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Theorem List for Metamath Proof Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmulcl 8701 Alias for ax-mulcl 8679, for naming consistency with mulcli 8722. (Contributed by NM, 10-Mar-2008.)

Theoremremulcl 8702 Alias for ax-mulrcl 8680, for naming consistency with remulcli 8731. (Contributed by NM, 10-Mar-2008.)

Theoremmulcom 8703 Alias for ax-mulcom 8681, for naming consistency with mulcomi 8723. (Contributed by NM, 10-Mar-2008.)

Theoremmulass 8705 Alias for ax-mulass 8683, for naming consistency with mulassi 8726. (Contributed by NM, 10-Mar-2008.)

Theoremadddi 8706 Alias for ax-distr 8684, for naming consistency with adddii 8727. (Contributed by NM, 10-Mar-2008.)

Theoremrecn 8707 A real number is a complex number. (Contributed by NM, 10-Aug-1999.)

Theoremreex 8708 The real numbers form a set. See also reexALT 10227. (Contributed by Mario Carneiro, 17-Nov-2014.)

Theoremelimne0 8709 Hypothesis for weak deduction theorem to eliminate . (Contributed by NM, 15-May-1999.)

Theoremadddir 8710 Distributive law for complex numbers. (Contributed by NM, 10-Oct-2004.)

Theorem0cn 8711 0 is a complex number. (Contributed by NM, 19-Feb-2005.)

Theoremc0ex 8712 0 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theorem1ex 8713 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)

Theoremmulid1 8714 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmulid2 8715 Identity law for multiplication. Note: see mulid1 8714 for commuted version. (Contributed by NM, 8-Oct-1999.)

Theoremcnre 8716* Alias for ax-cnre 8690, for naming consistency. (Contributed by Mario Carneiro, 3-Jan-2013.)

Theorem1re 8717 is a real number. This used to be one of our postulates for complex numbers, but Eric Schmidt discovered that it could be derived from a weaker postulate, ax-1cn 8675, by exploiting properties of the imaginary unit . (Contributed by Eric Schmidt, 11-Apr-2007.) (Revised by Scott Fenton, 3-Jan-2013.)

Theorem0re 8718 is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremmulid1i 8719 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)

Theoremmulid2i 8720 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)

Theoremmulcli 8722 Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)

Theoremmulcomi 8723 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)

Theoremmulcomli 8724 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)

Theoremmulassi 8726 Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)

Theoremadddii 8727 Distributive law. (Contributed by NM, 23-Nov-1994.)

Theoremadddiri 8728 Distributive law. (Contributed by NM, 16-Feb-1995.)

Theoremrecni 8729 A real number is a complex number. (Contributed by NM, 1-Mar-1995.)

Theoremreaddcli 8730 Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)

Theoremremulcli 8731 Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)

Theoremmulid1d 8732 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulid2d 8733 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulcld 8735 Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulcomd 8736 Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulassd 8738 Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadddid 8739 Distributive law. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadddird 8740 Distributive law. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremrecnd 8741 Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)

Theoremreaddcld 8742 Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremremulcld 8743 Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.)

5.2.2  Infinity and the extended real number system

Syntaxcpnf 8744 Plus infinity.

Syntaxcmnf 8745 Minus infinity.

Syntaxcxr 8746 The set of extended reals (includes plus and minus infinity).

Syntaxclt 8747 'Less than' predicate (extended to include the extended reals).

Syntaxcle 8748 Extend wff notation to include the 'less than or equal to' relation.

Definitiondf-pnf 8749 Define plus infinity. Note that the definition is arbitrary, requiring only that be a set not in and different from (df-mnf 8750). We use to make it independent of the construction of , and Cantor's Theorem will show that it is different from any member of and therefore . See pnfnre 8754, mnfnre 8755, and pnfnemnf 10338.

A simpler possibility is to define as and as , but that approach requires the Axiom of Regularity to show that and are different from each other and from all members of . (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)

Definitiondf-mnf 8750 Define minus infinity as the power set of plus infinity. Note that the definition is arbitrary, requiring only that be a set not in and different from (see mnfnre 8755 and pnfnemnf 10338). (Contributed by NM, 13-Oct-2005.) (New usage is discouraged.)

Definitiondf-xr 8751 Define the set of extended reals that includes plus and minus infinity. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 13-Oct-2005.)

Definitiondf-ltxr 8752* Define 'less than' on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. Note that in our postulates for complex numbers, is primitive and not necessarily a relation on . (Contributed by NM, 13-Oct-2005.)

Definitiondf-le 8753 Define 'less than or equal to' on the extended real subset of complex numbers. Theorem leloe 8788 relates it to 'less than' for reals. (Contributed by NM, 13-Oct-2005.)

Theorempnfnre 8754 Plus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)

Theoremmnfnre 8755 Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.)

Theoremressxr 8756 The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.)

Theoremrexr 8757 A standard real is an extended real. (Contributed by NM, 14-Oct-2005.)

Theorem0xr 8758 Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.)

Theoremrenepnf 8759 No (finite) real equals plus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremrenemnf 8760 No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremrexrd 8761 A standard real is an extended real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrenepnfd 8762 No (finite) real equals plus infinity. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrenemnfd 8763 No real equals minus infinity. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremrexri 8764 A standard real is an extended real (inference form.) (Contributed by David Moews, 28-Feb-2017.)

Theoremrenfdisj 8765 The reals and the infinities are disjoint. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltrelxr 8766 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremltrel 8767 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)

Theoremlerelxr 8768 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremlerel 8769 'Less or equal to' is a relation. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 28-Apr-2015.)

Theoremxrlenlt 8770 'Less than or equal to' expressed in terms of 'less than', for extended reals. (Contributed by NM, 14-Oct-2005.)

Theoremxrltnle 8771 'Less than' expressed in terms of 'less than or equal to', for extended reals. (Contributed by NM, 6-Feb-2007.)

Theoremssxr 8772 The three (non-exclusive) possibilities implied by a subset of extended reals. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltxrlt 8773 The standard less-than and the extended real less-than are identical when restricted to the non-extended reals . (Contributed by NM, 13-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

5.2.3  Restate the ordering postulates with extended real "less than"

Theoremaxlttri 8774 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-lttri 8691 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)

Theoremaxlttrn 8775 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-lttrn 8692 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)

Theoremaxltadd 8776 Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-ltadd 8693 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)

Theoremaxmulgt0 8777 The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-mulgt0 8694 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)

Theoremaxsup 8778* A non-empty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-sup 8695 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)

5.2.4  Ordering on reals

Theoremlttr 8779 Alias for axlttrn 8775, for naming consistency with lttri 8825. (Contributed by NM, 10-Mar-2008.)

Theoremmulgt0 8780 The product of two positive numbers is positive. (Contributed by NM, 10-Mar-2008.)

Theoremlenlt 8781 'Less than or equal to' expressed in terms of 'less than'. (Contributed by NM, 13-May-1999.)

Theoremltnle 8782 'Less than' expressed in terms of 'less than or equal to'. (Contributed by NM, 11-Jul-2005.)

Theoremltso 8783 'Less than' is a strict ordering. (Contributed by NM, 19-Jan-1997.)

Theoremlttri2 8784 Consequence of trichotomy. (Contributed by NM, 9-Oct-1999.)

Theoremlttri3 8785 Trichotomy law for 'less than'. (Contributed by NM, 5-May-1999.)

Theoremlttri4 8786 Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremletri3 8787 Trichotomy law. (Contributed by NM, 14-May-1999.)

Theoremleloe 8788 'Less than or equal to' expressed in terms of 'less than' or 'equals'. (Contributed by NM, 13-May-1999.)

Theoremeqlelt 8789 Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.)

Theoremltle 8790 'Less than' implies 'less than or equal to'. (Contributed by NM, 25-Aug-1999.)

Theoremleltne 8791 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by NM, 27-Jul-1999.)

Theoremlelttr 8792 Transitive law. (Contributed by NM, 23-May-1999.)

Theoremltletr 8793 Transitive law. (Contributed by NM, 25-Aug-1999.)

Theoremletr 8794 Transitive law. (Contributed by NM, 12-Nov-1999.)

Theoremltnr 8795 'Less than' is irreflexive. (Contributed by NM, 18-Aug-1999.)

Theoremleid 8796 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)

Theoremltne 8797 'Less than' implies not equal. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 16-Sep-2015.)

TheoremltneOLD 8798 'Less than' implies not equal. (Contributed by NM, 9-Oct-1999.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremltnsym 8799 'Less than' is not symmetric. (Contributed by NM, 8-Jan-2002.)

Theoremltnsym2 8800 'Less than' is antisymmetric and irreflexive. (Contributed by NM, 13-Aug-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

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