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Theorem List for Intuitionistic Logic Explorer - 7301-7400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremleidd 7301 'Less than or equal to' is reflexive. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   =>     <_
 
Theoremlt0neg1d 7302 Comparison of a number and its negative to zero. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   =>     <  0  0  <  -u
 
Theoremlt0neg2d 7303 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   =>    
 0  <  -u  <  0
 
Theoremle0neg1d 7304 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   =>     <_  0  0  <_  -u
 
Theoremle0neg2d 7305 Comparison of a number and its negative to zero. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   =>    
 0  <_  -u  <_  0
 
Theoremaddgegt0d 7306 Addition of nonnegative and positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     0  <_    &     0  <    =>     0  <  +
 
Theoremaddgt0d 7307 Addition of 2 positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     0  <    &     0  <    =>     0  <  +
 
Theoremaddge0d 7308 Addition of 2 nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     0  <_    &     0  <_    =>     0  <_  +
 
Theoremltnegd 7309 Negative of both sides of 'less than'. Theorem I.23 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>     <  -u  < 
 -u
 
Theoremlenegd 7310 Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>     <_  -u  <_ 
 -u
 
Theoremltnegcon1d 7311 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     -u  <    =>     -u  <
 
Theoremltnegcon2d 7312 Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     < 
 -u   =>     < 
 -u
 
Theoremlenegcon1d 7313 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     -u  <_    =>     -u  <_
 
Theoremlenegcon2d 7314 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     <_ 
 -u   =>     <_ 
 -u
 
Theoremltaddposd 7315 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>    
 0  <  <  +
 
Theoremltaddpos2d 7316 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>    
 0  <  <  +
 
Theoremltsubposd 7317 Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>    
 0  < 
 -  <
 
Theoremposdifd 7318 Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>     <  0  <  -
 
Theoremaddge01d 7319 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>    
 0  <_  <_  +
 
Theoremaddge02d 7320 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>    
 0  <_  <_  +
 
Theoremsubge0d 7321 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>    
 0  <_  -  <_
 
Theoremsuble0d 7322 Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>     -  <_  0  <_
 
Theoremsubge02d 7323 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>    
 0  <_ 
 -  <_
 
Theoremltadd1d 7324 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     <  +  C  <  +  C
 
Theoremleadd1d 7325 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     <_  +  C  <_  +  C
 
Theoremleadd2d 7326 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     <_  C  +  <_  C  +
 
Theoremltsubaddd 7327 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     -  <  C  <  C  +
 
Theoremlesubaddd 7328 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     -  <_  C  <_  C  +
 
Theoremltsubadd2d 7329 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     -  <  C  <  +  C
 
Theoremlesubadd2d 7330 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     -  <_  C  <_  +  C
 
Theoremltaddsubd 7331 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     +  <  C  <  C  -
 
Theoremltaddsub2d 7332 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)
 RR   &     RR   &     C  RR   =>     +  <  C  <  C  -
 
Theoremleaddsub2d 7333 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     +  <_  C  <_  C  -
 
Theoremsubled 7334 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   &     -  <_  C   =>     -  C  <_
 
Theoremlesubd 7335 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   &     <_  -  C   =>     C  <_  -
 
Theoremltsub23d 7336 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   &     -  <  C   =>     -  C  <
 
Theoremltsub13d 7337 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   &     <  -  C   =>     C  <  -
 
Theoremlesub1d 7338 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     <_ 
 -  C  <_  -  C
 
Theoremlesub2d 7339 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     <_  C  -  <_  C  -
 
Theoremltsub1d 7340 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     < 
 -  C  <  -  C
 
Theoremltsub2d 7341 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     <  C  -  <  C  -
 
Theoremltadd1dd 7342 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)
 RR   &     RR   &     C  RR   &     <    =>     +  C  <  +  C
 
Theoremltsub1dd 7343 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
 RR   &     RR   &     C  RR   &     <    =>     -  C  <  -  C
 
Theoremltsub2dd 7344 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
 RR   &     RR   &     C  RR   &     <    =>     C  -  <  C  -
 
Theoremleadd1dd 7345 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
 RR   &     RR   &     C  RR   &     <_    =>     +  C  <_  +  C
 
Theoremleadd2dd 7346 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
 RR   &     RR   &     C  RR   &     <_    =>     C  +  <_  C  +
 
Theoremlesub1dd 7347 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
 RR   &     RR   &     C  RR   &     <_    =>     -  C  <_  -  C
 
Theoremlesub2dd 7348 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
 RR   &     RR   &     C  RR   &     <_    =>     C  -  <_  C  -
 
Theoremle2addd 7349 Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   &     D  RR   &     <_  C   &     <_  D   =>     +  <_  C  +  D
 
Theoremle2subd 7350 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   &     D  RR   &     <_  C   &     <_  D   =>     -  D  <_  C  -
 
Theoremltleaddd 7351 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   &     D  RR   &     <  C   &     <_  D   =>     +  <  C  +  D
 
Theoremleltaddd 7352 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   &     D  RR   &     <_  C   &     <  D   =>     +  <  C  +  D
 
Theoremlt2addd 7353 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   &     D  RR   &     <  C   &     <  D   =>     +  <  C  +  D
 
Theoremlt2subd 7354 Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   &     D  RR   &     <  C   &     <  D   =>     -  D  <  C  -
 
Theoremltaddsublt 7355 Addition and subtraction on one side of 'less than'. (Contributed by AV, 24-Nov-2018.)
 RR  RR  C  RR  <  C  +  -  C  <
 
Theorem1le1 7356  1  <_  1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
 1  <_  1
 
Theoremgt0add 7357 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.)
 RR  RR  0  <  +  0  <  0  <
 
3.3.5  Real Apartness
 
Syntaxcreap 7358 Class of real apartness relation.
#
 
Definitiondf-reap 7359* Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although # is an apartness relation on the reals (see df-ap 7366 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 7371). (Contributed by Jim Kingdon, 26-Jan-2020.)
# 
 { <. ,  >.  |  RR  RR  <  <  }
 
Theoremreapval 7360 Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 7372 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 29-Jan-2020.)
 RR  RR # 
 <  <
 
Theoremreapirr 7361 Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 7389 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
 RR #
 
Theoremrecexre 7362* Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.)
 RR #  0  RR  x.  1
 
Theoremreapti 7363 Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 7406. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)
 RR  RR #
 
Theoremrecexgt0 7364* Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.)
 RR  0  <  RR  0  <  x.  1
 
3.3.6  Complex Apartness
 
Syntaxcap 7365 Class of complex apartness relation.
#
 
Definitiondf-ap 7366* 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 7440 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 7389), symmetry (apsym 7390), and cotransitivity (apcotr 7391). Apartness implies negated equality, as seen at apne 7407, 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 7406).

(Contributed by Jim Kingdon, 26-Jan-2020.)

#  { <. ,  >.  |  r 
 RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  }
 
Theoremixi 7367  _i times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 _i  x.  _i  -u 1
 
Theoreminelr 7368 The imaginary unit  _i is not a real number. (Contributed by NM, 6-May-1999.)
 _i  RR
 
Theoremrimul 7369 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.)
 RR  _i  x.  RR  0
 
Theoremrereim 7370 Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.)
 RR  RR  C  RR  +  _i  x.  C  C  0
 
Theoremapreap 7371 Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.)
 RR  RR # #
 
Theoremreaplt 7372 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.)
 RR  RR #  <  <
 
Theoremreapltxor 7373 Real apartness in terms of less than (exclusive-or version). (Contributed by Jim Kingdon, 23-Mar-2020.)
 RR  RR #  <  \/_ 
 <
 
Theorem1ap0 7374 One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
 1 #  0
 
Theoremltmul1a 7375 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.)
 RR  RR  C  RR  0  <  C  <  x.  C  <  x.  C
 
Theoremltmul1 7376 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.)
 RR  RR  C  RR  0  <  C 
 <  x.  C  <  x.  C
 
Theoremlemul1 7377 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.)
 RR  RR  C  RR  0  <  C 
 <_  x.  C  <_  x.  C
 
Theoremreapmul1lem 7378 Lemma for reapmul1 7379. (Contributed by Jim Kingdon, 8-Feb-2020.)
 RR  RR  C  RR  0  <  C #  x.  C #  x.  C
 
Theoremreapmul1 7379 Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 7546. (Contributed by Jim Kingdon, 8-Feb-2020.)
 RR  RR  C  RR  C #  0 #  x.  C #  x.  C
 
Theoremreapadd1 7380 Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
 RR  RR  C  RR #  +  C #  +  C
 
Theoremreapneg 7381 Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
 RR  RR #  -u #  -u
 
Theoremreapcotr 7382 Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)
 RR  RR  C  RR # #  C #  C
 
Theoremremulext1 7383 Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.)
 RR  RR  C  RR  x.  C #  x.  C #
 
Theoremremulext2 7384 Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
 RR  RR  C  RR  C  x. #  C  x. #
 
Theoremapsqgt0 7385 The square of a real number apart from zero is positive. (Contributed by Jim Kingdon, 7-Feb-2020.)
 RR #  0  0  <  x.
 
Theoremcru 7386 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.)
 RR  RR  C  RR  D  RR  +  _i  x.  C  +  _i  x.  D  C  D
 
Theoremapreim 7387 Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.)
 RR  RR  C  RR  D  RR  +  _i  x. #  C  +  _i  x.  D #  C #  D
 
Theoremmulreim 7388 Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.)
 RR  RR  C  RR  D  RR  +  _i  x.  x.  C  +  _i  x.  D  x.  C  +  -u  x.  D  +  _i  x.  C  x.  +  D  x.
 
Theoremapirr 7389 Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
 CC #
 
Theoremapsym 7390 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.)
 CC  CC # #
 
Theoremapcotr 7391 Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.)
 CC  CC  C  CC # #  C #  C
 
Theoremapadd1 7392 Addition respects apartness. Analogue of addcan 6988 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
 CC  CC  C  CC #  +  C #  +  C
 
Theoremapadd2 7393 Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.)
 CC  CC  C  CC #  C  + #  C  +
 
Theoremaddext 7394 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.)
 CC  CC  C  CC  D  CC  + #  C  +  D #  C #  D
 
Theoremapneg 7395 Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
 CC  CC #  -u #  -u
 
Theoremmulext1 7396 Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
 CC  CC  C  CC  x.  C #  x.  C #
 
Theoremmulext2 7397 Right extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
 CC  CC  C  CC  C  x. #  C  x. #
 
Theoremmulext 7398 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.)
 CC  CC  C  CC  D  CC  x. #  C  x.  D #  C #  D
 
Theoremmulap0r 7399 A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 24-Feb-2020.)
 CC  CC  x. #  0 #  0 #  0
 
Theoremmsqge0 7400 A square is nonnegative. Lemma 2.35 of [Geuvers], p. 9. (Contributed by NM, 23-May-2007.) (Revised by Mario Carneiro, 27-May-2016.)
 RR  0  <_  x.
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