set, it spans the set, and no nonzero multiple of an element of the basis is in the span of the rest of the family. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 14-Jan-2015.)

Hypotheses

Ref	Expression
islbs.v	⊢ 𝑉 = (Base‘𝑊)
islbs.f	⊢ 𝐹 = (Scalar‘𝑊)
islbs.s	⊢ · = ( ·𝑠 ‘𝑊)
islbs.k	⊢ 𝐾 = (Base‘𝐹)
islbs.j	⊢ 𝐽 = (LBasis‘𝑊)
islbs.n	⊢ 𝑁 = (LSpan‘𝑊)
islbs.z	⊢ 0 = (0g‘𝐹)

Assertion

Ref	Expression
islbs	⊢ (𝑊 ∈ 𝑋 → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑦,𝐾   𝑥,𝑁,𝑦   𝑥,𝑊,𝑦   𝑥,𝐹,𝑦   𝑦, 0

Allowed substitution hints:   · (𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑥)   𝑉(𝑥,𝑦)   𝑋(𝑥,𝑦)   0 (𝑥)

Proof of Theorem islbs

Dummy variables 𝑏 𝑓 𝑛 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. . . 4 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2		islbs.j	. . . . 5 ⊢ 𝐽 = (LBasis‘𝑊)
3		fveq2 6191	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4		islbs.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6	5	pweqd 4163	. . . . . . 7 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7		fvexd 6203	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) ∈ V)
8		fveq2 6191	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
9		islbs.n	. . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑊)
10	8, 9	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
11		fvexd 6203	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (Scalar‘𝑤) ∈ V)
12		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
13	12	adantr 481	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (Scalar‘𝑤) = (Scalar‘𝑊))
14		islbs.f	. . . . . . . . . 10 ⊢ 𝐹 = (Scalar‘𝑊)
15	13, 14	syl6eqr 2674	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (Scalar‘𝑤) = 𝐹)
16		simplr 792	. . . . . . . . . . . 12 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → 𝑛 = 𝑁)
17	16	fveq1d 6193	. . . . . . . . . . 11 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑛‘𝑏) = (𝑁‘𝑏))
18	5	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑤) = 𝑉)
19	17, 18	eqeq12d 2637	. . . . . . . . . 10 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((𝑛‘𝑏) = (Base‘𝑤) ↔ (𝑁‘𝑏) = 𝑉))
20		simpr 477	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
21	20	fveq2d 6195	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
22		islbs.k	. . . . . . . . . . . . . 14 ⊢ 𝐾 = (Base‘𝐹)
23	21, 22	syl6eqr 2674	. . . . . . . . . . . . 13 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑓) = 𝐾)
24	20	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (0g‘𝑓) = (0g‘𝐹))
25		islbs.z	. . . . . . . . . . . . . . 15 ⊢ 0 = (0g‘𝐹)
26	24, 25	syl6eqr 2674	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (0g‘𝑓) = 0 )
27	26	sneqd 4189	. . . . . . . . . . . . 13 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → {(0g‘𝑓)} = { 0 })
28	23, 27	difeq12d 3729	. . . . . . . . . . . 12 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((Base‘𝑓) ∖ {(0g‘𝑓)}) = (𝐾 ∖ { 0 }))
29		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
30		islbs.s	. . . . . . . . . . . . . . . . 17 ⊢ · = ( ·𝑠 ‘𝑊)
31	29, 30	syl6eqr 2674	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
32	31	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ( ·𝑠 ‘𝑤) = · )
33	32	oveqd 6667	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑦( ·𝑠 ‘𝑤)𝑥) = (𝑦 · 𝑥))
34	16	fveq1d 6193	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑛‘(𝑏 ∖ {𝑥})) = (𝑁‘(𝑏 ∖ {𝑥})))
35	33, 34	eleq12d 2695	. . . . . . . . . . . . 13 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((𝑦( ·𝑠 ‘𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})) ↔ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))))
36	35	notbid 308	. . . . . . . . . . . 12 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (¬ (𝑦( ·𝑠 ‘𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})) ↔ ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))))
37	28, 36	raleqbidv 3152	. . . . . . . . . . 11 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (∀𝑦 ∈ ((Base‘𝑓) ∖ {(0g‘𝑓)}) ¬ (𝑦( ·𝑠 ‘𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})) ↔ ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))))
38	37	ralbidv 2986	. . . . . . . . . 10 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ ((Base‘𝑓) ∖ {(0g‘𝑓)}) ¬ (𝑦( ·𝑠 ‘𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))))
39	19, 38	anbi12d 747	. . . . . . . . 9 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (((𝑛‘𝑏) = (Base‘𝑤) ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ ((Base‘𝑓) ∖ {(0g‘𝑓)}) ¬ (𝑦( ·𝑠 ‘𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥}))) ↔ ((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))))
40	11, 15, 39	sbcied2 3473	. . . . . . . 8 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → ([(Scalar‘𝑤) / 𝑓]((𝑛‘𝑏) = (Base‘𝑤) ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ ((Base‘𝑓) ∖ {(0g‘𝑓)}) ¬ (𝑦( ·𝑠 ‘𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥}))) ↔ ((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))))
41	7, 10, 40	sbcied2 3473	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ([(LSpan‘𝑤) / 𝑛][(Scalar‘𝑤) / 𝑓]((𝑛‘𝑏) = (Base‘𝑤) ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ ((Base‘𝑓) ∖ {(0g‘𝑓)}) ¬ (𝑦( ·𝑠 ‘𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥}))) ↔ ((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))))
42	6, 41	rabeqbidv 3195	. . . . . 6 ⊢ (𝑤 = 𝑊 → {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ [(LSpan‘𝑤) / 𝑛][(Scalar‘𝑤) / 𝑓]((𝑛‘𝑏) = (Base‘𝑤) ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ ((Base‘𝑓) ∖ {(0g‘𝑓)}) ¬ (𝑦( ·𝑠 ‘𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})))} = {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))})
43		df-lbs 19075	. . . . . 6 ⊢ LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ [(LSpan‘𝑤) / 𝑛][(Scalar‘𝑤) / 𝑓]((𝑛‘𝑏) = (Base‘𝑤) ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ ((Base‘𝑓) ∖ {(0g‘𝑓)}) ¬ (𝑦( ·𝑠 ‘𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})))})
44		fvex 6201	. . . . . . . . 9 ⊢ (Base‘𝑊) ∈ V
45	4, 44	eqeltri 2697	. . . . . . . 8 ⊢ 𝑉 ∈ V
46	45	pwex 4848	. . . . . . 7 ⊢ 𝒫 𝑉 ∈ V
47	46	rabex 4813	. . . . . 6 ⊢ {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))} ∈ V
48	42, 43, 47	fvmpt 6282	. . . . 5 ⊢ (𝑊 ∈ V → (LBasis‘𝑊) = {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))})
49	2, 48	syl5eq 2668	. . . 4 ⊢ (𝑊 ∈ V → 𝐽 = {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))})
50	1, 49	syl 17	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐽 = {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))})
51	50	eleq2d 2687	. 2 ⊢ (𝑊 ∈ 𝑋 → (𝐵 ∈ 𝐽 ↔ 𝐵 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))}))
52	45	elpw2 4828	. . . 4 ⊢ (𝐵 ∈ 𝒫 𝑉 ↔ 𝐵 ⊆ 𝑉)
53	52	anbi1i 731	. . 3 ⊢ ((𝐵 ∈ 𝒫 𝑉 ∧ ((𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))) ↔ (𝐵 ⊆ 𝑉 ∧ ((𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
54		fveq2 6191	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑁‘𝑏) = (𝑁‘𝐵))
55	54	eqeq1d 2624	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝑁‘𝑏) = 𝑉 ↔ (𝑁‘𝐵) = 𝑉))
56		difeq1 3721	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑏 ∖ {𝑥}) = (𝐵 ∖ {𝑥}))
57	56	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑁‘(𝑏 ∖ {𝑥})) = (𝑁‘(𝐵 ∖ {𝑥})))
58	57	eleq2d 2687	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → ((𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
59	58	notbid 308	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
60	59	ralbidv 2986	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
61	60	raleqbi1dv 3146	. . . . 5 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
62	55, 61	anbi12d 747	. . . 4 ⊢ (𝑏 = 𝐵 → (((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))) ↔ ((𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
63	62	elrab 3363	. . 3 ⊢ (𝐵 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))} ↔ (𝐵 ∈ 𝒫 𝑉 ∧ ((𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
64		3anass 1042	. . 3 ⊢ ((𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))) ↔ (𝐵 ⊆ 𝑉 ∧ ((𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))
65	53, 63, 64	3bitr4i 292	. 2 ⊢ (𝐵 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))} ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
66	51, 65	syl6bb 276	1 ⊢ (𝑊 ∈ 𝑋 → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200  [wsbc 3435   ∖ cdif 3571   ⊆ wss 3574  𝒫 cpw 4158  {csn 4177  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  Scalarcsca 15944   ·_𝑠 cvsca 15945  0_gc0g 16100  LSpanclspn 18971  LBasisclbs 19074

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-lbs 19075

This theorem is referenced by:  lbsss  19077  lbssp  19079  lbsind  19080  lbspropd  19099  islbs2  19154  frlmlbs  20136  islbs4  20171