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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ltsub2d 7301 | Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltadd1dd 7302 | Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | ltsub1dd 7303 | Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | ltsub2dd 7304 | Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | leadd1dd 7305 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | leadd2dd 7306 | Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | lesub1dd 7307 | Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | lesub2dd 7308 | Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
Theorem | le2addd 7309 | Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | le2subd 7310 | Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltleaddd 7311 | Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | leltaddd 7312 | Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lt2addd 7313 | Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | lt2subd 7314 | Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | ltaddsublt 7315 | Addition and subtraction on one side of 'less than'. (Contributed by AV, 24-Nov-2018.) |
Theorem | 1le1 7316 | . Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.) |
Theorem | gt0add 7317 | 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 7318 | Class of real apartness relation. |
#_{ℝ} | ||
Definition | df-reap 7319* | 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 7326 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 7331). (Contributed by Jim Kingdon, 26-Jan-2020.) |
#_{ℝ} | ||
Theorem | reapval 7320 | Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 7332 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.) |
#_{ℝ} | ||
Theorem | reapirr 7321 | Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 7349 instead. (Contributed by Jim Kingdon, 26-Jan-2020.) |
#_{ℝ} | ||
Theorem | recexre 7322* | Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.) |
#_{ℝ} | ||
Theorem | reapti 7323 | Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 7366. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.) |
#_{ℝ} | ||
Theorem | recexgt0 7324* | Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.) |
Syntax | cap 7325 | Class of complex apartness relation. |
# | ||
Definition | df-ap 7326* |
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 7400 which says that a number apart from zero has a reciprocal). The defining characteristics of an apartness are irreflexivity (apirr 7349), symmetry (apsym 7350), and cotransitivity (apcotr 7351). Apartness implies negated equality, as seen at apne 7367, 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 7366). (Contributed by Jim Kingdon, 26-Jan-2020.) |
# #_{ℝ} #_{ℝ} | ||
Theorem | ixi 7327 | times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Theorem | inelr 7328 | The imaginary unit is not a real number. (Contributed by NM, 6-May-1999.) |
Theorem | rimul 7329 | 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 7330 | Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.) |
Theorem | apreap 7331 | Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.) |
# #_{ℝ} | ||
Theorem | reaplt 7332 | 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 7333 | Real apartness in terms of less than (exclusive-or version). (Contributed by Jim Kingdon, 23-Mar-2020.) |
# | ||
Theorem | 1ap0 7334 | One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.) |
# | ||
Theorem | ltmul1a 7335 | 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 7336 | 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 7337 | Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.) |
Theorem | reapmul1lem 7338 | Lemma for reapmul1 7339. (Contributed by Jim Kingdon, 8-Feb-2020.) |
# # | ||
Theorem | reapmul1 7339 | Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 7506. (Contributed by Jim Kingdon, 8-Feb-2020.) |
# # # | ||
Theorem | reapadd1 7340 | Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
# # | ||
Theorem | reapneg 7341 | Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
# # | ||
Theorem | reapcotr 7342 | Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.) |
# # # | ||
Theorem | remulext1 7343 | Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.) |
# # | ||
Theorem | remulext2 7344 | Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
# # | ||
Theorem | apsqgt0 7345 | The square of a real number apart from zero is positive. (Contributed by Jim Kingdon, 7-Feb-2020.) |
# | ||
Theorem | cru 7346 | 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 7347 | Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.) |
# # # | ||
Theorem | mulreim 7348 | Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.) |
Theorem | apirr 7349 | Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.) |
# | ||
Theorem | apsym 7350 | 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 7351 | Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.) |
# # # | ||
Theorem | apadd1 7352 | Addition respects apartness. Analogue of addcan 6948 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.) |
# # | ||
Theorem | apadd2 7353 | Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.) |
# # | ||
Theorem | addext 7354 | Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5464. For us, it is proved a different way. (Contributed by Jim Kingdon, 15-Feb-2020.) |
# # # | ||
Theorem | apneg 7355 | Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.) |
# # | ||
Theorem | mulext1 7356 | Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
# # | ||
Theorem | mulext2 7357 | Right extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
# # | ||
Theorem | mulext 7358 | Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5464. For us, it is proved a different way. (Contributed by Jim Kingdon, 23-Feb-2020.) |
# # # | ||
Theorem | mulap0r 7359 | A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 24-Feb-2020.) |
# # # | ||
Theorem | msqge0 7360 | 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 7361 | A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Theorem | msqge0d 7362 | A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | mulge0 7363 | The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
Theorem | mulge0i 7364 | The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.) |
Theorem | mulge0d 7365 | The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.) |
Theorem | apti 7366 | Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.) |
# | ||
Theorem | apne 7367 | 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 | gt0ap0 7368 | Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.) |
# | ||
Theorem | gt0ap0i 7369 | Positive means apart from zero (useful for ordering theorems involving division). (Contributed by Jim Kingdon, 27-Feb-2020.) |
# | ||
Theorem | gt0ap0ii 7370 | Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.) |
# | ||
Theorem | gt0ap0d 7371 | 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 7372 | A number is apart from zero iff its negative is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.) |
# # | ||
Theorem | ltleap 7373 | Less than in terms of non-strict order and apartness. (Contributed by Jim Kingdon, 28-Feb-2020.) |
# | ||
Theorem | recextlem1 7374 | Lemma for recexap 7376. (Contributed by Eric Schmidt, 23-May-2007.) |
Theorem | recexaplem2 7375 | Lemma for recexap 7376. (Contributed by Jim Kingdon, 20-Feb-2020.) |
# # | ||
Theorem | recexap 7376* | Existence of reciprocal of nonzero complex number. (Contributed by Jim Kingdon, 20-Feb-2020.) |
# | ||
Theorem | mulap0 7377 | The product of two numbers apart from zero is apart from zero. Lemma 2.15 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 22-Feb-2020.) |
# # # | ||
Theorem | mulap0b 7378 | The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.) |
# # # | ||
Theorem | mulap0i 7379 | The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.) |
# # # | ||
Theorem | mulap0bd 7380 | The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.) |
# # # | ||
Theorem | mulap0d 7381 | The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.) |
# # # | ||
Theorem | mulap0bad 7382 | A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 7381 and consequence of mulap0bd 7380. (Contributed by Jim Kingdon, 24-Feb-2020.) |
# # | ||
Theorem | mulap0bbd 7383 | A factor of a complex number apart from zero is apart from zero. Partial converse of mulap0d 7381 and consequence of mulap0bd 7380. (Contributed by Jim Kingdon, 24-Feb-2020.) |
# # | ||
Theorem | mulcanapd 7384 | Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.) |
# | ||
Theorem | mulcanap2d 7385 | Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.) |
# | ||
Theorem | mulcanapad 7386 | Cancellation of a nonzero factor on the left in an equation. One-way deduction form of mulcanapd 7384. (Contributed by Jim Kingdon, 21-Feb-2020.) |
# | ||
Theorem | mulcanap2ad 7387 | Cancellation of a nonzero factor on the right in an equation. One-way deduction form of mulcanap2d 7385. (Contributed by Jim Kingdon, 21-Feb-2020.) |
# | ||
Theorem | mulcanap 7388 | Cancellation law for multiplication (full theorem form). (Contributed by Jim Kingdon, 21-Feb-2020.) |
# | ||
Theorem | mulcanap2 7389 | Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.) |
# | ||
Theorem | mulcanapi 7390 | Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.) |
# | ||
Theorem | muleqadd 7391 | Property of numbers whose product equals their sum. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 13-Nov-2006.) |
Theorem | receuap 7392* | Existential uniqueness of reciprocals. (Contributed by Jim Kingdon, 21-Feb-2020.) |
# | ||
Syntax | cdiv 7393 | Extend class notation to include division. |
Definition | df-div 7394* | Define division. Theorem divmulap 7396 relates it to multiplication, and divclap 7399 and redivclap 7449 prove its closure laws. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.) |
Theorem | divvalap 7395* | Value of division: the (unique) element such that . This is meaningful only when is apart from zero. (Contributed by Jim Kingdon, 21-Feb-2020.) |
# | ||
Theorem | divmulap 7396 | Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
# | ||
Theorem | divmulap2 7397 | Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
# | ||
Theorem | divmulap3 7398 | Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
# | ||
Theorem | divclap 7399 | Closure law for division. (Contributed by Jim Kingdon, 22-Feb-2020.) |
# | ||
Theorem | recclap 7400 | Closure law for reciprocal. (Contributed by Jim Kingdon, 22-Feb-2020.) |
# |
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