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Theorem List for Metamath Proof Explorer - 9801-9900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmulpqnq 9801 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 9802 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 9803 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 9804 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 9805 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 9806 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 9807 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 9808 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 9809 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 9810 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 9811 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 9812 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 9813 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 9814 Lemma for adderpq 9816. (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 9815 Lemma for mulerpq 9817. (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 9816 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 9817 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 9818 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 9819 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 9820 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 9821 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 9822 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 9823 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 9824 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 9825 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 9826 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 9827 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 9828 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 9829 '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 9830 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 9831 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 9832 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 9833 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 9834 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 9835* 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 9836* 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 9837* 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 9838* 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 9839 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 9840* 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 9841* 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 9980, 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 9842 Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 9980, 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 9843* Define addition on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 9980, 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 9844* Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 9980, 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 9845* Define ordering on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 9980, 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 9846 The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
P ∈ V

Theoremelnp 9847* 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 9848* 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 9849 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 9850 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 9851 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 9852 The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) (New usage is discouraged.)
¬ ∅ ∈ P

Theoremprcdnq 9853 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 9854 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 9855* 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 9856 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 9853 and nsmallnq 9837). (Contributed by Mario Carneiro, 11-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) (New usage is discouraged.)
(𝐴P → ω ∈ V)

Theoremprnmadd 9857* 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 9858 Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
<P ⊆ (P × P)

Theoremgenpv 9859* 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 9860* 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 9861* 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 9862* 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 9863* 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 9864* 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 9865* 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 9866* 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 9867* 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 9868* 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 9869* 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 9870* 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 9871* 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 9872 Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
dom +P = (P × P)

Theoremdmmp 9873 Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
dom ·P = (P × P)

Theoremnqpr 9874* 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 9875 The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
1PP

Theoremaddclprlem1 9876 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 9877* 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 9878 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 9879* 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 9880 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 9881 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 9882 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 9883 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 9884 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 9885 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)))

Theoremdistrlem4pr 9886* Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
(((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))

Theoremdistrlem5pr 9887 Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
((𝐴P𝐵P𝐶P) → ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ⊆ (𝐴 ·P (𝐵 +P 𝐶)))

Theoremdistrpr 9888 Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
(𝐴 ·P (𝐵 +P 𝐶)) = ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))

Theorem1idpr 9889 1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124. (Contributed by NM, 2-Apr-1996.) (New usage is discouraged.)
(𝐴P → (𝐴 ·P 1P) = 𝐴)

Theoremltprord 9890 Positive real 'less than' in terms of proper subset. (Contributed by NM, 20-Feb-1996.) (New usage is discouraged.)
((𝐴P𝐵P) → (𝐴<P 𝐵𝐴𝐵))

Theorempsslinpr 9891 Proper subset is a linear ordering on positive reals. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
((𝐴P𝐵P) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))

Theoremltsopr 9892 Positive real 'less than' is a strict ordering. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
<P Or P

Theoremprlem934 9893* Lemma 9-3.4 of [Gleason] p. 122. (Contributed by NM, 25-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
𝐵 ∈ V       (𝐴P → ∃𝑥𝐴 ¬ (𝑥 +Q 𝐵) ∈ 𝐴)

Theoremltaddpr 9894 The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123. (Contributed by NM, 26-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
((𝐴P𝐵P) → 𝐴<P (𝐴 +P 𝐵))

Theoremltaddpr2 9895 The sum of two positive reals is greater than one of them. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
(𝐶P → ((𝐴 +P 𝐵) = 𝐶𝐴<P 𝐶))

Theoremltexprlem1 9896* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 3-Apr-1996.) (New usage is discouraged.)
𝐶 = {𝑥 ∣ ∃𝑦𝑦𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)}       (𝐵P → (𝐴𝐵𝐶 ≠ ∅))

Theoremltexprlem2 9897* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 3-Apr-1996.) (New usage is discouraged.)
𝐶 = {𝑥 ∣ ∃𝑦𝑦𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)}       (𝐵P𝐶Q)

Theoremltexprlem3 9898* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)
𝐶 = {𝑥 ∣ ∃𝑦𝑦𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)}       (𝐵P → (𝑥𝐶 → ∀𝑧(𝑧 <Q 𝑥𝑧𝐶)))

Theoremltexprlem4 9899* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)
𝐶 = {𝑥 ∣ ∃𝑦𝑦𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)}       (𝐵P → (𝑥𝐶 → ∃𝑧(𝑧𝐶𝑥 <Q 𝑧)))

Theoremltexprlem5 9900* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)
𝐶 = {𝑥 ∣ ∃𝑦𝑦𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)}       ((𝐵P𝐴𝐵) → 𝐶P)

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