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Type | Label | Description |
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Statement | ||
Theorem | lenegcon2d 8201 | Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltaddposd 8202 | Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltaddpos2d 8203 | Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltsubposd 8204 | Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | posdifd 8205 | Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | addge01d 8206 | A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | addge02d 8207 | A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | subge0d 8208 | Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | suble0d 8209 | Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | subge02d 8210 | Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltadd1d 8211 | Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | leadd1d 8212 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | leadd2d 8213 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltsubaddd 8214 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | lesubaddd 8215 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltsubadd2d 8216 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | lesubadd2d 8217 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltaddsubd 8218 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltaddsub2d 8219 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.) |
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Theorem | leaddsub2d 8220 | 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | subled 8221 | Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | lesubd 8222 | Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltsub23d 8223 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltsub13d 8224 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | lesub1d 8225 | Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | lesub2d 8226 | Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltsub1d 8227 | Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltsub2d 8228 | Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltadd1dd 8229 | Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | ltsub1dd 8230 | Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | ltsub2dd 8231 | Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | leadd1dd 8232 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | leadd2dd 8233 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | lesub1dd 8234 | Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | lesub2dd 8235 | Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | le2addd 8236 | Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | le2subd 8237 | Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltleaddd 8238 | Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | leltaddd 8239 | Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | lt2addd 8240 | Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | lt2subd 8241 | Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | possumd 8242 | Condition for a positive sum. (Contributed by Scott Fenton, 16-Dec-2017.) |
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Theorem | sublt0d 8243 | When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ltaddsublt 8244 | Addition and subtraction on one side of 'less than'. (Contributed by AV, 24-Nov-2018.) |
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Theorem | 1le1 8245 |
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Theorem | gt0add 8246 | A positive sum must have a positive addend. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 26-Jan-2020.) |
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Syntax | creap 8247 | Class of real apartness relation. |
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Definition | df-reap 8248* | Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although #ℝ is an apartness relation on the reals (see df-ap 8255 for more discussion of apartness relations), for our purposes it is just a stepping stone to defining # which is an apartness relation on complex numbers. On the reals, #ℝ and # agree (apreap 8260). (Contributed by Jim Kingdon, 26-Jan-2020.) |
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Theorem | reapval 8249 | Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8261 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.) |
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Theorem | reapirr 8250 | Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 8278 instead. (Contributed by Jim Kingdon, 26-Jan-2020.) |
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Theorem | recexre 8251* | Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.) |
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Theorem | reapti 8252 | Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8295. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.) |
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Theorem | recexgt0 8253* | Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.) |
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Syntax | cap 8254 | Class of complex apartness relation. |
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Definition | df-ap 8255* |
Define complex apartness. Definition 6.1 of [Geuvers], p. 17.
Two numbers are considered apart if it is possible to separate them. One common usage is that we can divide by a number if it is apart from zero (see for example recclap 8345 which says that a number apart from zero has a reciprocal). The defining characteristics of an apartness are irreflexivity (apirr 8278), symmetry (apsym 8279), and cotransitivity (apcotr 8280). Apartness implies negated equality, as seen at apne 8296, and the converse would also follow if we assumed excluded middle. In addition, apartness of complex numbers is tight, which means that two numbers which are not apart are equal (apti 8295). (Contributed by Jim Kingdon, 26-Jan-2020.) |
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Theorem | ixi 8256 |
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Theorem | inelr 8257 |
The imaginary unit ![]() |
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Theorem | rimul 8258 | A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
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Theorem | rereim 8259 | Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.) |
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Theorem | apreap 8260 | Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.) |
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Theorem | reaplt 8261 | Real apartness in terms of less than. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 1-Feb-2020.) |
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Theorem | reapltxor 8262 | Real apartness in terms of less than (exclusive-or version). (Contributed by Jim Kingdon, 23-Mar-2020.) |
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Theorem | 1ap0 8263 | One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.) |
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Theorem | ltmul1a 8264 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 15-May-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
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Theorem | ltmul1 8265 | Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
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Theorem | lemul1 8266 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.) |
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Theorem | reapmul1lem 8267 | Lemma for reapmul1 8268. (Contributed by Jim Kingdon, 8-Feb-2020.) |
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Theorem | reapmul1 8268 | Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 8454. (Contributed by Jim Kingdon, 8-Feb-2020.) |
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Theorem | reapadd1 8269 | Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
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Theorem | reapneg 8270 | Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
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Theorem | reapcotr 8271 | Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.) |
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Theorem | remulext1 8272 | Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.) |
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Theorem | remulext2 8273 | Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
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Theorem | apsqgt0 8274 | The square of a real number apart from zero is positive. (Contributed by Jim Kingdon, 7-Feb-2020.) |
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Theorem | cru 8275 | The representation of complex numbers in terms of real and imaginary parts is unique. Proposition 10-1.3 of [Gleason] p. 130. (Contributed by NM, 9-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
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Theorem | apreim 8276 | Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.) |
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Theorem | mulreim 8277 | Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.) |
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Theorem | apirr 8278 | Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.) |
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Theorem | apsym 8279 | Apartness is symmetric. This theorem for real numbers is part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.) |
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Theorem | apcotr 8280 | Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.) |
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Theorem | apadd1 8281 | Addition respects apartness. Analogue of addcan 7858 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
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Theorem | apadd2 8282 | Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.) |
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Theorem | addext 8283 | Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5735. For us, it is proved a different way. (Contributed by Jim Kingdon, 15-Feb-2020.) |
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Theorem | apneg 8284 | Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.) |
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Theorem | mulext1 8285 | Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
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Theorem | mulext2 8286 | Right extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
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Theorem | mulext 8287 | Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5735. For us, it is proved a different way. (Contributed by Jim Kingdon, 23-Feb-2020.) |
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Theorem | mulap0r 8288 | A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 24-Feb-2020.) |
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Theorem | msqge0 8289 | A square is nonnegative. Lemma 2.35 of [Geuvers], p. 9. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.) |
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Theorem | msqge0i 8290 | A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
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Theorem | msqge0d 8291 | A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | mulge0 8292 | The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
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Theorem | mulge0i 8293 | The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.) |
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Theorem | mulge0d 8294 | The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | apti 8295 | Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.) |
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Theorem | apne 8296 | Apartness implies negated equality. We cannot in general prove the converse, which is the whole point of having separate notations for apartness and negated equality. (Contributed by Jim Kingdon, 21-Feb-2020.) |
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Theorem | apcon4bid 8297 | Contrapositive law deduction for apartness. (Contributed by Jim Kingdon, 31-Jul-2023.) |
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Theorem | leltap 8298 |
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Theorem | gt0ap0 8299 | Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.) |
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Theorem | gt0ap0i 8300 | Positive means apart from zero (useful for ordering theorems involving division). (Contributed by Jim Kingdon, 27-Feb-2020.) |
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