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Theorem List for Metamath Proof Explorer - 9201-9300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlesub0i 9201 Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltaddposi 9202 Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999.)

Theoremposdifi 9203 Comparison of two numbers whose difference is positive. (Contributed by NM, 19-Aug-2001.)

Theoremltnegcon1i 9204 Contraposition of negative in 'less than'. (Contributed by NM, 14-May-1999.)

Theoremlenegcon1i 9205 Contraposition of negative in 'less than or equal to'. (Contributed by NM, 6-Apr-2005.)

Theoremsubge0i 9206 Nonnegative subtraction. (Contributed by NM, 13-Aug-2000.)

Theoremltadd1i 9207 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by NM, 21-Jan-1997.)

Theoremleadd1i 9208 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)

Theoremleadd2i 9209 Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999.)

Theoremltsubaddi 9210 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremlesubaddi 9211 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 30-Sep-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremltsubadd2i 9212 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)

Theoremlesubadd2i 9213 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 3-Aug-1999.)

Theoremltaddsubi 9214 'Less than' relationship between subtraction and addition. (Contributed by NM, 14-May-1999.)

Theoremlt2addi 9215 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)

Theoremle2addi 9216 Adding both side of two inequalities. (Contributed by NM, 16-Sep-1999.)

Theoremgt0ne0d 9217 Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt0ne0d 9218 Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017.)

Theoremleidd 9219 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmsqgt0d 9220 A nonzero square is positive. Theorem I.20 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmsqge0d 9221 A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt0neg1d 9222 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt0neg2d 9223 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremle0neg1d 9224 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremle0neg2d 9225 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddgegt0d 9226 Addition of nonnegative and positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddgt0d 9227 Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddge0d 9228 Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulge0d 9229 The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnegd 9230 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenegd 9231 Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnegcon1d 9232 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltnegcon2d 9233 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenegcon1d 9234 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlenegcon2d 9235 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddposd 9236 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddpos2d 9237 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsubposd 9238 Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremposdifd 9239 Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddge01d 9240 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddge02d 9241 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubge0d 9242 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsuble0d 9243 Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubge02d 9244 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltadd1d 9245 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleadd1d 9246 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleadd2d 9247 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsubaddd 9248 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesubaddd 9249 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsubadd2d 9250 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesubadd2d 9251 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddsubd 9252 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddsub2d 9253 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremleaddsub2d 9254 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubled 9255 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesubd 9256 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub23d 9257 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub13d 9258 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesub1d 9259 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesub2d 9260 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub1d 9261 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub2d 9262 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltadd1dd 9263 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltsub1dd 9264 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltsub2dd 9265 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremleadd1dd 9266 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremleadd2dd 9267 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlesub1dd 9268 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlesub2dd 9269 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremle2addd 9270 Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremle2subd 9271 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltleaddd 9272 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleltaddd 9273 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2addd 9274 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2subd 9275 Adding both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.)

Theorem1le1 9276 . Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)

5.3.5  Reciprocals

Theoremixi 9277 times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremrecextlem1 9278 Lemma for recex 9280. (Contributed by Eric Schmidt, 23-May-2007.)

Theoremrecextlem2 9279 Lemma for recex 9280. (Contributed by Eric Schmidt, 23-May-2007.)

Theoremrecex 9280* Existence of reciprocal of nonzero complex number. (Contributed by Eric Schmidt, 22-May-2007.)

Theoremmulcand 9281 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 26-Jan-1995.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulcan2d 9282 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulcanad 9283 Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcand 9281. (Contributed by David Moews, 28-Feb-2017.)

Theoremmulcan2ad 9284 Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcan2d 9282. (Contributed by David Moews, 28-Feb-2017.)

Theoremmulcan 9285 Cancellation law for multiplication (full theorem form). Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 29-Jan-1995.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulcan2 9286 Cancellation law for multiplication. (Contributed by NM, 21-Jan-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulcani 9287 Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 26-Jan-1995.)

Theoremmul0or 9288 If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulne0b 9289 The product of two nonzero numbers is nonzero. (Contributed by NM, 1-Aug-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremmulne0 9290 The product of two nonzero numbers is nonzero. (Contributed by NM, 30-Dec-2007.)

Theoremmulne0i 9291 The product of two nonzero numbers is nonzero. (Contributed by NM, 15-Feb-1995.)

Theoremmuleqadd 9292 Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.)

Theoremreceu 9293* Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by NM, 29-Jan-1995.) (Revised by Mario Carneiro, 17-Feb-2014.)

Theoremmulnzcnopr 9294 Multiplication maps nonzero complex numbers to nonzero complex numbers. (Contributed by Steve Rodriguez, 23-Feb-2007.)

Theoremmsq0i 9295 A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by NM, 28-Jul-1999.)

Theoremmul0ori 9296 If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by NM, 7-Oct-1999.)

Theoremmsq0d 9297 A number is zero iff its square is zero (where square is represented using multiplication). (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul0ord 9298 If a product is zero, one of its factors must be zero. Theorem I.11 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulne0bd 9299 The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulne0d 9300 The product of two nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

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