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Theorem List for Intuitionistic Logic Explorer - 8201-8300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlenegcon2d 8201 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddposd 8202 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddpos2d 8203 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsubposd 8204 Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremposdifd 8205 Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddge01d 8206 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddge02d 8207 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubge0d 8208 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsuble0d 8209 Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubge02d 8210 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltadd1d 8211 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleadd1d 8212 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleadd2d 8213 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsubaddd 8214 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesubaddd 8215 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsubadd2d 8216 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesubadd2d 8217 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddsubd 8218 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltaddsub2d 8219 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)

Theoremleaddsub2d 8220 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubled 8221 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesubd 8222 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub23d 8223 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub13d 8224 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesub1d 8225 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlesub2d 8226 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub1d 8227 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltsub2d 8228 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltadd1dd 8229 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltsub1dd 8230 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremltsub2dd 8231 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremleadd1dd 8232 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremleadd2dd 8233 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlesub1dd 8234 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlesub2dd 8235 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremle2addd 8236 Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremle2subd 8237 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremltleaddd 8238 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremleltaddd 8239 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2addd 8240 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremlt2subd 8241 Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempossumd 8242 Condition for a positive sum. (Contributed by Scott Fenton, 16-Dec-2017.)

Theoremsublt0d 8243 When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Theoremltaddsublt 8244 Addition and subtraction on one side of 'less than'. (Contributed by AV, 24-Nov-2018.)

Theorem1le1 8245 . Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)

Theoremgt0add 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.)

3.3.5  Real Apartness

Syntaxcreap 8247 Class of real apartness relation.
#

Definitiondf-reap 8248* Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although # 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, # and # agree (apreap 8260). (Contributed by Jim Kingdon, 26-Jan-2020.)
#

Theoremreapval 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.)
#

Theoremreapirr 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.)
#

Theoremrecexre 8251* Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.)
#

Theoremreapti 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.)
#

Theoremrecexgt0 8253* Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.)

3.3.6  Complex Apartness

Syntaxcap 8254 Class of complex apartness relation.
#

Definitiondf-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.)

# # #

Theoremixi 8256 times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoreminelr 8257 The imaginary unit is not a real number. (Contributed by NM, 6-May-1999.)

Theoremrimul 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.)

Theoremrereim 8259 Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.)

Theoremapreap 8260 Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.)
# #

Theoremreaplt 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.)
#

Theoremreapltxor 8262 Real apartness in terms of less than (exclusive-or version). (Contributed by Jim Kingdon, 23-Mar-2020.)
#

Theorem1ap0 8263 One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
#

Theoremltmul1a 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.)

Theoremltmul1 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.)

Theoremlemul1 8266 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.)

Theoremreapmul1lem 8267 Lemma for reapmul1 8268. (Contributed by Jim Kingdon, 8-Feb-2020.)
# #

Theoremreapmul1 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.)
# # #

# #

Theoremreapneg 8270 Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
# #

Theoremreapcotr 8271 Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)
# # #

Theoremremulext1 8272 Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.)
# #

Theoremremulext2 8273 Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
# #

Theoremapsqgt0 8274 The square of a real number apart from zero is positive. (Contributed by Jim Kingdon, 7-Feb-2020.)
#

Theoremcru 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.)

Theoremapreim 8276 Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.)
# # #

Theoremmulreim 8277 Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.)

Theoremapirr 8278 Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
#

Theoremapsym 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.)
# #

Theoremapcotr 8280 Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.)
# # #

# #

# #

Theoremaddext 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.)
# # #

Theoremapneg 8284 Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
# #

Theoremmulext1 8285 Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
# #

Theoremmulext2 8286 Right extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
# #

Theoremmulext 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.)
# # #

Theoremmulap0r 8288 A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 24-Feb-2020.)
# # #

Theoremmsqge0 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.)

Theoremmsqge0i 8290 A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremmsqge0d 8291 A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmulge0 8292 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremmulge0i 8293 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.)

Theoremmulge0d 8294 The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremapti 8295 Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)
#

Theoremapne 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.)
#

Theoremapcon4bid 8297 Contrapositive law deduction for apartness. (Contributed by Jim Kingdon, 31-Jul-2023.)
# #

Theoremleltap 8298 implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
#

Theoremgt0ap0 8299 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
#

Theoremgt0ap0i 8300 Positive means apart from zero (useful for ordering theorems involving division). (Contributed by Jim Kingdon, 27-Feb-2020.)
#

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