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Theorem List for Metamath Proof Explorer - 9601-9700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmulasspr 9601 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 9602 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 9603* 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 9604 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 9605 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 9606 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 9607 Positive real 'less than' in terms of proper subset. (Contributed by NM, 20-Feb-1996.) (New usage is discouraged.)
((𝐴P𝐵P) → (𝐴<P 𝐵𝐴𝐵))
 
Theorempsslinpr 9608 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 9609 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 9610* 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 9611 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 9612 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 9613* 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 9614* 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 9615* 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 9616* 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 9617* 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 9618* 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 9619* 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 9620* 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 9621 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 9622 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 9623 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 9624* 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 9625* 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 9626* 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 9627* 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 9628* 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 9629* 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 9630* 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 9631* 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 9632* Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9697, 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 9633 Define class of signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9697, 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 9634* Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9697, 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 9635* Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9697, 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 9636* Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9697, 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 9637 Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers df-c 9697, 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 9638 Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 9697, 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 9639 Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers df-c 9697, 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
 
Theoremenrbreq 9640 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)))
 
Theoremenrer 9641 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)
 
Theoremenreceq 9642 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 9643 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
~R ∈ V
 
Theoremltrelsr 9644 Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
<R ⊆ (R × R)
 
Theoremaddcmpblnr 9645 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 9646 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 9647 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 9648* Decomposing signed reals into positive reals. Lemma for addsrpr 9651 and mulsrpr 9652. (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 9649* 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 9650* 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 9651 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 9652 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 9653 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 9654 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 9655 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 9656 The constant 0R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
0RR
 
Theorem1sr 9657 The constant 1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
1RR
 
Theoremm1r 9658 The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
-1RR
 
Theoremaddclsr 9659 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
((𝐴R𝐵R) → (𝐴 +R 𝐵) ∈ R)
 
Theoremmulclsr 9660 Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
((𝐴R𝐵R) → (𝐴 ·R 𝐵) ∈ R)
 
Theoremdmaddsr 9661 Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
dom +R = (R × R)
 
Theoremdmmulsr 9662 Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
dom ·R = (R × R)
 
Theoremaddcomsr 9663 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 9664 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 9665 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)
 
Theoremmulasssr 9666 Multiplication 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))
 
Theoremdistrsr 9667 Multiplication of signed reals is distributive. (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 (𝐴 ·R 𝐶))
 
Theoremm1p1sr 9668 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
(-1R +R 1R) = 0R
 
Theoremm1m1sr 9669 Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
(-1R ·R -1R) = 1R
 
Theoremltsosr 9670 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.)
<R Or R
 
Theorem0lt1sr 9671 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
0R <R 1R
 
Theorem1ne0sr 9672 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
¬ 1R = 0R
 
Theorem0idsr 9673 The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
(𝐴R → (𝐴 +R 0R) = 𝐴)
 
Theorem1idsr 9674 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
(𝐴R → (𝐴 ·R 1R) = 𝐴)
 
Theorem00sr 9675 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
(𝐴R → (𝐴 ·R 0R) = 0R)
 
Theoremltasr 9676 Ordering property of addition. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
(𝐶R → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵)))
 
Theorempn0sr 9677 A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
(𝐴R → (𝐴 +R (𝐴 ·R -1R)) = 0R)
 
Theoremnegexsr 9678* Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
(𝐴R → ∃𝑥R (𝐴 +R 𝑥) = 0R)
 
Theoremrecexsrlem 9679* The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
(0R <R 𝐴 → ∃𝑥R (𝐴 ·R 𝑥) = 1R)
 
Theoremaddgt0sr 9680 The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 +R 𝐵))
 
Theoremmulgt0sr 9681 The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵))
 
Theoremsqgt0sr 9682 The square of a nonzero signed real is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
((𝐴R𝐴 ≠ 0R) → 0R <R (𝐴 ·R 𝐴))
 
Theoremrecexsr 9683* The reciprocal of a nonzero signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
((𝐴R𝐴 ≠ 0R) → ∃𝑥R (𝐴 ·R 𝑥) = 1R)
 
Theoremmappsrpr 9684 Mapping from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
𝐶R       ((𝐶 +R -1R) <R (𝐶 +R [⟨𝐴, 1P⟩] ~R ) ↔ 𝐴P)
 
Theoremltpsrpr 9685 Mapping of order from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
𝐶R       ((𝐶 +R [⟨𝐴, 1P⟩] ~R ) <R (𝐶 +R [⟨𝐵, 1P⟩] ~R ) ↔ 𝐴<P 𝐵)
 
Theoremmap2psrpr 9686* Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
𝐶R       ((𝐶 +R -1R) <R 𝐴 ↔ ∃𝑥P (𝐶 +R [⟨𝑥, 1P⟩] ~R ) = 𝐴)
 
Theoremsupsrlem 9687* Lemma for supremum theorem. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
𝐵 = {𝑤 ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}    &   𝐶R       ((𝐶𝐴 ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))
 
Theoremsupsr 9688* A nonempty, bounded set of signed reals has a supremum. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
((𝐴 ≠ ∅ ∧ ∃𝑥R𝑦𝐴 𝑦 <R 𝑥) → ∃𝑥R (∀𝑦𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦R (𝑦 <R 𝑥 → ∃𝑧𝐴 𝑦 <R 𝑧)))
 
Syntaxcc 9689 Class of complex numbers.
class
 
Syntaxcr 9690 Class of real numbers.
class
 
Syntaxcc0 9691 Extend class notation to include the complex number 0.
class 0
 
Syntaxc1 9692 Extend class notation to include the complex number 1.
class 1
 
Syntaxci 9693 Extend class notation to include the complex number i.
class i
 
Syntaxcaddc 9694 Addition on complex numbers.
class +
 
Syntaxcltrr 9695 'Less than' predicate (defined over real subset of complex numbers).
class <
 
Syntaxcmul 9696 Multiplication on complex numbers. The token · is a center dot.
class ·
 
Definitiondf-c 9697 Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 9724. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
ℂ = (R × R)
 
Definitiondf-0 9698 Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
0 = ⟨0R, 0R
 
Definitiondf-1 9699 Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
1 = ⟨1R, 0R
 
Definitiondf-i 9700 Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
i = ⟨0R, 1R
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