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Theorem List for Metamath Proof Explorer - 8901-9000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremaddasspr 8901 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.)

Theoremmulcompr 8902 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.)

Theoremmulasspr 8903 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.)

Theoremdistrlem1pr 8904 Lemma for distributive law for positive reals. (Contributed by NM, 1-May-1996.) (Revised by Mario Carneiro, 13-Jun-2013.) (New usage is discouraged.)

Theoremdistrlem4pr 8905* Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)

Theoremdistrlem5pr 8906 Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)

Theoremdistrpr 8907 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.)

Theorem1idpr 8908 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.)

Theoremltprord 8909 Positive real 'less than' in terms of proper subset. (Contributed by NM, 20-Feb-1996.) (New usage is discouraged.)

Theorempsslinpr 8910 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.)

Theoremltsopr 8911 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.)

Theoremprlem934 8912* 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.)

Theoremltaddpr 8913 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.)

Theoremltaddpr2 8914 The sum of two positive reals is greater than one of them. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)

Theoremltexprlem1 8915* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 3-Apr-1996.) (New usage is discouraged.)

Theoremltexprlem2 8916* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 3-Apr-1996.) (New usage is discouraged.)

Theoremltexprlem3 8917* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)

Theoremltexprlem4 8918* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)

Theoremltexprlem5 8919* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)

Theoremltexprlem6 8920* 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.)

Theoremltexprlem7 8921* 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.)

Theoremltexpri 8922* 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.)

Theoremltaprlem 8923 Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)

Theoremltapr 8924 Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)

Theoremaddcanpr 8925 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.)

Theoremprlem936 8926* 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.)

Theoremreclem2pr 8927* 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.)

Theoremreclem3pr 8928* 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.)

Theoremreclem4pr 8929* Lemma for Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (New usage is discouraged.)

Theoremrecexpr 8930* 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.)

Theoremsuplem1pr 8931* The union of a non-empty, 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.)

Theoremsuplem2pr 8932* 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.)

Theoremsupexpr 8933* The union of a non-empty, 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.)

Definitiondf-plpr 8934* Define pre-addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8998, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)

Definitiondf-mpr 8935* Define pre-multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8998, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)

Definitiondf-enr 8936* Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8998, 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.)

Definitiondf-nr 8937 Define class of signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8998, 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.)

Definitiondf-plr 8938* Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8998, 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.)

Definitiondf-mr 8939* Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8998, 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.)

Definitiondf-ltr 8940* Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8998, 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.)

Definitiondf-0r 8941 Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers df-c 8998, 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.)

Definitiondf-1r 8942 Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 8998, 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.)

Definitiondf-m1r 8943 Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers df-c 8998, and is intended to be used only by the construction. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)

Theoremenrbreq 8944 Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)

Theoremenrer 8945 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.)

Theoremenreceq 8946 Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) (New usage is discouraged.)

Theoremenrex 8947 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)

Theoremltrelsr 8948 Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)

Theoremaddcmpblnr 8949 Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)

Theoremmulcmpblnrlem 8950 Lemma used in lemma showing compatibility of multiplication. (Contributed by NM, 4-Sep-1995.) (New usage is discouraged.)

Theoremmulcmpblnr 8951 Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) (New usage is discouraged.)

Theoremaddsrpr 8952 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.)

Theoremmulsrpr 8953 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.)

Theoremltsrpr 8954 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.)

Theoremgt0srpr 8955 Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)

Theorem0nsr 8956 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.)

Theorem0r 8957 The constant is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)

Theorem1sr 8958 The constant is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)

Theoremm1r 8959 The constant is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)

Theoremaddclsr 8960 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)

Theoremmulclsr 8961 Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)

Theoremdmaddsr 8962 Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)

Theoremdmmulsr 8963 Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)

Theoremaddcomsr 8964 Addition of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)

Theoremaddasssr 8965 Addition of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)

Theoremmulcomsr 8966 Multiplication of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)

Theoremmulasssr 8967 Multiplication of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)

Theoremdistrsr 8968 Multiplication of signed reals is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)

Theoremm1p1sr 8969 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)

Theoremm1m1sr 8970 Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)

Theoremltsosr 8971 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.)

Theorem0lt1sr 8972 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)

Theorem1ne0sr 8973 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)

Theorem0idsr 8974 The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)

Theorem1idsr 8975 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)

Theorem00sr 8976 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)

Theoremltasr 8977 Ordering property of addition. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)

Theorempn0sr 8978 A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)

Theoremnegexsr 8979* 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.)

Theoremrecexsrlem 8980* 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.)

Theoremaddgt0sr 8981 The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)

Theoremmulgt0sr 8982 The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)

Theoremsqgt0sr 8983 The square of a nonzero signed real is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)

Theoremrecexsr 8984* 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.)

Theoremmappsrpr 8985 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.)

Theoremltpsrpr 8986 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.)

Theoremmap2psrpr 8987* Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)

Theoremsupsrlem 8988* Lemma for supremum theorem. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)

Theoremsupsr 8989* A non-empty, bounded set of signed reals has a supremum. (Cotributed by Mario Carneiro, 15-Jun-2013.) (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)

Syntaxcc 8990 Class of complex numbers.

Syntaxcr 8991 Class of real numbers.

Syntaxcc0 8992 Extend class notation to include the complex number 0.

Syntaxc1 8993 Extend class notation to include the complex number 1.

Syntaxci 8994 Extend class notation to include the complex number i.

Syntaxcltrr 8996 'Less than' predicate (defined over real subset of complex numbers).

Syntaxcmul 8997 Multiplication on complex numbers. The token is a center dot.

Definitiondf-c 8998 Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 9025. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)

Definitiondf-0 8999 Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)

Definitiondf-1 9000 Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)

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