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Theorem List for Metamath Proof Explorer - 9901-10000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltexprlem6 9901* 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 9902* 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 9903* 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 9904 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 9905 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 9906 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 9907* 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 9908* 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 9909* 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 9910* 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 9911* 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 9912* 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 9913* 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 9914* 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 9915* Define equivalence relation for signed 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 9-4.1 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 9916 Define class of signed 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 9-4.2 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
R = ((P × P) / ~R )

Definitiondf-plr 9917* Define addition on signed 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 9-4.3 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 9918* Define multiplication on signed 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 9-4.3 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 9919* Define ordering relation on signed 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 9-4.4 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 9920 Define signed real constant 0. 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 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
0R = [⟨1P, 1P⟩] ~R

Definitiondf-1r 9921 Define signed 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. 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 9922 Define signed 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. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
-1R = [⟨1P, (1P +P 1P)⟩] ~R

Theoremenrbreq 9923 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 9924 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 9925 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 9926 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
~R ∈ V

Theoremltrelsr 9927 Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
<R ⊆ (R × R)

Theoremaddcmpblnr 9928 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 9929 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 9930 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 9931* Decomposing signed reals into positive reals. Lemma for addsrpr 9934 and mulsrpr 9935. (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 9932* 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 9933* 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 9934 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 9935 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 9936 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 9937 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 9938 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 9939 The constant 0R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
0RR

Theorem1sr 9940 The constant 1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
1RR

Theoremm1r 9941 The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
-1RR

Theoremaddclsr 9942 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
((𝐴R𝐵R) → (𝐴 +R 𝐵) ∈ R)

Theoremmulclsr 9943 Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
((𝐴R𝐵R) → (𝐴 ·R 𝐵) ∈ R)

Theoremdmaddsr 9944 Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
dom +R = (R × R)

Theoremdmmulsr 9945 Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
dom ·R = (R × R)

Theoremaddcomsr 9946 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 9947 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 9948 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 9949 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 9950 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 9951 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
(-1R +R 1R) = 0R

Theoremm1m1sr 9952 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 9953 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.)
<R Or R

Theorem0lt1sr 9954 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
0R <R 1R

Theorem1ne0sr 9955 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
¬ 1R = 0R

Theorem0idsr 9956 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 9957 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
(𝐴R → (𝐴 ·R 1R) = 𝐴)

Theorem00sr 9958 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
(𝐴R → (𝐴 ·R 0R) = 0R)

Theoremltasr 9959 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 9960 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 9961* 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 9962* 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 9963 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 9964 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 9965 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 9966* 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 9967 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 9968 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 9969* 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 9970* 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 9971* 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 9972 Class of complex numbers.
class

Syntaxcr 9973 Class of real numbers.
class

Syntaxcc0 9974 Extend class notation to include the complex number 0.
class 0

Syntaxc1 9975 Extend class notation to include the complex number 1.
class 1

Syntaxci 9976 Extend class notation to include the complex number i.
class i

class +

Syntaxcltrr 9978 'Less than' predicate (defined over real subset of complex numbers).
class <

Syntaxcmul 9979 Multiplication on complex numbers. The token · is a center dot.
class ·

Definitiondf-c 9980 Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 10007. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
ℂ = (R × R)

Definitiondf-0 9981 Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
0 = ⟨0R, 0R

Definitiondf-1 9982 Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
1 = ⟨1R, 0R

Definitiondf-i 9983 Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
i = ⟨0R, 1R

Definitiondf-r 9984 Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
ℝ = (R × {0R})

Definitiondf-add 9985* Define addition over complex numbers. (Contributed by NM, 28-May-1995.) (New usage is discouraged.)
+ = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨(𝑤 +R 𝑢), (𝑣 +R 𝑓)⟩))}

Definitiondf-mul 9986* Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
· = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩))}

Definitiondf-lt 9987* Define 'less than' on the real subset of complex numbers. Proofs should typically use < instead; see df-ltxr 10117. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
< = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧𝑤((𝑥 = ⟨𝑧, 0R⟩ ∧ 𝑦 = ⟨𝑤, 0R⟩) ∧ 𝑧 <R 𝑤))}

Theoremopelcn 9988 Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
(⟨𝐴, 𝐵⟩ ∈ ℂ ↔ (𝐴R𝐵R))

Theoremopelreal 9989 Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
(⟨𝐴, 0R⟩ ∈ ℝ ↔ 𝐴R)

Theoremelreal 9990* Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
(𝐴 ∈ ℝ ↔ ∃𝑥R𝑥, 0R⟩ = 𝐴)

Theoremelreal2 9991 Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
(𝐴 ∈ ℝ ↔ ((1st𝐴) ∈ R𝐴 = ⟨(1st𝐴), 0R⟩))

Theorem0ncn 9992 The empty set is not a complex number. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
¬ ∅ ∈ ℂ

Theoremltrelre 9993 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
< ⊆ (ℝ × ℝ)

Theoremaddcnsr 9994 Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) (New usage is discouraged.)
(((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → (⟨𝐴, 𝐵⟩ + ⟨𝐶, 𝐷⟩) = ⟨(𝐴 +R 𝐶), (𝐵 +R 𝐷)⟩)

Theoremmulcnsr 9995 Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
(((𝐴R𝐵R) ∧ (𝐶R𝐷R)) → (⟨𝐴, 𝐵⟩ · ⟨𝐶, 𝐷⟩) = ⟨((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))⟩)

Theoremeqresr 9996 Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
𝐴 ∈ V       (⟨𝐴, 0R⟩ = ⟨𝐵, 0R⟩ ↔ 𝐴 = 𝐵)

Theoremaddresr 9997 Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
((𝐴R𝐵R) → (⟨𝐴, 0R⟩ + ⟨𝐵, 0R⟩) = ⟨(𝐴 +R 𝐵), 0R⟩)

Theoremmulresr 9998 Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
((𝐴R𝐵R) → (⟨𝐴, 0R⟩ · ⟨𝐵, 0R⟩) = ⟨(𝐴 ·R 𝐵), 0R⟩)

Theoremltresr 9999 Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
(⟨𝐴, 0R⟩ <𝐵, 0R⟩ ↔ 𝐴 <R 𝐵)

Theoremltresr2 10000 Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (1st𝐴) <R (1st𝐵)))

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