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Theorem List for Metamath Proof Explorer - 9701-9800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlttri 9701 'Less than' is transitive. Theorem I.17 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)

Theoremlelttri 9702 'Less than or equal to', 'less than' transitive law. (Contributed by NM, 14-May-1999.)

Theoremltletri 9703 'Less than', 'less than or equal to' transitive law. (Contributed by NM, 14-May-1999.)

Theoremletri 9704 'Less than or equal to' is transitive. (Contributed by NM, 14-May-1999.)

Theoremle2tri3i 9705 Extended trichotomy law for 'less than or equal to'. (Contributed by NM, 14-Aug-2000.)

Theoremltadd2i 9706 Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Paul Chapman, 27-Jan-2008.)

Theoremmulgt0i 9707 The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.)

Theoremmulgt0ii 9708 The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.)

Theoremltnrd 9709 'Less than' is irreflexive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremgtned 9710 'Less than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltned 9711 'Greater than' implies not equal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremne0gt0d 9712 A nonzero nonnegative number is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttrid 9713 Ordering on reals satisfies strict trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttri2d 9714 Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttri3d 9715 Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlttri4d 9716 Trichotomy law for 'less than'. (Contributed by NM, 20-Sep-2007.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremletri3d 9717 Consequence of trichotomy. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleloed 9718 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremeqleltd 9719 Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.)

Theoremltlend 9720 'Less than' expressed in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenltd 9721 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnled 9722 'Less than' in terms of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltled 9723 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnsymd 9724 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremletrid 9725 Trichotomy law for 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleltned 9726 'Less than or equal to' implies 'less than' is not 'equals'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulgt0d 9727 The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltadd2d 9728 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremletrd 9729 Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.)

Theoremlelttrd 9730 Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.)

Theoremltadd2dd 9731 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltletrd 9732 Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.)

Theoremlttrd 9733 Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006.)

Theoremdedekind 9734* The Dedekind cut theorem. This theorem, which may be used to replace ax-pre-sup 9561 with appropriate adjustments, states that, if completely preceeds , then there is some number separating the two of them. (Contributed by Scott Fenton, 13-Jun-2013.)

Theoremdedekindle 9735* The Dedekind cut theorem, with the hypothesis weakened to only require non-strict less than. (Contributed by Scott Fenton, 2-Jul-2013.)

5.2.5  Initial properties of the complex numbers

Theoremmul12 9736 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)

Theoremmul32 9737 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)

Theoremmul31 9738 Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul4 9739 Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)

Theoremmuladd11 9740 A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)

Theorem1p1times 9741 Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theorempeano2cn 9742 A theorem for complex numbers analogous the second Peano postulate peano2nn 10539. (Contributed by NM, 17-Aug-2005.)

Theorempeano2re 9743 A theorem for reals analogous the second Peano postulate peano2nn 10539. (Contributed by NM, 5-Jul-2005.)

Theoremreaddcan 9744 Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theorem00id 9745 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul02lem1 9746 Lemma for mul02 9748. If any real does not produce when multiplied by , then any complex is equal to double itself. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul02lem2 9747 Lemma for mul02 9748. Zero times a real is zero. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremmul02 9748 Multiplication by . Theorem I.6 of [Apostol] p. 18. Based on ideas by Eric Schmidt. (Contributed by NM, 10-Aug-1999.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremmul01 9749 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddid1 9750 is an additive identity. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremcnegex 9751* Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremcnegex2 9752* Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremaddid2 9753 is a left identity for addition. This used to be one of our complex number axioms, until it was discovered that it was dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremaddcan 9754 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremaddcan2 9755 Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddcom 9756 Addition commutes. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremaddid1i 9757 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddid2i 9758 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)

Theoremmul02i 9759 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.)

Theoremmul01i 9760 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddcomi 9761 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremaddcani 9763 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddcan2i 9764 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 14-May-2003.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremmul12i 9765 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremmul32i 9766 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)

Theoremmul4i 9767 Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)

Theoremmul02d 9768 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul01d 9769 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddid2d 9771 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcomd 9772 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremaddcand 9773 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcan2d 9774 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcanad 9775 Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 9773. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddcan2ad 9776 Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 9774. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddneintrd 9777 Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 9775. Consequence of addcand 9773. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddneintr2d 9778 Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 9776. Consequence of addcan2d 9774. (Contributed by David Moews, 28-Feb-2017.)

Theoremmul12d 9779 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul32d 9780 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul31d 9781 Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul4d 9782 Rearrangement of 4 factors. (Contributed by Mario Carneiro, 27-May-2016.)

5.3  Real and complex numbers - basic operations

Theoremadd12 9783 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.)

Theoremadd32 9784 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.)

Theoremadd32r 9785 Commutative/associative law that swaps the last two terms in a triple sum, rearranging the parentheses. (Contributed by Paul Chapman, 18-May-2007.)

Theoremadd4 9786 Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremadd42 9787 Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)

Theoremadd12i 9788 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)

Theoremadd32i 9789 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)

Theoremadd4i 9790 Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)

Theoremadd42i 9791 Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)

Theoremadd12d 9792 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadd32d 9793 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadd4d 9794 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadd42d 9795 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)

5.3.2  Subtraction

Syntaxcmin 9796 Extend class notation to include subtraction.

Syntaxcneg 9797 Extend class notation to include unary minus. The symbol is not a class by itself but part of a compound class definition. We do this rather than making it a formal function since it is so commonly used. Note: We use different symbols for unary minus () and subtraction cmin 9796 () to prevent syntax ambiguity. For example, looking at the syntax definition co 6277, if we used the same symbol then " " could mean either " " minus "", or it could represent the (meaningless) operation of classes " " and " " connected with "operation" "". On the other hand, " " is unambiguous.

Definitiondf-sub 9798* Define subtraction. Theorem subval 9802 shows its value (and describes how this definition works), theorem subaddi 9897 relates it to addition, and theorems subcli 9886 and resubcli 9872 prove its closure laws. (Contributed by NM, 26-Nov-1994.)

Definitiondf-neg 9799 Define the negative of a number (unary minus). We use different symbols for unary minus () and subtraction () to prevent syntax ambiguity. See cneg 9797 for a discussion of this. (Contributed by NM, 10-Feb-1995.)

Theorem0cnALT 9800 Alternate proof of 0cn 9579 which does not reference ax-1cn 9541. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

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