Metamath Proof Explorer - mmtheorems39

Jump to page: 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5787

Statement List for Metamath Proof Explorer - 3801-3900 - Page 39 of 58

Type	Label	Description
Statement
Theorem	pion 3801	A positive integer is an ordinal number.
⊢ (A ∈ N → A ∈ On)
Theorem	piord 3802	A positive integer is ordinal.
⊢ (A ∈ N → Ord A)
Theorem	niex 3803	The class of positive integers is a set.
⊢ N ∈ V
Theorem	0npi 3804	The empty set is not a positive integer.
⊢ ¬ ∅ ∈ N
Theorem	1pi 3805	Ordinal 'one' is a positive integer.
⊢ 1o ∈ N
Theorem	addpiord 3806	Positive integer addition in terms of ordinal addition.
⊢ ((A ∈ N ∧ B ∈ N) → (A +N B) = (A +o B))
Theorem	mulpiord 3807	Positive integer multiplication in terms of ordinal multiplication.
⊢ ((A ∈ N ∧ B ∈ N) → (A ·N B) = (A ·o B))
Theorem	mulidpi 3808	1 is an identity element for multiplication on positive integers.
⊢ (A ∈ N → (A ·N 1o) = A)
Theorem	ltpiord 3809	Positive integer 'less than' in terms of ordinal membership.
⊢ ((A ∈ N ∧ B ∈ N) → (A <N B ↔ A ∈ B))
Theorem	ltsopi 3810	Positive integer 'less than' is a strict ordering.
⊢ <N Or N
Theorem	ltrelpi 3811	Positive integer 'less than' is a relation on positive integers.
⊢ <N ⊆ (N × N)
Theorem	dmaddpi 3812	Domain of addition on positive integers.
⊢ dom +N = (N × N)
Theorem	dmmulpi 3813	Domain of multiplication on positive integers.
⊢ dom ·N = (N × N)
Theorem	addclpi 3814	Closure of addition of positive integers.
⊢ ((A ∈ N ∧ B ∈ N) → (A +N B) ∈ N)
Theorem	mulclpi 3815	Closure of multiplication of positive integers.
⊢ ((A ∈ N ∧ B ∈ N) → (A ·N B) ∈ N)
Theorem	addcompi 3816	Addition of positive integers is commutative.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (A +N B) = (B +N A)
Theorem	addasspi 3817	Addition of positive integers is associative.
⊢ B ∈ V    &   ⊢ C ∈ V    ⇒   ⊢ ((A +N B) +N C) = (A +N (B +N C))
Theorem	mulcompi 3818	Multiplication of positive integers is commutative.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (A ·N B) = (B ·N A)
Theorem	mulasspi 3819	Multiplication of positive integers is associative.
⊢ B ∈ V    &   ⊢ C ∈ V    ⇒   ⊢ ((A ·N B) ·N C) = (A ·N (B ·N C))
Theorem	distrpi 3820	Multiplication of positive integers is distributive.
⊢ B ∈ V    &   ⊢ C ∈ V    ⇒   ⊢ (A ·N (B +N C)) = ((A ·N B) +N (A ·N C))
Theorem	mulcanpi 3821	Multiplication cancellation law for positive integers.
⊢ C ∈ V    ⇒   ⊢ ((A ∈ N ∧ B ∈ N) → ((A ·N B) = (A ·N C) → B = C))
Theorem	addnidpi 3822	There is no identity element for addition on positive integers.
⊢ B ∈ V    ⇒   ⊢ (A ∈ N → ¬ (A +N B) = A)
Theorem	ltexpi 3823	Ordering on positive integers in terms of existence of sum.
⊢ ((A ∈ N ∧ B ∈ N) → (A <N B ↔ ∃x(x ∈ N ∧ (A +N x) = B)))
Theorem	ltapi 3824	Ordering property of multiplication for positive integers.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (C ∈ N → (A <N B ↔ (C +N A) <N (C +N B)))
Theorem	ltmpi 3825	Ordering property of multiplication for positive integers.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (C ∈ N → (A <N B ↔ (C ·N A) <N (C ·N B)))
Theorem	1lt2pi 3826	One is less than two (one plus one).
⊢ 1o <N (1o +N 1o)
Theorem	nlt1pi 3827	No positive integer is less than one.
⊢ ¬ A <N 1o
Theorem	indpi 3828	Principle of Finite Induction on positive integers.
⊢ (x = 1o → (φ ↔ ψ))    &   ⊢ (x = y → (φ ↔ χ))    &   ⊢ (x = (y +N 1o) → (φ ↔ θ))    &   ⊢ (x = A → (φ ↔ τ))    &   ⊢ ψ    &   ⊢ (y ∈ N → (χ → θ))    ⇒   ⊢ (A ∈ N → τ)
Definition	df-plpq 3829	Define pre-addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction. From Proposition 9-2.3 of [Gleason] p. 117.
⊢ +pQ = {⟨⟨x, y⟩, z⟩∣((x ∈ (N × N) ∧ y ∈ (N × N)) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = ⟨((w ·N f) +N (v ·N u)), (v ·N f)⟩))}
Definition	df-mpq 3830	Define pre-multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119.
⊢ ·pQ = {⟨⟨x, y⟩, z⟩∣((x ∈ (N × N) ∧ y ∈ (N × N)) ∧ ∃w∃v∃u∃f((x = ⟨w, v⟩ ∧ y = ⟨u, f⟩) ∧ z = ⟨(w ·N u), (v ·N f)⟩))}
Definition	df-enq 3831	Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction. From Proposition 9-2.1 of [Gleason] p. 117.
⊢ ~Q = {⟨x, y⟩∣((x ∈ (N × N) ∧ y ∈ (N × N)) ∧ ∃z∃w∃v∃u((x = ⟨z, w⟩ ∧ y = ⟨v, u⟩) ∧ (z ·N u) = (w ·N v)))}
Definition	df-nq 3832	Define class of positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117.
⊢ Q = ((N × N) / ~Q )
Definition	df-plq 3833	Define addition on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction. From Proposition 9-2.3 of [Gleason] p. 117.
⊢ +Q = {⟨⟨x, y⟩, z⟩∣((x ∈ Q ∧ y ∈ Q) ∧ ∃w∃v∃u∃f((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ +pQ ⟨u, f⟩)] ~Q ))}
Definition	df-mq 3834	Define multiplication on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction. From Proposition 9-2.4 of [Gleason] p. 119.
⊢ ·Q = {⟨⟨x, y⟩, z⟩∣((x ∈ Q ∧ y ∈ Q) ∧ ∃w∃v∃u∃f((x = [⟨w, v⟩] ~Q ∧ y = [⟨u, f⟩] ~Q ) ∧ z = [(⟨w, v⟩ ·pQ ⟨u, f⟩)] ~Q ))}
Definition	df-rq 3835	Define reciprocal on positive fractions. It means the same thing as one divided by the argument (although we don't define full division since we will never need it). This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction. From Proposition 9-2.5 of [Gleason] p. 119, who uses an asterisk to denote this unary operation.
⊢ *Q = {⟨x, y⟩∣(x ∈ Q ∧ (x ·Q y) = 1Q)}
Definition	df-ltq 3836	Define ordering relation on positive fractions. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction. Similar to Definition 5 of [Suppes] p. 162.
⊢ <Q = {⟨x, y⟩∣((x ∈ Q ∧ y ∈ Q) ∧ ∃z∃w∃v∃u((x = [⟨z, w⟩] ~Q ∧ y = [⟨v, u⟩] ~Q ) ∧ (z ·N u) <N (w ·N v)))}
Definition	df-1q 3837	Define positive fraction constant 1. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction. From Proposition 9-2.2 of [Gleason] p. 117.
⊢ 1Q = [⟨1o, 1o⟩] ~Q
Theorem	enqbreq 3838	Equivalence relation for positive fractions in terms of positive integers.
⊢ (((A ∈ N ∧ B ∈ N) ∧ (C ∈ N ∧ D ∈ N)) → (⟨A, B⟩ ~Q ⟨C, D⟩ ↔ (A ·N D) = (B ·N C)))
Theorem	dmenq 3839	Domain of equivalence relation for positive fractions.
⊢ dom ~Q = (N × N)
Theorem	enqer 3840	The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of [Gleason] p. 117.
⊢ Er ~Q
Theorem	enqeceq 3841	Equivalence class equality of positive fractions in terms of positive integers.
⊢ (((A ∈ N ∧ B ∈ N) ∧ (C ∈ N ∧ D ∈ N)) → ([⟨A, B⟩] ~Q = [⟨C, D⟩] ~Q ↔ (A ·N D) = (B ·N C)))
Theorem	enqex 3842	The equivalence relation for positive fractions exists.
⊢ ~Q ∈ V
Theorem	nqex 3843	The class of positive fractions exists.
⊢ Q ∈ V
Theorem	0npq 3844	The empty set is not a positive fraction.
⊢ ¬ ∅ ∈ Q
Theorem	ltrelpq 3845	Positive fraction 'less than' is a relation on positive fractions.
⊢ <Q ⊆ (Q × Q)
Theorem	addcmpblnq 3846	Lemma showing compatibility of addition.
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ C ∈ V    &   ⊢ D ∈ V    &   ⊢ F ∈ V    &   ⊢ G ∈ V    &   ⊢ R ∈ V    &   ⊢ S ∈ V    ⇒   ⊢ ((((A ∈ N ∧ B ∈ N) ∧ (C ∈ N ∧ D ∈ N)) ∧ ((F ∈ N ∧ G ∈ N) ∧ (R ∈ N ∧ S ∈ N))) → (((A ·N D) = (B ·N C) ∧ (F ·N S) = (G ·N R)) → ⟨((A ·N G) +N (B ·N F)), (B ·N G)⟩ ~Q ⟨((C ·N S) +N (D ·N R)), (D ·N S)⟩))
Theorem	mulcmpblnq 3847	Lemma showing compatibility of multiplication.
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ C ∈ V    &   ⊢ D ∈ V    &   ⊢ F ∈ V    &   ⊢ G ∈ V    &   ⊢ R ∈ V    &   ⊢ S ∈ V    ⇒   ⊢ ((((A ∈ N ∧ B ∈ N) ∧ (C ∈ N ∧ D ∈ N)) ∧ ((F ∈ N ∧ G ∈ N) ∧ (R ∈ N ∧ S ∈ N))) → (((A ·N D) = (B ·N C) ∧ (F ·N S) = (G ·N R)) → ⟨(A ·N F), (B ·N G)⟩ ~Q ⟨(C ·N R), (D ·N S)⟩))
Theorem	addpipq 3848	Addition of positive fractions in terms of positive integers.
⊢ (((A ∈ N ∧ B ∈ N) ∧ (C ∈ N ∧ D ∈ N)) → ([⟨A, B⟩] ~Q +Q [⟨C, D⟩] ~Q ) = [⟨((A ·N D) +N (B ·N C)), (B ·N D)⟩] ~Q )
Theorem	mulpipq 3849	Multiplication of positive fractions in terms of positive integers.
⊢ (((A ∈ N ∧ B ∈ N) ∧ (C ∈ N ∧ D ∈ N)) → ([⟨A, B⟩] ~Q ·Q [⟨C, D⟩] ~Q ) = [⟨(A ·N C), (B ·N D)⟩] ~Q )
Theorem	ordpipq 3850	Ordering of positive fractions in terms of positive integers.
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ C ∈ V    &   ⊢ D ∈ V    ⇒   ⊢ ([⟨A, B⟩] ~Q <Q [⟨C, D⟩] ~Q ↔ (A ·N D) <N (B ·N C))
Theorem	1q 3851	The positive fraction 'one'.
⊢ 1Q ∈ Q
Theorem	addclpq 3852	Closure of addition on positive fractions.
⊢ ((A ∈ Q ∧ B ∈ Q) → (A +Q B) ∈ Q)
Theorem	dmaddpq 3853	Domain of addition on positive fractions.
⊢ dom +Q = (Q × Q)
Theorem	mulclpq 3854	Closure of multiplication on positive fractions.
⊢ ((A ∈ Q ∧ B ∈ Q) → (A ·Q B) ∈ Q)
Theorem	dmmulpq 3855	Domain of multiplication on positive fractions.
⊢ dom ·Q = (Q × Q)
Theorem	addcompq 3856	Addition of positive fractions is commutative.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (A +Q B) = (B +Q A)
Theorem	addasspq 3857	Addition of positive fractions is associative.
⊢ B ∈ V    &   ⊢ C ∈ V    ⇒   ⊢ ((A +Q B) +Q C) = (A +Q (B +Q C))
Theorem	mulcompq 3858	Multiplication of positive fractions is commutative.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (A ·Q B) = (B ·Q A)
Theorem	mulasspq 3859	Multiplication of positive fractions is associative.
⊢ B ∈ V    &   ⊢ C ∈ V    ⇒   ⊢ ((A ·Q B) ·Q C) = (A ·Q (B ·Q C))
Theorem	distrpqlem 3860	Lemma for distributive law: cancellation of common factor.
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ C ∈ V    ⇒   ⊢ ((A ∈ N ∧ B ∈ N ∧ C ∈ N) → [⟨(A ·N B), (A ·N C)⟩] ~Q = [⟨B, C⟩] ~Q )
Theorem	distrpq 3861	Multiplication of positive fractions is distributive.
⊢ B ∈ V    &   ⊢ C ∈ V    ⇒   ⊢ (A ·Q (B +Q C)) = ((A ·Q B) +Q (A ·Q C))
Theorem	1qec 3862	The equivalence class of ratio 1.
⊢ A ∈ V    ⇒   ⊢ (A ∈ N → 1Q = [⟨A, A⟩] ~Q )
Theorem	mulidpq 3863	Multiplication identity element for positive fractions.
⊢ (A ∈ Q → (A ·Q 1Q) = A)
Theorem	recmulpq 3864	Relationship between reciprocal and multiplication on positive fractions.
⊢ B ∈ V    ⇒   ⊢ (A ∈ Q → ((*Q ‘A) = B ↔ (A ·Q B) = 1Q))
Theorem	recidpq 3865	A positive fraction times its reciprocal is 1.
⊢ (A ∈ Q → (A ·Q (*Q ‘A)) = 1Q)
Theorem	recclpq 3866	Closure law for positive fraction reciprocal.
⊢ (A ∈ Q → (*Q ‘A) ∈ Q)
Theorem	recrecpq 3867	Reciprocal of reciprocal of positive fraction.
⊢ A ∈ V    ⇒   ⊢ (A ∈ Q → (*Q ‘(*Q ‘A)) = A)
Theorem	dmrecpq 3868	Domain of reciprocal on positive fractions.
⊢ dom *Q = Q
Theorem	ltsopq 3869	'Less than' is a strict ordering on positive fractions.
⊢ <Q Or Q
Theorem	ltapq 3870	Ordering property of addition for positive fractions. Proposition 9-2.6(ii) of [Gleason] p. 120.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (C ∈ Q → (A <Q B ↔ (C +Q A) <Q (C +Q B)))
Theorem	ltmpq 3871	Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (C ∈ Q → (A <Q B ↔ (C ·Q A) <Q (C ·Q B)))
Theorem	1lt2pq 3872	One is less than two (one plus one).
⊢ 1Q <Q (1Q +Q 1Q)
Theorem	ltaddpq 3873	The sum of two fractions is greater than one of them.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ ((A ∈ Q ∧ B ∈ Q) → A <Q (A +Q B))
Theorem	ltexpq 3874	Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119.
⊢ A ∈ V    ⇒   ⊢ ((A ∈ Q ∧ B ∈ Q) → (A <Q B ↔ ∃x(A +Q x) = B))
Theorem	ltexpq2 3875	Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119.
⊢ A ∈ V    ⇒   ⊢ ((A ∈ Q ∧ B ∈ Q) → (A <Q B ↔ ∃x(x ∈ Q ∧ (A +Q x) = B)))
Theorem	halfpq 3876	One-half of any positive fraction exists. Lemma for Proposition 9-2.6(i) of [Gleason] p. 120.
⊢ (A ∈ Q → ∃x(x +Q x) = A)
Theorem	nsmallpq 3877	The is no smallest positive fraction.
⊢ (A ∈ Q → ∃x x <Q A)
Theorem	ltbtwnpq 3878	There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (A <Q B → ∃x(A <Q x ∧ x <Q B))
Theorem	ltrpq 3879	Ordering property of reciprocal for positive fractions. Proposition 9-2.6(iv) of [Gleason] p. 120.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (A <Q B → (*Q ‘B) <Q (*Q ‘A))
Definition	df-np 3880	Define the class of positive reals. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction. Definition 9-3.1 of [Gleason] p. 121.
⊢ P = {x∣((∅ ⊂ x ∧ x ⊂ Q) ∧ ∀y ∈ x (∀z(z <Q y → z ∈ x) ∧ ∃z ∈ x y <Q z))}
Definition	df-1p 3881	Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction. Definition of [Gleason] p. 122.
⊢ 1P = {x∣x <Q 1Q}
Definition	df-plp 3882	Define addition on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction. From Proposition 9-3.5 of [Gleason] p. 123.
⊢ +P = {⟨⟨x, y⟩, z⟩∣((x ∈ P ∧ y ∈ P) ∧ z = {w∣∃v ∈ x ∃u ∈ y w = (v +Q u)})}
Definition	df-mp 3883	Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction. From Proposition 9-3.7 of [Gleason] p. 124.
⊢ ·P = {⟨⟨x, y⟩, z⟩∣((x ∈ P ∧ y ∈ P) ∧ z = {w∣∃v ∈ x ∃u ∈ y w = (v ·Q u)})}
Definition	df-ltp 3884	Define ordering on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction. From Proposition 9-3.2 of [Gleason] p. 122.
⊢ <P = {⟨x, y⟩∣((x ∈ P ∧ y ∈ P) ∧ x ⊂ y)}
Theorem	npex 3885	The class of positive reals is a set.
⊢ P ∈ V
Theorem	elnp 3886	Membership in positive reals.
⊢ (A ∈ P ↔ ((∅ ⊂ A ∧ A ⊂ Q) ∧ ∀x ∈ A (∀y(y <Q x → y ∈ A) ∧ ∃y ∈ A x <Q y)))
Theorem	prn0 3887	A positive real is not empty.
⊢ (A ∈ P → ¬ A = ∅)
Theorem	prpssnq 3888	A positive real is a subset of the positive fractions.
⊢ (A ∈ P → A ⊂ Q)
Theorem	elprpq 3889	A positive real is a set of positive fractions.
⊢ ((A ∈ P ∧ B ∈ A) → B ∈ Q)
Theorem	0npr 3890	The empty set is not a positive real.
⊢ ¬ ∅ ∈ P
Theorem	prcdpq 3891	A positive real is closed downwards under the positive fractions. Definition 9-3.1 (ii) of [Gleason] p. 121.
⊢ ((A ∈ P ∧ B ∈ A) → (C <Q B → C ∈ A))
Theorem	prub 3892	A positive fraction not in a positive real is an upper bound. Remark (1) of [Gleason] p. 122.
⊢ (((A ∈ P ∧ B ∈ A) ∧ C ∈ Q) → (¬ C ∈ A → B <Q C))
Theorem	prnmax 3893	A positive real has no largest member. Definition 9-3.1(iii) of [Gleason] p. 121.
⊢ ((A ∈ P ∧ B ∈ A) → ∃x(x ∈ A ∧ B <Q x))
Theorem	prnmadd 3894	A positive real has no largest member. Addition version.
⊢ B ∈ V    ⇒   ⊢ ((A ∈ P ∧ B ∈ A) → ∃x(B +Q x) ∈ A)
Theorem	ltrelpr 3895	Positive real 'less than' is a relation on positive reals.
⊢ <P ⊆ (P × P)
Theorem	genpv 3896	Value of general operation (addition or multiplication) on positive reals.
⊢ F = {⟨⟨w, v⟩, u⟩∣((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)})}    ⇒   ⊢ ((A ∈ P ∧ B ∈ P) → (AFB) = {f∣∃g∃h((g ∈ A ∧ h ∈ B) ∧ f = (gGh))})
Theorem	genpelv 3897	Membership in value of general operation (addition or multiplication) on positive reals.
⊢ F = {⟨⟨w, v⟩, u⟩∣((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)})}    &   ⊢ C ∈ V    ⇒   ⊢ ((A ∈ P ∧ B ∈ P) → (C ∈ (AFB) ↔ ∃f∃g((f ∈ A ∧ g ∈ B) ∧ C = (fGg))))
Theorem	genpprecl 3898	Pre-closure law for general operation on positive reals.
⊢ F = {⟨⟨w, v⟩, u⟩∣((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)})}    ⇒   ⊢ ((A ∈ P ∧ B ∈ P) → ((C ∈ A ∧ D ∈ B) → (CGD) ∈ (AFB)))
Theorem	genpdm 3899	Domain of general operation on positive reals.
⊢ F = {⟨⟨w, v⟩, u⟩∣((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)})}    ⇒   ⊢ dom F = (P × P)
Theorem	genpn0 3900	The result of an operation on positive reals is not empty.
⊢ F = {⟨⟨w, v⟩, u⟩∣((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (yGz)})}    ⇒   ⊢ ((A ∈ P ∧ B ∈ P) → ∅ ⊂ (AFB))