Theorem islbs 20409

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 3459	. . . 4 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2		islbs.j	. . . . 5 ⊢ 𝐽 = (LBasis‘𝑊)
3		fveq2 6809	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4		islbs.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2795	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6	5	pweqd 4560	. . . . . . 7 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7		fvexd 6824	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) ∈ V)
8		fveq2 6809	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
9		islbs.n	. . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑊)
10	8, 9	eqtr4di 2795	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
11		fvexd 6824	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (Scalar‘𝑤) ∈ V)
12		fveq2 6809	. . . . . . . . . . 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	eqtr4di 2795	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (Scalar‘𝑤) = 𝐹)
16		simplr 766	. . . . . . . . . . . 12 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → 𝑛 = 𝑁)
17	16	fveq1d 6811	. . . . . . . . . . 11 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑛‘𝑏) = (𝑁‘𝑏))
18	5	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑤) = 𝑉)
19	17, 18	eqeq12d 2753	. . . . . . . . . 10 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((𝑛‘𝑏) = (Base‘𝑤) ↔ (𝑁‘𝑏) = 𝑉))
20		simpr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
21	20	fveq2d 6813	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
22		islbs.k	. . . . . . . . . . . . . 14 ⊢ 𝐾 = (Base‘𝐹)
23	21, 22	eqtr4di 2795	. . . . . . . . . . . . 13 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑓) = 𝐾)
24	20	fveq2d 6813	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (0g‘𝑓) = (0g‘𝐹))
25		islbs.z	. . . . . . . . . . . . . . 15 ⊢ 0 = (0g‘𝐹)
26	24, 25	eqtr4di 2795	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (0g‘𝑓) = 0 )
27	26	sneqd 4581	. . . . . . . . . . . . 13 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → {(0g‘𝑓)} = { 0 })
28	23, 27	difeq12d 4068	. . . . . . . . . . . 12 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((Base‘𝑓) ∖ {(0g‘𝑓)}) = (𝐾 ∖ { 0 }))
29		fveq2 6809	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
30		islbs.s	. . . . . . . . . . . . . . . . 17 ⊢ · = ( ·𝑠 ‘𝑊)
31	29, 30	eqtr4di 2795	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
32	31	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ( ·𝑠 ‘𝑤) = · )
33	32	oveqd 7330	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑦( ·𝑠 ‘𝑤)𝑥) = (𝑦 · 𝑥))
34	16	fveq1d 6811	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑛‘(𝑏 ∖ {𝑥})) = (𝑁‘(𝑏 ∖ {𝑥})))
35	33, 34	eleq12d 2832	. . . . . . . . . . . . 13 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((𝑦( ·𝑠 ‘𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})) ↔ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))))
36	35	notbid 317	. . . . . . . . . . . 12 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (¬ (𝑦( ·𝑠 ‘𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})) ↔ ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))))
37	28, 36	raleqbidv 3316	. . . . . . . . . . 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 3171	. . . . . . . . . 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 631	. . . . . . . . 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 3772	. . . . . . . 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 3772	. . . . . . 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 3419	. . . . . 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 20408	. . . . . 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	4	fvexi 6823	. . . . . . . 8 ⊢ 𝑉 ∈ V
45	44	pwex 5316	. . . . . . 7 ⊢ 𝒫 𝑉 ∈ V
46	45	rabex 5269	. . . . . 6 ⊢ {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))} ∈ V
47	42, 43, 46	fvmpt 6912	. . . . 5 ⊢ (𝑊 ∈ V → (LBasis‘𝑊) = {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))})
48	2, 47	eqtrid 2789	. . . 4 ⊢ (𝑊 ∈ V → 𝐽 = {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))})
49	1, 48	syl 17	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐽 = {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))})
50	49	eleq2d 2823	. 2 ⊢ (𝑊 ∈ 𝑋 → (𝐵 ∈ 𝐽 ↔ 𝐵 ∈ {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))}))
51	44	elpw2 5282	. . . 4 ⊢ (𝐵 ∈ 𝒫 𝑉 ↔ 𝐵 ⊆ 𝑉)
52	51	anbi1i 624	. . 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})))))
53		fveqeq2 6818	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝑁‘𝑏) = 𝑉 ↔ (𝑁‘𝐵) = 𝑉))
54		difeq1 4060	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑏 ∖ {𝑥}) = (𝐵 ∖ {𝑥}))
55	54	fveq2d 6813	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑁‘(𝑏 ∖ {𝑥})) = (𝑁‘(𝐵 ∖ {𝑥})))
56	55	eleq2d 2823	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → ((𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
57	56	notbid 317	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
58	57	ralbidv 3171	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
59	58	raleqbi1dv 3304	. . . . 5 ⊢ (𝑏 = 𝐵 → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
60	53, 59	anbi12d 631	. . . 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})))))
61	60	elrab 3633	. . 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})))))
62		3anass 1094	. . 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})))))
63	52, 61, 62	3bitr4i 302	. 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}))))
64	50, 63	bitrdi 286	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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3062  {crab 3404  Vcvv 3441  [wsbc 3725   ∖ cdif 3893   ⊆ wss 3896  𝒫 cpw 4543  {csn 4569  ‘cfv 6463  (class class class)co 7313  Basecbs 16979  Scalarcsca 17032   ·_𝑠 cvsca 17033  0_gc0g 17217  LSpanclspn 20304  LBasisclbs 20407

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-sbc 3726  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-iota 6415  df-fun 6465  df-fv 6471  df-ov 7316  df-lbs 20408

This theorem is referenced by:  lbsss  20410  lbssp  20412  lbsind  20413  lbspropd  20432  islbs2  20487  frlmlbs  21075  islbs4  21110