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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | leadd1d 8301 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | leadd2d 8302 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsubaddd 8303 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lesubaddd 8304 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsubadd2d 8305 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lesubadd2d 8306 | 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltaddsubd 8307 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltaddsub2d 8308 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.) |
Theorem | leaddsub2d 8309 | 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | subled 8310 | Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lesubd 8311 | Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsub23d 8312 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsub13d 8313 | 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lesub1d 8314 | Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lesub2d 8315 | Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsub1d 8316 | Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltsub2d 8317 | Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltadd1dd 8318 | Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | ltsub1dd 8319 | Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | ltsub2dd 8320 | Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | leadd1dd 8321 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | leadd2dd 8322 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | lesub1dd 8323 | Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | lesub2dd 8324 | Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | le2addd 8325 | Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | le2subd 8326 | Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltleaddd 8327 | Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | leltaddd 8328 | Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lt2addd 8329 | Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lt2subd 8330 | Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | possumd 8331 | Condition for a positive sum. (Contributed by Scott Fenton, 16-Dec-2017.) |
Theorem | sublt0d 8332 | When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Theorem | ltaddsublt 8333 | Addition and subtraction on one side of 'less than'. (Contributed by AV, 24-Nov-2018.) |
Theorem | 1le1 8334 | . Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.) |
Theorem | gt0add 8335 | 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.) |
Syntax | creap 8336 | Class of real apartness relation. |
#ℝ | ||
Definition | df-reap 8337* | 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 8344 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 8349). (Contributed by Jim Kingdon, 26-Jan-2020.) |
#ℝ | ||
Theorem | reapval 8338 | Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8350 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.) |
#ℝ | ||
Theorem | reapirr 8339 | Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 8367 instead. (Contributed by Jim Kingdon, 26-Jan-2020.) |
#ℝ | ||
Theorem | recexre 8340* | Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.) |
#ℝ | ||
Theorem | reapti 8341 | Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8384. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.) |
#ℝ | ||
Theorem | recexgt0 8342* | Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.) |
Syntax | cap 8343 | Class of complex apartness relation. |
# | ||
Definition | df-ap 8344* |
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 8439 which says that a number apart from zero has a reciprocal). The defining characteristics of an apartness are irreflexivity (apirr 8367), symmetry (apsym 8368), and cotransitivity (apcotr 8369). Apartness implies negated equality, as seen at apne 8385, 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 8384). (Contributed by Jim Kingdon, 26-Jan-2020.) |
# #ℝ #ℝ | ||
Theorem | ixi 8345 | times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Theorem | inelr 8346 | The imaginary unit is not a real number. (Contributed by NM, 6-May-1999.) |
Theorem | rimul 8347 | 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.) |
Theorem | rereim 8348 | Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.) |
Theorem | apreap 8349 | Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.) |
# #ℝ | ||
Theorem | reaplt 8350 | 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.) |
# | ||
Theorem | reapltxor 8351 | Real apartness in terms of less than (exclusive-or version). (Contributed by Jim Kingdon, 23-Mar-2020.) |
# | ||
Theorem | 1ap0 8352 | One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.) |
# | ||
Theorem | ltmul1a 8353 | 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.) |
Theorem | ltmul1 8354 | 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.) |
Theorem | lemul1 8355 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.) |
Theorem | reapmul1lem 8356 | Lemma for reapmul1 8357. (Contributed by Jim Kingdon, 8-Feb-2020.) |
# # | ||
Theorem | reapmul1 8357 | Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 8548. (Contributed by Jim Kingdon, 8-Feb-2020.) |
# # # | ||
Theorem | reapadd1 8358 | Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
# # | ||
Theorem | reapneg 8359 | Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
# # | ||
Theorem | reapcotr 8360 | Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.) |
# # # | ||
Theorem | remulext1 8361 | Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.) |
# # | ||
Theorem | remulext2 8362 | Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
# # | ||
Theorem | apsqgt0 8363 | The square of a real number apart from zero is positive. (Contributed by Jim Kingdon, 7-Feb-2020.) |
# | ||
Theorem | cru 8364 | 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.) |
Theorem | apreim 8365 | Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.) |
# # # | ||
Theorem | mulreim 8366 | Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.) |
Theorem | apirr 8367 | Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.) |
# | ||
Theorem | apsym 8368 | 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.) |
# # | ||
Theorem | apcotr 8369 | Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.) |
# # # | ||
Theorem | apadd1 8370 | Addition respects apartness. Analogue of addcan 7942 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
# # | ||
Theorem | apadd2 8371 | Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.) |
# # | ||
Theorem | addext 8372 | Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5783. For us, it is proved a different way. (Contributed by Jim Kingdon, 15-Feb-2020.) |
# # # | ||
Theorem | apneg 8373 | Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.) |
# # | ||
Theorem | mulext1 8374 | Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
# # | ||
Theorem | mulext2 8375 | Right extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
# # | ||
Theorem | mulext 8376 | Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5783. For us, it is proved a different way. (Contributed by Jim Kingdon, 23-Feb-2020.) |
# # # | ||
Theorem | mulap0r 8377 | A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 24-Feb-2020.) |
# # # | ||
Theorem | msqge0 8378 | A square is nonnegative. Lemma 2.35 of [Geuvers], p. 9. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.) |
Theorem | msqge0i 8379 | A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Theorem | msqge0d 8380 | A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | mulge0 8381 | The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
Theorem | mulge0i 8382 | The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.) |
Theorem | mulge0d 8383 | The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | apti 8384 | Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.) |
# | ||
Theorem | apne 8385 | 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.) |
# | ||
Theorem | apcon4bid 8386 | Contrapositive law deduction for apartness. (Contributed by Jim Kingdon, 31-Jul-2023.) |
# # | ||
Theorem | leltap 8387 | implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.) |
# | ||
Theorem | gt0ap0 8388 | Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.) |
# | ||
Theorem | gt0ap0i 8389 | Positive means apart from zero (useful for ordering theorems involving division). (Contributed by Jim Kingdon, 27-Feb-2020.) |
# | ||
Theorem | gt0ap0ii 8390 | Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.) |
# | ||
Theorem | gt0ap0d 8391 | Positive implies apart from zero. Because of the way we define #, must be an element of , not just . (Contributed by Jim Kingdon, 27-Feb-2020.) |
# | ||
Theorem | negap0 8392 | A number is apart from zero iff its negative is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.) |
# # | ||
Theorem | negap0d 8393 | The negative of a number apart from zero is apart from zero. (Contributed by Jim Kingdon, 25-Feb-2024.) |
# # | ||
Theorem | ltleap 8394 | Less than in terms of non-strict order and apartness. (Contributed by Jim Kingdon, 28-Feb-2020.) |
# | ||
Theorem | ltap 8395 | 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.) |
# | ||
Theorem | gtapii 8396 | 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.) |
# | ||
Theorem | ltapii 8397 | 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.) |
# | ||
Theorem | ltapi 8398 | 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.) |
# | ||
Theorem | gtapd 8399 | 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.) |
# | ||
Theorem | ltapd 8400 | 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.) |
# |
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