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Theorem List for Metamath Proof Explorer - 9501-9600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltrelnq 9501 Positive fraction 'less than' is a relation on positive fractions. (Contributed by NM, 14-Feb-1996.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)
<Q ⊆ (Q × Q)
 
Theorempinq 9502 The representatives of positive integers as positive fractions. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
(𝐴N → ⟨𝐴, 1𝑜⟩ ∈ Q)
 
Theorem1nq 9503 The positive fraction 'one'. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
1QQ
 
Theoremnqereu 9504* There is a unique element of Q equivalent to each element of N × N. (Contributed by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
(𝐴 ∈ (N × N) → ∃!𝑥Q 𝑥 ~Q 𝐴)
 
Theoremnqerf 9505 Corollary of nqereu 9504: the function [Q] is actually a function. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
[Q]:(N × N)⟶Q
 
Theoremnqercl 9506 Corollary of nqereu 9504: closure of [Q]. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
(𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
 
Theoremnqerrel 9507 Any member of (N × N) relates to the representative of its equivalence class. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
(𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))
 
Theoremnqerid 9508 Corollary of nqereu 9504: the function [Q] acts as the identity on members of Q. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
(𝐴Q → ([Q]‘𝐴) = 𝐴)
 
Theoremenqeq 9509 Corollary of nqereu 9504: if two fractions are both reduced and equivalent, then they are equal. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
((𝐴Q𝐵Q𝐴 ~Q 𝐵) → 𝐴 = 𝐵)
 
Theoremnqereq 9510 The function [Q] acts as a substitute for equivalence classes, and it satisfies the fundamental requirement for equivalence representatives: the representatives are equal iff the members are equivalent. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ([Q]‘𝐴) = ([Q]‘𝐵)))
 
Theoremaddpipq2 9511 Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 +pQ 𝐵) = ⟨(((1st𝐴) ·N (2nd𝐵)) +N ((1st𝐵) ·N (2nd𝐴))), ((2nd𝐴) ·N (2nd𝐵))⟩)
 
Theoremaddpipq 9512 Addition of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
(((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → (⟨𝐴, 𝐵⟩ +pQ𝐶, 𝐷⟩) = ⟨((𝐴 ·N 𝐷) +N (𝐶 ·N 𝐵)), (𝐵 ·N 𝐷)⟩)
 
Theoremaddpqnq 9513 Addition of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 26-Dec-2014.) (New usage is discouraged.)
((𝐴Q𝐵Q) → (𝐴 +Q 𝐵) = ([Q]‘(𝐴 +pQ 𝐵)))
 
Theoremmulpipq2 9514 Multiplication of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ·pQ 𝐵) = ⟨((1st𝐴) ·N (1st𝐵)), ((2nd𝐴) ·N (2nd𝐵))⟩)
 
Theoremmulpipq 9515 Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
(((𝐴N𝐵N) ∧ (𝐶N𝐷N)) → (⟨𝐴, 𝐵⟩ ·pQ𝐶, 𝐷⟩) = ⟨(𝐴 ·N 𝐶), (𝐵 ·N 𝐷)⟩)
 
Theoremmulpqnq 9516 Multiplication of positive fractions in terms of positive integers. (Contributed by NM, 28-Aug-1995.) (Revised by Mario Carneiro, 26-Dec-2014.) (New usage is discouraged.)
((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) = ([Q]‘(𝐴 ·pQ 𝐵)))
 
Theoremordpipq 9517 Ordering of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
(⟨𝐴, 𝐵⟩ <pQ𝐶, 𝐷⟩ ↔ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))
 
Theoremordpinq 9518 Ordering of positive fractions in terms of positive integers. (Contributed by NM, 13-Feb-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
((𝐴Q𝐵Q) → (𝐴 <Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
 
Theoremaddpqf 9519 Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
+pQ :((N × N) × (N × N))⟶(N × N)
 
Theoremaddclnq 9520 Closure of addition on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
((𝐴Q𝐵Q) → (𝐴 +Q 𝐵) ∈ Q)
 
Theoremmulpqf 9521 Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
·pQ :((N × N) × (N × N))⟶(N × N)
 
Theoremmulclnq 9522 Closure of multiplication on positive fractions. (Contributed by NM, 29-Aug-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) ∈ Q)
 
Theoremaddnqf 9523 Domain of addition on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
+Q :(Q × Q)⟶Q
 
Theoremmulnqf 9524 Domain of multiplication on positive fractions. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
·Q :(Q × Q)⟶Q
 
Theoremaddcompq 9525 Addition of positive fractions is commutative. (Contributed by NM, 30-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
(𝐴 +pQ 𝐵) = (𝐵 +pQ 𝐴)
 
Theoremaddcomnq 9526 Addition of positive fractions is commutative. (Contributed by NM, 30-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
(𝐴 +Q 𝐵) = (𝐵 +Q 𝐴)
 
Theoremmulcompq 9527 Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
(𝐴 ·pQ 𝐵) = (𝐵 ·pQ 𝐴)
 
Theoremmulcomnq 9528 Multiplication of positive fractions is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
(𝐴 ·Q 𝐵) = (𝐵 ·Q 𝐴)
 
Theoremadderpqlem 9529 Lemma for adderpq 9531. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ (𝐴 +pQ 𝐶) ~Q (𝐵 +pQ 𝐶)))
 
Theoremmulerpqlem 9530 Lemma for mulerpq 9532. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ (𝐴 ·pQ 𝐶) ~Q (𝐵 ·pQ 𝐶)))
 
Theoremadderpq 9531 Addition is compatible with the equivalence relation. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
(([Q]‘𝐴) +Q ([Q]‘𝐵)) = ([Q]‘(𝐴 +pQ 𝐵))
 
Theoremmulerpq 9532 Multiplication is compatible with the equivalence relation. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
(([Q]‘𝐴) ·Q ([Q]‘𝐵)) = ([Q]‘(𝐴 ·pQ 𝐵))
 
Theoremaddassnq 9533 Addition of positive fractions is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
((𝐴 +Q 𝐵) +Q 𝐶) = (𝐴 +Q (𝐵 +Q 𝐶))
 
Theoremmulassnq 9534 Multiplication of positive fractions is associative. (Contributed by NM, 1-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
((𝐴 ·Q 𝐵) ·Q 𝐶) = (𝐴 ·Q (𝐵 ·Q 𝐶))
 
Theoremmulcanenq 9535 Lemma for distributive law: cancellation of common factor. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
((𝐴N𝐵N𝐶N) → ⟨(𝐴 ·N 𝐵), (𝐴 ·N 𝐶)⟩ ~Q𝐵, 𝐶⟩)
 
Theoremdistrnq 9536 Multiplication of positive fractions is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
(𝐴 ·Q (𝐵 +Q 𝐶)) = ((𝐴 ·Q 𝐵) +Q (𝐴 ·Q 𝐶))
 
Theorem1nqenq 9537 The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
(𝐴N → 1Q ~Q𝐴, 𝐴⟩)
 
Theoremmulidnq 9538 Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.)
(𝐴Q → (𝐴 ·Q 1Q) = 𝐴)
 
Theoremrecmulnq 9539 Relationship between reciprocal and multiplication on positive fractions. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
(𝐴Q → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))
 
Theoremrecidnq 9540 A positive fraction times its reciprocal is 1. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
(𝐴Q → (𝐴 ·Q (*Q𝐴)) = 1Q)
 
Theoremrecclnq 9541 Closure law for positive fraction reciprocal. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
(𝐴Q → (*Q𝐴) ∈ Q)
 
Theoremrecrecnq 9542 Reciprocal of reciprocal of positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.) (New usage is discouraged.)
(𝐴Q → (*Q‘(*Q𝐴)) = 𝐴)
 
Theoremdmrecnq 9543 Domain of reciprocal on positive fractions. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
dom *Q = Q
 
Theoremltsonq 9544 'Less than' is a strict ordering on positive fractions. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 4-May-2013.) (New usage is discouraged.)
<Q Or Q
 
Theoremlterpq 9545 Compatibility of ordering on equivalent fractions. (Contributed by Mario Carneiro, 9-May-2013.) (New usage is discouraged.)
(𝐴 <pQ 𝐵 ↔ ([Q]‘𝐴) <Q ([Q]‘𝐵))
 
Theoremltanq 9546 Ordering property of addition for positive fractions. Proposition 9-2.6(ii) of [Gleason] p. 120. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
(𝐶Q → (𝐴 <Q 𝐵 ↔ (𝐶 +Q 𝐴) <Q (𝐶 +Q 𝐵)))
 
Theoremltmnq 9547 Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
(𝐶Q → (𝐴 <Q 𝐵 ↔ (𝐶 ·Q 𝐴) <Q (𝐶 ·Q 𝐵)))
 
Theorem1lt2nq 9548 One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
1Q <Q (1Q +Q 1Q)
 
Theoremltaddnq 9549 The sum of two fractions is greater than one of them. (Contributed by NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
((𝐴Q𝐵Q) → 𝐴 <Q (𝐴 +Q 𝐵))
 
Theoremltexnq 9550* Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by NM, 24-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
(𝐵Q → (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
 
Theoremhalfnq 9551* One-half of any positive fraction exists. Lemma for Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 16-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
(𝐴Q → ∃𝑥(𝑥 +Q 𝑥) = 𝐴)
 
Theoremnsmallnq 9552* The is no smallest positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
(𝐴Q → ∃𝑥 𝑥 <Q 𝐴)
 
Theoremltbtwnnq 9553* There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 17-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
(𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 <Q 𝑥𝑥 <Q 𝐵))
 
Theoremltrnq 9554 Ordering property of reciprocal for positive fractions. Proposition 9-2.6(iv) of [Gleason] p. 120. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
(𝐴 <Q 𝐵 ↔ (*Q𝐵) <Q (*Q𝐴))
 
Theoremarchnq 9555* For any fraction, there is an integer that is greater than it. This is also known as the "archimedean property". (Contributed by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
(𝐴Q → ∃𝑥N 𝐴 <Q𝑥, 1𝑜⟩)
 
Definitiondf-np 9556* Define the set of positive reals. A "Dedekind cut" is a partition of the positive rational numbers into two classes such that all the numbers of one class are less than all the numbers of the other. A positive real is defined as the lower class of a Dedekind cut. Definition 9-3.1 of [Gleason] p. 121. (Note: This is a "temporary" definition used in the construction of complex numbers df-c 9695, and is intended to be used only by the construction.) (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
P = {𝑥 ∣ ((∅ ⊊ 𝑥𝑥Q) ∧ ∀𝑦𝑥 (∀𝑧(𝑧 <Q 𝑦𝑧𝑥) ∧ ∃𝑧𝑥 𝑦 <Q 𝑧))}
 
Definitiondf-1p 9557 Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 9695, and is intended to be used only by the construction. Definition of [Gleason] p. 122. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
1P = {𝑥𝑥 <Q 1Q}
 
Definitiondf-plp 9558* Define addition on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 9695, and is intended to be used only by the construction. From Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
+P = (𝑥P, 𝑦P ↦ {𝑤 ∣ ∃𝑣𝑥𝑢𝑦 𝑤 = (𝑣 +Q 𝑢)})
 
Definitiondf-mp 9559* Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 9695, and is intended to be used only by the construction. From Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
·P = (𝑥P, 𝑦P ↦ {𝑤 ∣ ∃𝑣𝑥𝑢𝑦 𝑤 = (𝑣 ·Q 𝑢)})
 
Definitiondf-ltp 9560* Define ordering on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 9695, and is intended to be used only by the construction. From Proposition 9-3.2 of [Gleason] p. 122. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
<P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ 𝑥𝑦)}
 
Theoremnpex 9561 The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
P ∈ V
 
Theoremelnp 9562* Membership in positive reals. (Contributed by NM, 16-Feb-1996.) (New usage is discouraged.)
(𝐴P ↔ ((∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
 
Theoremelnpi 9563* Membership in positive reals. (Contributed by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
(𝐴P ↔ ((𝐴 ∈ V ∧ ∅ ⊊ 𝐴𝐴Q) ∧ ∀𝑥𝐴 (∀𝑦(𝑦 <Q 𝑥𝑦𝐴) ∧ ∃𝑦𝐴 𝑥 <Q 𝑦)))
 
Theoremprn0 9564 A positive real is not empty. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
(𝐴P𝐴 ≠ ∅)
 
Theoremprpssnq 9565 A positive real is a subset of the positive fractions. (Contributed by NM, 29-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
(𝐴P𝐴Q)
 
Theoremelprnq 9566 A positive real is a set of positive fractions. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
((𝐴P𝐵𝐴) → 𝐵Q)
 
Theorem0npr 9567 The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) (New usage is discouraged.)
¬ ∅ ∈ P
 
Theoremprcdnq 9568 A positive real is closed downwards under the positive fractions. Definition 9-3.1 (ii) of [Gleason] p. 121. (Contributed by NM, 25-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
((𝐴P𝐵𝐴) → (𝐶 <Q 𝐵𝐶𝐴))
 
Theoremprub 9569 A positive fraction not in a positive real is an upper bound. Remark (1) of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
(((𝐴P𝐵𝐴) ∧ 𝐶Q) → (¬ 𝐶𝐴𝐵 <Q 𝐶))
 
Theoremprnmax 9570* A positive real has no largest member. Definition 9-3.1(iii) of [Gleason] p. 121. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
((𝐴P𝐵𝐴) → ∃𝑥𝐴 𝐵 <Q 𝑥)
 
Theoremnpomex 9571 A simplifying observation, and an indication of why any attempt to develop a theory of the real numbers without the Axiom of Infinity is doomed to failure: since every member of P is an infinite set, the negation of Infinity implies that P, and hence , is empty. (Note that this proof, which used the fact that Dedekind cuts have no maximum, could just as well have used that they have no minimum, since they are downward-closed by prcdnq 9568 and nsmallnq 9552). (Contributed by Mario Carneiro, 11-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) (New usage is discouraged.)
(𝐴P → ω ∈ V)
 
Theoremprnmadd 9572* A positive real has no largest member. Addition version. (Contributed by NM, 7-Apr-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
((𝐴P𝐵𝐴) → ∃𝑥(𝐵 +Q 𝑥) ∈ 𝐴)
 
Theoremltrelpr 9573 Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
<P ⊆ (P × P)
 
Theoremgenpv 9574* Value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)})
 
Theoremgenpelv 9575* Membership in value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺)))
 
Theoremgenpprecl 9576* Pre-closure law for general operation on positive reals. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → ((𝐶𝐴𝐷𝐵) → (𝐶𝐺𝐷) ∈ (𝐴𝐹𝐵)))
 
Theoremgenpdm 9577* Domain of general operation on positive reals. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       dom 𝐹 = (P × P)
 
Theoremgenpn0 9578* The result of an operation on positive reals is not empty. (Contributed by NM, 28-Feb-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → ∅ ⊊ (𝐴𝐹𝐵))
 
Theoremgenpss 9579* The result of an operation on positive reals is a subset of the positive fractions. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)       ((𝐴P𝐵P) → (𝐴𝐹𝐵) ⊆ Q)
 
Theoremgenpnnp 9580* The result of an operation on positive reals is different from the set of positive fractions. (Contributed by NM, 29-Feb-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   (𝑧Q → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))    &   (𝑥𝐺𝑦) = (𝑦𝐺𝑥)       ((𝐴P𝐵P) → ¬ (𝐴𝐹𝐵) = Q)
 
Theoremgenpcd 9581* Downward closure of an operation on positive reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔𝐺) → 𝑥 ∈ (𝐴𝐹𝐵)))       ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) → (𝑥 <Q 𝑓𝑥 ∈ (𝐴𝐹𝐵))))
 
Theoremgenpnmax 9582* An operation on positive reals has no largest member. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   (𝑣Q → (𝑧 <Q 𝑤 ↔ (𝑣𝐺𝑧) <Q (𝑣𝐺𝑤)))    &   (𝑧𝐺𝑤) = (𝑤𝐺𝑧)       ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))
 
Theoremgenpcl 9583* Closure of an operation on reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   (Q → (𝑓 <Q 𝑔 ↔ (𝐺𝑓) <Q (𝐺𝑔)))    &   (𝑥𝐺𝑦) = (𝑦𝐺𝑥)    &   ((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔𝐺) → 𝑥 ∈ (𝐴𝐹𝐵)))       ((𝐴P𝐵P) → (𝐴𝐹𝐵) ∈ P)
 
Theoremgenpass 9584* Associativity of an operation on reals. (Contributed by NM, 18-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})    &   ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)    &   dom 𝐹 = (P × P)    &   ((𝑓P𝑔P) → (𝑓𝐹𝑔) ∈ P)    &   ((𝑓𝐺𝑔)𝐺) = (𝑓𝐺(𝑔𝐺))       ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))
 
Theoremplpv 9585* Value of addition on positive reals. (Contributed by NM, 28-Feb-1996.) (New usage is discouraged.)
((𝐴P𝐵P) → (𝐴 +P 𝐵) = {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦 +Q 𝑧)})
 
Theoremmpv 9586* Value of multiplication on positive reals. (Contributed by NM, 28-Feb-1996.) (New usage is discouraged.)
((𝐴P𝐵P) → (𝐴 ·P 𝐵) = {𝑥 ∣ ∃𝑦𝐴𝑧𝐵 𝑥 = (𝑦 ·Q 𝑧)})
 
Theoremdmplp 9587 Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
dom +P = (P × P)
 
Theoremdmmp 9588 Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
dom ·P = (P × P)
 
Theoremnqpr 9589* The canonical embedding of the rationals into the reals. (Contributed by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
(𝐴Q → {𝑥𝑥 <Q 𝐴} ∈ P)
 
Theorem1pr 9590 The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
1PP
 
Theoremaddclprlem1 9591 Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
(((𝐴P𝑔𝐴) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ))) ·Q 𝑔) ∈ 𝐴))
 
Theoremaddclprlem2 9592* Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 +Q ) → 𝑥 ∈ (𝐴 +P 𝐵)))
 
Theoremaddclpr 9593 Closure of addition on positive reals. First statement of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
 
Theoremmulclprlem 9594* Lemma to prove downward closure in positive real multiplication. Part of proof of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 14-Mar-1996.) (New usage is discouraged.)
((((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) ∧ 𝑥Q) → (𝑥 <Q (𝑔 ·Q ) → 𝑥 ∈ (𝐴 ·P 𝐵)))
 
Theoremmulclpr 9595 Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
 
Theoremaddcompr 9596 Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by NM, 19-Nov-1995.) (New usage is discouraged.)
(𝐴 +P 𝐵) = (𝐵 +P 𝐴)
 
Theoremaddasspr 9597 Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (Contributed by NM, 18-Mar-1996.) (New usage is discouraged.)
((𝐴 +P 𝐵) +P 𝐶) = (𝐴 +P (𝐵 +P 𝐶))
 
Theoremmulcompr 9598 Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by NM, 19-Nov-1995.) (New usage is discouraged.)
(𝐴 ·P 𝐵) = (𝐵 ·P 𝐴)
 
Theoremmulasspr 9599 Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124. (Contributed by NM, 18-Mar-1996.) (New usage is discouraged.)
((𝐴 ·P 𝐵) ·P 𝐶) = (𝐴 ·P (𝐵 ·P 𝐶))
 
Theoremdistrlem1pr 9600 Lemma for distributive law for positive reals. (Contributed by NM, 1-May-1996.) (Revised by Mario Carneiro, 13-Jun-2013.) (New usage is discouraged.)
((𝐴P𝐵P𝐶P) → (𝐴 ·P (𝐵 +P 𝐶)) ⊆ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))
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