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Theorem List for Metamath Proof Explorer - 10401-10500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprpssnq 10401 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 10402 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 10403 The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) (New usage is discouraged.)
¬ ∅ ∈ P
 
Theoremprcdnq 10404 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 10405 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 10406* 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 10407 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 10404 and nsmallnq 10388). (Contributed by Mario Carneiro, 11-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) (New usage is discouraged.)
(𝐴P → ω ∈ V)
 
Theoremprnmadd 10408* 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 10409 Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
<P ⊆ (P × P)
 
Theoremgenpv 10410* 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 10411* 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 10412* 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 10413* 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 10414* 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 10415* 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 10416* 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 10417* 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 10418* 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 10419* 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 10420* 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 10421* 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 10422* 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 10423 Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
dom +P = (P × P)
 
Theoremdmmp 10424 Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
dom ·P = (P × P)
 
Theoremnqpr 10425* 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 10426 The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
1PP
 
Theoremaddclprlem1 10427 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 10428* 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 10429 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 10430* 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 10431 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 10432 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 10433 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 10434 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 10435 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 10436 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 10437* 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 10438 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 10439 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 10440 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 10441 Positive real 'less than' in terms of proper subset. (Contributed by NM, 20-Feb-1996.) (New usage is discouraged.)
((𝐴P𝐵P) → (𝐴<P 𝐵𝐴𝐵))
 
Theorempsslinpr 10442 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 10443 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 10444* 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 10445 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 10446 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 10447* 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 10448* 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 10449* 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 10450* 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 10451* 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)
 
Theoremltexprlem6 10452* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
𝐶 = {𝑥 ∣ ∃𝑦𝑦𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)}       (((𝐴P𝐵P) ∧ 𝐴𝐵) → (𝐴 +P 𝐶) ⊆ 𝐵)
 
Theoremltexprlem7 10453* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
𝐶 = {𝑥 ∣ ∃𝑦𝑦𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)}       (((𝐴P𝐵P) ∧ 𝐴𝐵) → 𝐵 ⊆ (𝐴 +P 𝐶))
 
Theoremltexpri 10454* Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
(𝐴<P 𝐵 → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
 
Theoremltaprlem 10455 Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)
(𝐶P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
 
Theoremltapr 10456 Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)
(𝐶P → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
 
Theoremaddcanpr 10457 Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by NM, 9-Apr-1996.) (New usage is discouraged.)
((𝐴P𝐵P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶))
 
Theoremprlem936 10458* Lemma 9-3.6 of [Gleason] p. 124. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
((𝐴P ∧ 1Q <Q 𝐵) → ∃𝑥𝐴 ¬ (𝑥 ·Q 𝐵) ∈ 𝐴)
 
Theoremreclem2pr 10459* Lemma for Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
𝐵 = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}       (𝐴P𝐵P)
 
Theoremreclem3pr 10460* Lemma for Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
𝐵 = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}       (𝐴P → 1P ⊆ (𝐴 ·P 𝐵))
 
Theoremreclem4pr 10461* Lemma for Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (New usage is discouraged.)
𝐵 = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}       (𝐴P → (𝐴 ·P 𝐵) = 1P)
 
Theoremrecexpr 10462* The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
(𝐴P → ∃𝑥P (𝐴 ·P 𝑥) = 1P)
 
Theoremsuplem1pr 10463* The union of a nonempty, bounded set of positive reals is a positive real. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
((𝐴 ≠ ∅ ∧ ∃𝑥P𝑦𝐴 𝑦<P 𝑥) → 𝐴P)
 
Theoremsuplem2pr 10464* The union of a set of positive reals (if a positive real) is its supremum (the least upper bound). Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
(𝐴P → ((𝑦𝐴 → ¬ 𝐴<P 𝑦) ∧ (𝑦<P 𝐴 → ∃𝑧𝐴 𝑦<P 𝑧)))
 
Theoremsupexpr 10465* The union of a nonempty, bounded set of positive reals has a supremum. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (New usage is discouraged.)
((𝐴 ≠ ∅ ∧ ∃𝑥P𝑦𝐴 𝑦<P 𝑥) → ∃𝑥P (∀𝑦𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦P (𝑦<P 𝑥 → ∃𝑧𝐴 𝑦<P 𝑧)))
 
Definitiondf-enr 10466* Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers df-c 10532, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
~R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (P × P) ∧ 𝑦 ∈ (P × P)) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣)))}
 
Definitiondf-nr 10467 Define class of signed reals. This is a "temporary" set used in the construction of complex numbers df-c 10532, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
R = ((P × P) / ~R )
 
Definitiondf-plr 10468* Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 10532, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
+R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑓)⟩] ~R ))}
 
Definitiondf-mr 10469* Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 10532, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
·R = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = [⟨𝑤, 𝑣⟩] ~R𝑦 = [⟨𝑢, 𝑓⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))⟩] ~R ))}
 
Definitiondf-ltr 10470* Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 10532, and is intended to be used only by the construction. From Proposition 9-4.4 of [Gleason] p. 127. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
<R = {⟨𝑥, 𝑦⟩ ∣ ((𝑥R𝑦R) ∧ ∃𝑧𝑤𝑣𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ~R𝑦 = [⟨𝑣, 𝑢⟩] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))}
 
Definitiondf-0r 10471 Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers df-c 10532, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
0R = [⟨1P, 1P⟩] ~R
 
Definitiondf-1r 10472 Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 10532, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
1R = [⟨(1P +P 1P), 1P⟩] ~R
 
Definitiondf-m1r 10473 Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers df-c 10532, and is intended to be used only by the construction. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
-1R = [⟨1P, (1P +P 1P)⟩] ~R
 
Theoremenrer 10474 The equivalence relation for signed reals is an equivalence relation. Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) (New usage is discouraged.)
~R Er (P × P)
 
Theoremnrex1 10475 The class of signed reals is a set. Note that a shorter proof is possible using qsex 8346 (and not requiring enrer 10474), but it would add a dependency on ax-rep 5182. (Contributed by Mario Carneiro, 17-Nov-2014.) Extract proof from that of axcnex 10558. (Revised by BJ, 4-Feb-2023.) (New usage is discouraged.)
R ∈ V
 
Theoremenrbreq 10476 Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
(((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → (⟨𝐴, 𝐵⟩ ~R𝐶, 𝐷⟩ ↔ (𝐴 +P 𝐷) = (𝐵 +P 𝐶)))
 
Theoremenreceq 10477 Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) (New usage is discouraged.)
(((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R = [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷) = (𝐵 +P 𝐶)))
 
Theoremenrex 10478 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
~R ∈ V
 
Theoremltrelsr 10479 Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
<R ⊆ (R × R)
 
Theoremaddcmpblnr 10480 Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ⟨(𝐴 +P 𝐹), (𝐵 +P 𝐺)⟩ ~R ⟨(𝐶 +P 𝑅), (𝐷 +P 𝑆)⟩))
 
Theoremmulcmpblnrlem 10481 Lemma used in lemma showing compatibility of multiplication. (Contributed by NM, 4-Sep-1995.) (New usage is discouraged.)
(((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆)))))
 
Theoremmulcmpblnr 10482 Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) (New usage is discouraged.)
((((𝐴P𝐵P) ∧ (𝐶P𝐷P)) ∧ ((𝐹P𝐺P) ∧ (𝑅P𝑆P))) → (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ⟨((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)), ((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹))⟩ ~R ⟨((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆)), ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅))⟩))
 
Theoremprsrlem1 10483* Decomposing signed reals into positive reals. Lemma for addsrpr 10486 and mulsrpr 10487. (Contributed by Jim Kingdon, 30-Dec-2019.)
(((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ (𝐴 = [⟨𝑠, 𝑓⟩] ~R𝐵 = [⟨𝑔, ⟩] ~R ))) → ((((𝑤P𝑣P) ∧ (𝑠P𝑓P)) ∧ ((𝑢P𝑡P) ∧ (𝑔PP))) ∧ ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ) = (𝑡 +P 𝑔))))
 
Theoremaddsrmo 10484* There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨(𝑤 +P 𝑢), (𝑣 +P 𝑡)⟩] ~R ))
 
Theoremmulsrmo 10485* There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧𝑤𝑣𝑢𝑡((𝐴 = [⟨𝑤, 𝑣⟩] ~R𝐵 = [⟨𝑢, 𝑡⟩] ~R ) ∧ 𝑧 = [⟨((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))⟩] ~R ))
 
Theoremaddsrpr 10486 Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
(((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R +R [⟨𝐶, 𝐷⟩] ~R ) = [⟨(𝐴 +P 𝐶), (𝐵 +P 𝐷)⟩] ~R )
 
Theoremmulsrpr 10487 Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
(((𝐴P𝐵P) ∧ (𝐶P𝐷P)) → ([⟨𝐴, 𝐵⟩] ~R ·R [⟨𝐶, 𝐷⟩] ~R ) = [⟨((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))⟩] ~R )
 
Theoremltsrpr 10488 Ordering of signed reals in terms of positive reals. (Contributed by NM, 20-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
([⟨𝐴, 𝐵⟩] ~R <R [⟨𝐶, 𝐷⟩] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶))
 
Theoremgt0srpr 10489 Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
(0R <R [⟨𝐴, 𝐵⟩] ~R𝐵<P 𝐴)
 
Theorem0nsr 10490 The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
¬ ∅ ∈ R
 
Theorem0r 10491 The constant 0R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
0RR
 
Theorem1sr 10492 The constant 1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
1RR
 
Theoremm1r 10493 The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
-1RR
 
Theoremaddclsr 10494 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
((𝐴R𝐵R) → (𝐴 +R 𝐵) ∈ R)
 
Theoremmulclsr 10495 Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
((𝐴R𝐵R) → (𝐴 ·R 𝐵) ∈ R)
 
Theoremdmaddsr 10496 Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
dom +R = (R × R)
 
Theoremdmmulsr 10497 Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
dom ·R = (R × R)
 
Theoremaddcomsr 10498 Addition of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
(𝐴 +R 𝐵) = (𝐵 +R 𝐴)
 
Theoremaddasssr 10499 Addition of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
((𝐴 +R 𝐵) +R 𝐶) = (𝐴 +R (𝐵 +R 𝐶))
 
Theoremmulcomsr 10500 Multiplication of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
(𝐴 ·R 𝐵) = (𝐵 ·R 𝐴)
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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