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Type | Label | Description |
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Statement | ||
Theorem | lenegcon1d 8101 | Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | lenegcon2d 8102 | Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltaddposd 8103 | Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltaddpos2d 8104 | Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltsubposd 8105 | Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | posdifd 8106 | Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | addge01d 8107 | 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 8108 | 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 8109 | Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | suble0d 8110 | Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | subge02d 8111 | Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltadd1d 8112 | 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 8113 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | leadd2d 8114 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltsubaddd 8115 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | lesubaddd 8116 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltsubadd2d 8117 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | lesubadd2d 8118 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltaddsubd 8119 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltaddsub2d 8120 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.) |
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Theorem | leaddsub2d 8121 | 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | subled 8122 | Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | lesubd 8123 | Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltsub23d 8124 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltsub13d 8125 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | lesub1d 8126 | Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | lesub2d 8127 | Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltsub1d 8128 | Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltsub2d 8129 | Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltadd1dd 8130 | 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 8131 | Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | ltsub2dd 8132 | Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | leadd1dd 8133 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | leadd2dd 8134 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | lesub1dd 8135 | Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | lesub2dd 8136 | Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
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Theorem | le2addd 8137 | Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | le2subd 8138 | Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | ltleaddd 8139 | Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | leltaddd 8140 | Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | lt2addd 8141 | 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 8142 | Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | possumd 8143 | Condition for a positive sum. (Contributed by Scott Fenton, 16-Dec-2017.) |
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Theorem | sublt0d 8144 | When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ltaddsublt 8145 | Addition and subtraction on one side of 'less than'. (Contributed by AV, 24-Nov-2018.) |
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Theorem | 1le1 8146 |
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Theorem | gt0add 8147 | 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 8148 | Class of real apartness relation. |
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Definition | df-reap 8149* | 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 8156 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 8161). (Contributed by Jim Kingdon, 26-Jan-2020.) |
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Theorem | reapval 8150 | Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8162 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.) |
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Theorem | reapirr 8151 | Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 8179 instead. (Contributed by Jim Kingdon, 26-Jan-2020.) |
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Theorem | recexre 8152* | Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.) |
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Theorem | reapti 8153 | Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8196. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.) |
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Theorem | recexgt0 8154* | Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.) |
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Syntax | cap 8155 | Class of complex apartness relation. |
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Definition | df-ap 8156* |
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 8243 which says that a number apart from zero has a reciprocal). The defining characteristics of an apartness are irreflexivity (apirr 8179), symmetry (apsym 8180), and cotransitivity (apcotr 8181). Apartness implies negated equality, as seen at apne 8197, 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 8196). (Contributed by Jim Kingdon, 26-Jan-2020.) |
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Theorem | ixi 8157 |
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Theorem | inelr 8158 |
The imaginary unit ![]() |
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Theorem | rimul 8159 | 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 8160 | 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 8161 | Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.) |
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Theorem | reaplt 8162 | 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 8163 | Real apartness in terms of less than (exclusive-or version). (Contributed by Jim Kingdon, 23-Mar-2020.) |
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Theorem | 1ap0 8164 | One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.) |
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Theorem | ltmul1a 8165 | 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 8166 | 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 8167 | 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 8168 | Lemma for reapmul1 8169. (Contributed by Jim Kingdon, 8-Feb-2020.) |
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Theorem | reapmul1 8169 | Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 8352. (Contributed by Jim Kingdon, 8-Feb-2020.) |
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Theorem | reapadd1 8170 | Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
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Theorem | reapneg 8171 | Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
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Theorem | reapcotr 8172 | 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 8173 | Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.) |
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Theorem | remulext2 8174 | Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
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Theorem | apsqgt0 8175 | The square of a real number apart from zero is positive. (Contributed by Jim Kingdon, 7-Feb-2020.) |
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Theorem | cru 8176 | 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 8177 | Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.) |
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Theorem | mulreim 8178 | Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.) |
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Theorem | apirr 8179 | Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.) |
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Theorem | apsym 8180 | 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 8181 | Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.) |
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Theorem | apadd1 8182 | Addition respects apartness. Analogue of addcan 7759 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
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Theorem | apadd2 8183 | Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.) |
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Theorem | addext 8184 | Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5699. For us, it is proved a different way. (Contributed by Jim Kingdon, 15-Feb-2020.) |
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Theorem | apneg 8185 | Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.) |
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Theorem | mulext1 8186 | Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
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Theorem | mulext2 8187 | Right extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
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Theorem | mulext 8188 | Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5699. For us, it is proved a different way. (Contributed by Jim Kingdon, 23-Feb-2020.) |
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Theorem | mulap0r 8189 | 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 8190 | 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 8191 | A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
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Theorem | msqge0d 8192 | A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | mulge0 8193 | 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 8194 | The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.) |
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Theorem | mulge0d 8195 | The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) |
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Theorem | apti 8196 | Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.) |
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Theorem | apne 8197 | 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 | leltap 8198 | '<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.) |
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Theorem | gt0ap0 8199 | Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.) |
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Theorem | gt0ap0i 8200 | Positive means apart from zero (useful for ordering theorems involving division). (Contributed by Jim Kingdon, 27-Feb-2020.) |
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