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Theorem List for Metamath Proof Explorer - 8501-8600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremprnmadd 8501* 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.)

Theoremltrelpr 8502 Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)

Theoremgenpv 8503* 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.)

Theoremgenpelv 8504* 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.)

Theoremgenpprecl 8505* 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.)

Theoremgenpdm 8506* Domain of general operation on positive reals. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)

Theoremgenpn0 8507* 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.)

Theoremgenpss 8508* 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.)

Theoremgenpnnp 8509* 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.)

Theoremgenpcd 8510* 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.)

Theoremgenpnmax 8511* 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.)

Theoremgenpcl 8512* Closure of an operation on reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)

Theoremgenpass 8513* Associativity of an operation on reals. (Contributed by NM, 18-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)

Theoremplpv 8514* Value of addition on positive reals. (Contributed by NM, 28-Feb-1996.) (New usage is discouraged.)

Theoremmpv 8515* Value of multiplication on positive reals. (Contributed by NM, 28-Feb-1996.) (New usage is discouraged.)

Theoremdmplp 8516 Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)

Theoremdmmp 8517 Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)

Theoremnqpr 8518* The canonical embedding of the rationals into the reals. (Contributed by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)

Theorem1pr 8519 The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)

Theoremaddclprlem1 8520 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.)

Theoremaddclprlem2 8521* 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.)

Theoremaddclpr 8522 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.)

Theoremmulclprlem 8523* 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.)

Theoremmulclpr 8524 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.)

Theoremaddcompr 8525 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.)

Theoremaddasspr 8526 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 8527 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 8528 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 8529 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 8530* 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 8531 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 8532 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 8533 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 8534 Positive real 'less than' in terms of proper subset. (Contributed by NM, 20-Feb-1996.) (New usage is discouraged.)

Theorempsslinpr 8535 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 8536 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 8537* 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 8538 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 8539 The sum of two positive reals is greater than one of them. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)

Theoremltexprlem1 8540* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 3-Apr-1996.) (New usage is discouraged.)

Theoremltexprlem2 8541* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 3-Apr-1996.) (New usage is discouraged.)

Theoremltexprlem3 8542* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)

Theoremltexprlem4 8543* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)

Theoremltexprlem5 8544* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)

Theoremltexprlem6 8545* 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 8546* 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 8547* 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 8548 Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)

Theoremltapr 8549 Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)

Theoremaddcanpr 8550 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 8551* 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 8552* 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 8553* 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 8554* Lemma for Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (New usage is discouraged.)

Theoremrecexpr 8555* 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 8556* 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 8557* The union of a set of positive reals (if a positive real) is its supremum (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 8558* 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 8559* Define pre-addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8560* Define pre-multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8561* Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8562 Define class of signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8563* Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8564* Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8565* Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8566 Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8567 Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8568 Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers df-c 8623, and is intended to be used only by the construction. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)

Theoremenrbreq 8569 Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)

Theoremenrer 8570 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 8571 Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) (New usage is discouraged.)

Theoremenrex 8572 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)

Theoremltrelsr 8573 Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)

Theoremaddcmpblnr 8574 Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)

Theoremmulcmpblnrlem 8575 Lemma used in lemma showing compatibility of multiplication. (Contributed by NM, 4-Sep-1995.) (New usage is discouraged.)

Theoremmulcmpblnr 8576 Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) (New usage is discouraged.)

Theoremaddsrpr 8577 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 8578 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 8579 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 8580 Greater then zero in terms of positive reals. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)

Theorem0nsr 8581 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 8582 The constant is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)

Theorem1sr 8583 The constant is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)

Theoremm1r 8584 The constant is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)

Theoremaddclsr 8585 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)

Theoremmulclsr 8586 Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)

Theoremdmaddsr 8587 Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)

Theoremdmmulsr 8588 Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)

Theoremaddcomsr 8589 Addition of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)

Theoremaddasssr 8590 Addition of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)

Theoremmulcomsr 8591 Multiplication of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)

Theoremmulasssr 8592 Multiplication of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)

Theoremdistrsr 8593 Multiplication of signed reals is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)

Theoremm1p1sr 8594 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)

Theoremm1m1sr 8595 Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)

Theoremltsosr 8596 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.)

Theorem0lt1sr 8597 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)

Theorem1ne0sr 8598 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)

Theorem0idsr 8599 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 8600 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)

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