Theorem islbs 21098

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 3509	. . . 4 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
2		islbs.j	. . . . 5 ⊢ 𝐽 = (LBasis‘𝑊)
3		fveq2 6920	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4		islbs.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2798	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6	5	pweqd 4639	. . . . . . 7 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘𝑤) = 𝒫 𝑉)
7		fvexd 6935	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) ∈ V)
8		fveq2 6920	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
9		islbs.n	. . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑊)
10	8, 9	eqtr4di 2798	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
11		fvexd 6935	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (Scalar‘𝑤) ∈ V)
12		fveq2 6920	. . . . . . . . . . 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 2798	. . . . . . . . 9 ⊢ ((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) → (Scalar‘𝑤) = 𝐹)
16		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → 𝑛 = 𝑁)
17	16	fveq1d 6922	. . . . . . . . . . 11 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑛‘𝑏) = (𝑁‘𝑏))
18	5	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑤) = 𝑉)
19	17, 18	eqeq12d 2756	. . . . . . . . . 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 6924	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹))
22		islbs.k	. . . . . . . . . . . . . 14 ⊢ 𝐾 = (Base‘𝐹)
23	21, 22	eqtr4di 2798	. . . . . . . . . . . . 13 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (Base‘𝑓) = 𝐾)
24	20	fveq2d 6924	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (0g‘𝑓) = (0g‘𝐹))
25		islbs.z	. . . . . . . . . . . . . . 15 ⊢ 0 = (0g‘𝐹)
26	24, 25	eqtr4di 2798	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (0g‘𝑓) = 0 )
27	26	sneqd 4660	. . . . . . . . . . . . 13 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → {(0g‘𝑓)} = { 0 })
28	23, 27	difeq12d 4150	. . . . . . . . . . . 12 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ((Base‘𝑓) ∖ {(0g‘𝑓)}) = (𝐾 ∖ { 0 }))
29		fveq2 6920	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
30		islbs.s	. . . . . . . . . . . . . . . . 17 ⊢ · = ( ·𝑠 ‘𝑊)
31	29, 30	eqtr4di 2798	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
32	31	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → ( ·𝑠 ‘𝑤) = · )
33	32	oveqd 7465	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑦( ·𝑠 ‘𝑤)𝑥) = (𝑦 · 𝑥))
34	16	fveq1d 6922	. . . . . . . . . . . . . 14 ⊢ (((𝑤 = 𝑊 ∧ 𝑛 = 𝑁) ∧ 𝑓 = 𝐹) → (𝑛‘(𝑏 ∖ {𝑥})) = (𝑁‘(𝑏 ∖ {𝑥})))
35	33, 34	eleq12d 2838	. . . . . . . . . . . . 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 3354	. . . . . . . . . . 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 3184	. . . . . . . . . 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 3852	. . . . . . . 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 3852	. . . . . . 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 3462	. . . . . 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 21097	. . . . . 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 6934	. . . . . . . 8 ⊢ 𝑉 ∈ V
45	44	pwex 5398	. . . . . . 7 ⊢ 𝒫 𝑉 ∈ V
46	45	rabex 5357	. . . . . 6 ⊢ {𝑏 ∈ 𝒫 𝑉 ∣ ((𝑁‘𝑏) = 𝑉 ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})))} ∈ V
47	42, 43, 46	fvmpt 7029	. . . . 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 2792	. . . 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 2830	. 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 5352	. . . 4 ⊢ (𝐵 ∈ 𝒫 𝑉 ↔ 𝐵 ⊆ 𝑉)
52	51	anbi1i 623	. . 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 6929	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝑁‘𝑏) = 𝑉 ↔ (𝑁‘𝐵) = 𝑉))
54		difeq1 4142	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑏 ∖ {𝑥}) = (𝐵 ∖ {𝑥}))
55	54	fveq2d 6924	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑁‘(𝑏 ∖ {𝑥})) = (𝑁‘(𝐵 ∖ {𝑥})))
56	55	eleq2d 2830	. . . . . . . 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 3184	. . . . . 6 ⊢ (𝑏 = 𝐵 → (∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝑏 ∖ {𝑥})) ↔ ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))
59	58	raleqbi1dv 3346	. . . . 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 3708	. . 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 1095	. . 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 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443  Vcvv 3488  [wsbc 3804   ∖ cdif 3973   ⊆ wss 3976  𝒫 cpw 4622  {csn 4648  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  Scalarcsca 17314   ·_𝑠 cvsca 17315  0_gc0g 17499  LSpanclspn 20992  LBasisclbs 21096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-lbs 21097

This theorem is referenced by:  lbsss  21099  lbssp  21101  lbsind  21102  lbspropd  21121  islbs2  21179  frlmlbs  21840  islbs4  21875