Theorem islbs 21028

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 3461	. . . 4 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2		islbs.j	. . . . 5 ⊢ 𝐽 = (LBasis‘𝑊)
3		fveq2 6834	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4		islbs.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2789	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6	5	pweqd 4571	. . . . . . 7 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7		fvexd 6849	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) ∈ V)
8		fveq2 6834	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
9		islbs.n	. . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑊)
10	8, 9	eqtr4di 2789	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
11		fvexd 6849	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (Scalar‘𝑤) ∈ V)
12		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
13	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (Scalar‘𝑤) = (Scalar‘𝑊))
14		islbs.f	. . . . . . . . . 10 ⊢ 𝐹 = (Scalar‘𝑊)
15	13, 14	eqtr4di 2789	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (Scalar‘𝑤) = 𝐹)
16		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → 𝑛 = 𝑁)
17	16	fveq1d 6836	. . . . . . . . . . 11 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑛‘𝑏) = (𝑁‘𝑏))
18	5	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑤) = 𝑉)
19	17, 18	eqeq12d 2752	. . . . . . . . . 10 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((𝑛‘𝑏) = (Base‘𝑤) ↔ (𝑁‘𝑏) = 𝑉))
20		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
21	20	fveq2d 6838	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
22		islbs.k	. . . . . . . . . . . . . 14 ⊢ 𝐾 = (Base‘𝐹)
23	21, 22	eqtr4di 2789	. . . . . . . . . . . . 13 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑓) = 𝐾)
24	20	fveq2d 6838	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (0g‘𝑓) = (0g‘𝐹))
25		islbs.z	. . . . . . . . . . . . . . 15 ⊢ 0 = (0g‘𝐹)
26	24, 25	eqtr4di 2789	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (0g‘𝑓) = 0 )
27	26	sneqd 4592	. . . . . . . . . . . . 13 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → {(0g‘𝑓)} = { 0 })
28	23, 27	difeq12d 4079	. . . . . . . . . . . 12 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((Base‘𝑓) ∖ {(0g‘𝑓)}) = (𝐾 ∖ { 0 }))
29		fveq2 6834	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
30		islbs.s	. . . . . . . . . . . . . . . . 17 ⊢ · = ( ·𝑠 ‘𝑊)
31	29, 30	eqtr4di 2789	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
32	31	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ( ·𝑠 ‘𝑤) = · )
33	32	oveqd 7375	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑦( ·𝑠 ‘𝑤)𝑥) = (𝑦 · 𝑥))
34	16	fveq1d 6836	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑛‘(𝑏 ∖ {𝑥})) = (𝑁‘(𝑏 ∖ {𝑥})))
35	33, 34	eleq12d 2830	. . . . . . . . . . . . 13 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((𝑦( ·𝑠 ‘𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})) ↔ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥}))))
36	35	notbid 318	. . . . . . . . . . . 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 3159	. . . . . . . . . 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 632	. . . . . . . . 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 3785	. . . . . . . 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 3785	. . . . . . 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 3417	. . . . . 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 21027	. . . . . 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 6848	. . . . . . . 8 ⊢ 𝑉 ∈ V
45	44	pwex 5325	. . . . . . 7 ⊢ 𝒫 𝑉 ∈ V
46	45	rabex 5284	. . . . . 6 ⊢ {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))} ∈ V
47	42, 43, 46	fvmpt 6941	. . . . 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 2783	. . . 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 2822	. 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 5279	. . . 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 6843	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝑁‘𝑏) = 𝑉 ↔ (𝑁‘𝐵) = 𝑉))
54		difeq1 4071	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑏 ∖ {𝑥}) = (𝐵 ∖ {𝑥}))
55	54	fveq2d 6838	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑁‘(𝑏 ∖ {𝑥})) = (𝑁‘(𝐵 ∖ {𝑥})))
56	55	eleq2d 2822	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → ((𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
57	56	notbid 318	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
58	57	ralbidv 3159	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
59	58	raleqbi1dv 3308	. . . . 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 632	. . . 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 3646	. . 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 303	. 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 287	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051  {crab 3399  Vcvv 3440  [wsbc 3740   ∖ cdif 3898   ⊆ wss 3901  𝒫 cpw 4554  {csn 4580  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  Scalarcsca 17180   ·_𝑠 cvsca 17181  0_gc0g 17359  LSpanclspn 20922  LBasisclbs 21026

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-lbs 21027

This theorem is referenced by:  lbsss  21029  lbssp  21031  lbsind  21032  lbspropd  21051  islbs2  21109  frlmlbs  21752  islbs4  21787