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Theorem List for Metamath Proof Explorer - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmulm1 9101 Product with minus one is negative. (Contributed by NM, 16-Nov-1999.)

Theoremmulsub 9102 Product of two differences. (Contributed by NM, 14-Jan-2006.)

Theoremmulsub2 9103 Swap the order of subtraction in a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)

Theoremmulm1i 9104 Product with minus one is negative. (Contributed by NM, 31-Jul-1999.)

Theoremmulneg1i 9105 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 10-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulneg2i 9106 Product with negative is negative of product. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmul2negi 9107 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremsubdii 9108 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.)

Theoremsubdiri 9109 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 8-May-1999.)

Theoremmuladdi 9110 Product of two sums. (Contributed by NM, 17-May-1999.)

Theoremmulm1d 9111 Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulneg1d 9112 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulneg2d 9113 Product with negative is negative of product. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul2negd 9114 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubdid 9115 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubdird 9116 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmuladdd 9117 Product of two sums. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulsubd 9118 Product of two differences. (Contributed by Mario Carneiro, 27-May-2016.)

5.3.4  Ordering on reals (cont.)

Theoremgt0ne0 9119 Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlt0ne0 9120 A number which is less than zero is not zero. (Contributed by Stefan O'Rear, 13-Sep-2014.)

Theoremltadd1 9121 Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremleadd1 9122 Addition to both sides of 'less than or equal to'. (Contributed by NM, 18-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremleadd2 9123 Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999.)

Theoremltsubadd 9124 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltsubadd2 9125 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)

Theoremlesubadd 9126 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlesubadd2 9127 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 10-Aug-1999.)

Theoremltaddsub 9128 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)

Theoremltaddsub2 9129 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)

Theoremleaddsub 9130 'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.)

Theoremleaddsub2 9131 'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005.)

Theoremsuble 9132 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)

Theoremlesub 9133 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltsub23 9134 'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.)

Theoremltsub13 9135 'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.)

Theoremle2add 9136 Adding both sides of two 'less than or equal to' relations. (Contributed by NM, 17-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltleadd 9137 Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)

Theoremleltadd 9138 Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)

Theoremlt2add 9139 Adding both sides of two 'less than' relations. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 15-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremaddgt0 9140 The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddgegt0 9141 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddgtge0 9142 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddge0 9143 The sum of 2 nonnegative numbers is nonnegative. (Contributed by NM, 17-Mar-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltaddpos 9144 Adding a positive number to another number increases it. (Contributed by NM, 17-Nov-2004.)

Theoremltaddpos2 9145 Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005.)

Theoremltsubpos 9146 Subtracting a positive number from another number decreases it. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremposdif 9147 Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.)

Theoremlesub1 9148 Subtraction from both sides of 'less than or equal to'. (Contributed by NM, 13-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlesub2 9149 Subtraction of both sides of 'less than or equal to'. (Contributed by NM, 29-Sep-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremltsub1 9150 Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltsub2 9151 Subtraction of both sides of 'less than'. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlt2sub 9152 Adding both sides of two 'less than' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)

Theoremle2sub 9153 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)

Theoremltneg 9154 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 27-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremltnegcon1 9155 Contraposition of negative in 'less than'. (Contributed by NM, 8-Nov-2004.)

Theoremltnegcon2 9156 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015.)

Theoremleneg 9157 Negative of both sides of 'less than or equal to'. (Contributed by NM, 12-Sep-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremlenegcon1 9158 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004.)

Theoremlenegcon2 9159 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005.)

Theoremlt0neg1 9160 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)

Theoremlt0neg2 9161 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)

Theoremle0neg1 9162 Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.)

Theoremle0neg2 9163 Comparison of a number and its negative to zero. (Contributed by NM, 24-Aug-1999.)

Theoremaddge01 9164 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005.)

Theoremaddge02 9165 A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 27-Jul-2005.)

Theoremadd20 9166 Two nonnegative numbers are zero iff their sum is zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremsubge0 9167 Nonnegative subtraction. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremsuble0 9168 Nonpositive subtraction. (Contributed by NM, 20-Mar-2008.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremsubge02 9169 Nonnegative subtraction. (Contributed by NM, 27-Jul-2005.)

Theoremlesub0 9170 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremmulge0 9171 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulge0OLD 9172 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmullt0 9173 The product of two negative numbers is positive. (Contributed by Jeffrey Hankins, 8-Jun-2009.)

Theoremmsqgt0 9174 A nonzero square is positive. Theorem I.20 of [Apostol] p. 20. (Contributed by NM, 6-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremmsqge0 9175 A square is nonnegative. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.)

Theorem0lt1 9176 0 is less than 1. Theorem I.21 of [Apostol] p. 20. (Contributed by NM, 17-Jan-1997.)

Theorem0le1 9177 0 is less than or equal to 1. (Contributed by Mario Carneiro, 29-Apr-2015.)

Theoremltordlem 9178* Lemma for ltord1 9179. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremltord1 9179* Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremleord1 9180* Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremeqord1 9181* Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremltord2 9182* Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremleord2 9183* Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremeqord2 9184* Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)

Theoremwloglei 9185* Form of wlogle 9186 where both sides of the equivalence are proven rather than showing that they are equivalent to each other. (Contributed by Mario Carneiro, 9-Mar-2015.)

Theoremwlogle 9186* If the predicate is symmetric under interchange of , then "without loss of generality" we can assume that . (Contributed by Mario Carneiro, 18-Aug-2014.) (Revised by Mario Carneiro, 11-Sep-2014.)

Theoremleidi 9187 'Less than or equal to' is reflexive. (Contributed by NM, 18-Aug-1999.)

Theoremgt0ne0i 9188 Positive means nonzero (useful for ordering theorems involving division). (Contributed by NM, 16-Sep-1999.)

Theoremgt0ne0ii 9189 Positive implies nonzero. (Contributed by NM, 15-May-1999.)

Theoremmsqgt0i 9190 A nonzero square is positive. Theorem I.20 of [Apostol] p. 20. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmsqge0i 9191 A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddgt0i 9192 Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddge0i 9193 Addition of 2 nonnegative numbers is nonnegative. (Contributed by NM, 28-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremaddgegt0i 9194 Addition of nonnegative and positive numbers is positive. (Contributed by NM, 25-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremaddgt0ii 9195 Addition of 2 positive numbers is positive. (Contributed by NM, 18-May-1999.)

Theoremadd20i 9196 Two nonnegative numbers are zero iff their sum is zero. (Contributed by NM, 28-Jul-1999.)

Theoremltnegi 9197 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)

Theoremlenegi 9198 Negative of both sides of 'less than or equal to'. (Contributed by NM, 1-Aug-1999.)

Theoremltnegcon2i 9199 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)

Theoremmulge0i 9200 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.)

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