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Theorem List for Intuitionistic Logic Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlenegcon1d 8101 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  -u A  <_  B )   =>    |-  ( ph  ->  -u B  <_  A )
 
Theoremlenegcon2d 8102 Contraposition of negative in 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  -u B )   =>    |-  ( ph  ->  B  <_ 
 -u A )
 
Theoremltaddposd 8103 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <  A  <->  B  <  ( B  +  A ) ) )
 
Theoremltaddpos2d 8104 Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <  A  <->  B  <  ( A  +  B ) ) )
 
Theoremltsubposd 8105 Subtracting a positive number from another number decreases it. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <  A  <->  ( B  -  A )  <  B ) )
 
Theoremposdifd 8106 Comparison of two numbers whose difference is positive. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  0  <  ( B  -  A ) ) )
 
Theoremaddge01d 8107 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <_  B  <->  A  <_  ( A  +  B ) ) )
 
Theoremaddge02d 8108 A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <_  B  <->  A  <_  ( B  +  A ) ) )
 
Theoremsubge0d 8109 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <_  ( A  -  B )  <->  B  <_  A ) )
 
Theoremsuble0d 8110 Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( ( A  -  B )  <_  0  <->  A  <_  B ) )
 
Theoremsubge02d 8111 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <_  B  <->  ( A  -  B )  <_  A ) )
 
Theoremltadd1d 8112 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  ( A  +  C )  <  ( B  +  C ) ) )
 
Theoremleadd1d 8113 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( A  +  C )  <_  ( B  +  C ) ) )
 
Theoremleadd2d 8114 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( C  +  A )  <_  ( C  +  B ) ) )
 
Theoremltsubaddd 8115 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( A  -  B )  <  C  <->  A  <  ( C  +  B ) ) )
 
Theoremlesubaddd 8116 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( A  -  B )  <_  C  <->  A  <_  ( C  +  B ) ) )
 
Theoremltsubadd2d 8117 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( A  -  B )  <  C  <->  A  <  ( B  +  C ) ) )
 
Theoremlesubadd2d 8118 'Less than or equal to' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( A  -  B )  <_  C  <->  A  <_  ( B  +  C ) ) )
 
Theoremltaddsubd 8119 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( A  +  B )  <  C  <->  A  <  ( C  -  B ) ) )
 
Theoremltaddsub2d 8120 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 29-Dec-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( A  +  B )  <  C  <->  B  <  ( C  -  A ) ) )
 
Theoremleaddsub2d 8121 'Less than or equal to' relationship between and addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  (
 ( A  +  B )  <_  C  <->  B  <_  ( C  -  A ) ) )
 
Theoremsubled 8122 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  ( A  -  B )  <_  C )   =>    |-  ( ph  ->  ( A  -  C )  <_  B )
 
Theoremlesubd 8123 Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <_  ( B  -  C ) )   =>    |-  ( ph  ->  C  <_  ( B  -  A ) )
 
Theoremltsub23d 8124 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  ( A  -  B )  <  C )   =>    |-  ( ph  ->  ( A  -  C )  <  B )
 
Theoremltsub13d 8125 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  ( B  -  C ) )   =>    |-  ( ph  ->  C  <  ( B  -  A ) )
 
Theoremlesub1d 8126 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( A  -  C )  <_  ( B  -  C ) ) )
 
Theoremlesub2d 8127 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( A  <_  B  <->  ( C  -  B )  <_  ( C  -  A ) ) )
 
Theoremltsub1d 8128 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  ( A  -  C )  <  ( B  -  C ) ) )
 
Theoremltsub2d 8129 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( A  <  B  <->  ( C  -  B )  <  ( C  -  A ) ) )
 
Theoremltadd1dd 8130 Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( A  +  C )  <  ( B  +  C ) )
 
Theoremltsub1dd 8131 Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( A  -  C )  < 
 ( B  -  C ) )
 
Theoremltsub2dd 8132 Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( C  -  B )  < 
 ( C  -  A ) )
 
Theoremleadd1dd 8133 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( A  +  C )  <_  ( B  +  C ) )
 
Theoremleadd2dd 8134 Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( C  +  A )  <_  ( C  +  B ) )
 
Theoremlesub1dd 8135 Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( A  -  C )  <_  ( B  -  C ) )
 
Theoremlesub2dd 8136 Subtraction of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( C  -  B )  <_  ( C  -  A ) )
 
Theoremle2addd 8137 Adding both side of two inequalities. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  A 
 <_  C )   &    |-  ( ph  ->  B 
 <_  D )   =>    |-  ( ph  ->  ( A  +  B )  <_  ( C  +  D ) )
 
Theoremle2subd 8138 Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  A 
 <_  C )   &    |-  ( ph  ->  B 
 <_  D )   =>    |-  ( ph  ->  ( A  -  D )  <_  ( C  -  B ) )
 
Theoremltleaddd 8139 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  A  <  C )   &    |-  ( ph  ->  B  <_  D )   =>    |-  ( ph  ->  ( A  +  B )  <  ( C  +  D ) )
 
Theoremleltaddd 8140 Adding both sides of two orderings. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  A 
 <_  C )   &    |-  ( ph  ->  B  <  D )   =>    |-  ( ph  ->  ( A  +  B )  <  ( C  +  D ) )
 
Theoremlt2addd 8141 Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  A  <  C )   &    |-  ( ph  ->  B  <  D )   =>    |-  ( ph  ->  ( A  +  B )  <  ( C  +  D ) )
 
Theoremlt2subd 8142 Subtracting both sides of two 'less than' relations. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  D  e.  RR )   &    |-  ( ph  ->  A  <  C )   &    |-  ( ph  ->  B  <  D )   =>    |-  ( ph  ->  ( A  -  D )  < 
 ( C  -  B ) )
 
Theorempossumd 8143 Condition for a positive sum. (Contributed by Scott Fenton, 16-Dec-2017.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <  ( A  +  B )  <->  -u B  <  A ) )
 
Theoremsublt0d 8144 When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( ( A  -  B )  <  0  <->  A  <  B ) )
 
Theoremltaddsublt 8145 Addition and subtraction on one side of 'less than'. (Contributed by AV, 24-Nov-2018.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <  C  <->  ( ( A  +  B )  -  C )  <  A ) )
 
Theorem1le1 8146  1  <_  1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
 |-  1  <_  1
 
Theoremgt0add 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.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  0  <  ( A  +  B ) ) 
 ->  ( 0  <  A  \/  0  <  B ) )
 
3.3.5  Real Apartness
 
Syntaxcreap 8148 Class of real apartness relation.
 class #
 
Definitiondf-reap 8149* Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although # 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, # and # agree (apreap 8161). (Contributed by Jim Kingdon, 26-Jan-2020.)
 |- #  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  ( x  <  y  \/  y  <  x ) ) }
 
Theoremreapval 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.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( A  <  B  \/  B  <  A ) ) )
 
Theoremreapirr 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.)
 |-  ( A  e.  RR  ->  -.  A #  A )
 
Theoremrecexre 8152* Existence of reciprocal of real number. (Contributed by Jim Kingdon, 29-Jan-2020.)
 |-  ( ( A  e.  RR  /\  A #  0 )  ->  E. x  e.  RR  ( A  x.  x )  =  1 )
 
Theoremreapti 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.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B 
 <->  -.  A #  B ) )
 
Theoremrecexgt0 8154* Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.)
 |-  ( ( A  e.  RR  /\  0  <  A )  ->  E. x  e.  RR  ( 0  <  x  /\  ( A  x.  x )  =  1 )
 )
 
3.3.6  Complex Apartness
 
Syntaxcap 8155 Class of complex apartness relation.
 class #
 
Definitiondf-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.)

 |- # 
 =  { <. x ,  y >.  |  E. r  e.  RR  E. s  e. 
 RR  E. t  e.  RR  E. u  e.  RR  (
 ( x  =  ( r  +  ( _i 
 x.  s ) ) 
 /\  y  =  ( t  +  ( _i 
 x.  u ) ) )  /\  ( r #  t  \/  s #  u ) ) }
 
Theoremixi 8157  _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 8158 The imaginary unit  _i is not a real number. (Contributed by NM, 6-May-1999.)
 |- 
 -.  _i  e.  RR
 
Theoremrimul 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.)
 |-  ( ( A  e.  RR  /\  ( _i  x.  A )  e.  RR )  ->  A  =  0 )
 
Theoremrereim 8160 Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  A  =  ( B  +  ( _i  x.  C ) ) ) )  ->  ( B  =  A  /\  C  =  0 )
 )
 
Theoremapreap 8161 Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  A #  B ) )
 
Theoremreaplt 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.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( A  <  B  \/  B  <  A ) ) )
 
Theoremreapltxor 8163 Real apartness in terms of less than (exclusive-or version). (Contributed by Jim Kingdon, 23-Mar-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  ( A  <  B  \/_  B  <  A ) ) )
 
Theorem1ap0 8164 One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
 |-  1 #  0
 
Theoremltmul1a 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.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  A  <  B )  ->  ( A  x.  C )  <  ( B  x.  C ) )
 
Theoremltmul1 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.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )
 
Theoremlemul1 8167 Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <_  B  <->  ( A  x.  C )  <_  ( B  x.  C ) ) )
 
Theoremreapmul1lem 8168 Lemma for reapmul1 8169. (Contributed by Jim Kingdon, 8-Feb-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A #  B 
 <->  ( A  x.  C ) #  ( B  x.  C ) ) )
 
Theoremreapmul1 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.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  C #  0 ) ) 
 ->  ( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
 
Theoremreapadd1 8170 Real addition respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  <->  ( A  +  C ) #  ( B  +  C ) ) )
 
Theoremreapneg 8171 Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  -u A #  -u B ) )
 
Theoremreapcotr 8172 Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A #  B  ->  ( A #  C  \/  B #  C ) ) )
 
Theoremremulext1 8173 Left extensionality for multiplication. (Contributed by Jim Kingdon, 19-Feb-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  x.  C ) #  ( B  x.  C )  ->  A #  B ) )
 
Theoremremulext2 8174 Right extensionality for real multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( C  x.  A ) #  ( C  x.  B )  ->  A #  B ) )
 
Theoremapsqgt0 8175 The square of a real number apart from zero is positive. (Contributed by Jim Kingdon, 7-Feb-2020.)
 |-  ( ( A  e.  RR  /\  A #  0 ) 
 ->  0  <  ( A  x.  A ) )
 
Theoremcru 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.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  +  ( _i  x.  B ) )  =  ( C  +  ( _i  x.  D ) )  <->  ( A  =  C  /\  B  =  D ) ) )
 
Theoremapreim 8177 Complex apartness in terms of real and imaginary parts. (Contributed by Jim Kingdon, 12-Feb-2020.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  +  ( _i  x.  B ) ) #  ( C  +  ( _i  x.  D ) )  <->  ( A #  C  \/  B #  D ) ) )
 
Theoremmulreim 8178 Complex multiplication in terms of real and imaginary parts. (Contributed by Jim Kingdon, 23-Feb-2020.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  ( ( A  +  ( _i  x.  B ) )  x.  ( C  +  ( _i  x.  D ) ) )  =  ( ( ( A  x.  C )  +  -u ( B  x.  D ) )  +  ( _i  x.  (
 ( C  x.  B )  +  ( D  x.  A ) ) ) ) )
 
Theoremapirr 8179 Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
 |-  ( A  e.  CC  ->  -.  A #  A )
 
Theoremapsym 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.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A #  B  <->  B #  A ) )
 
Theoremapcotr 8181 Apartness is cotransitive. (Contributed by Jim Kingdon, 16-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A #  B  ->  ( A #  C  \/  B #  C ) ) )
 
Theoremapadd1 8182 Addition respects apartness. Analogue of addcan 7759 for apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A #  B  <->  ( A  +  C ) #  ( B  +  C ) ) )
 
Theoremapadd2 8183 Addition respects apartness. (Contributed by Jim Kingdon, 16-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A #  B  <->  ( C  +  A ) #  ( C  +  B ) ) )
 
Theoremaddext 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.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  +  B ) #  ( C  +  D )  ->  ( A #  C  \/  B #  D ) ) )
 
Theoremapneg 8185 Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A #  B  <->  -u A #  -u B ) )
 
Theoremmulext1 8186 Left extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  C ) #  ( B  x.  C )  ->  A #  B ) )
 
Theoremmulext2 8187 Right extensionality for complex multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( C  x.  A ) #  ( C  x.  B )  ->  A #  B ) )
 
Theoremmulext 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.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  x.  B ) #  ( C  x.  D )  ->  ( A #  C  \/  B #  D ) ) )
 
Theoremmulap0r 8189 A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 24-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( A  x.  B ) #  0 )  ->  ( A #  0  /\  B #  0
 ) )
 
Theoremmsqge0 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.)
 |-  ( A  e.  RR  ->  0  <_  ( A  x.  A ) )
 
Theoremmsqge0i 8191 A square is nonnegative. (Contributed by NM, 14-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  A  e.  RR   =>    |-  0  <_  ( A  x.  A )
 
Theoremmsqge0d 8192 A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  0  <_  ( A  x.  A ) )
 
Theoremmulge0 8193 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B ) )  -> 
 0  <_  ( A  x.  B ) )
 
Theoremmulge0i 8194 The product of two nonnegative numbers is nonnegative. (Contributed by NM, 30-Jul-1999.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( ( 0  <_  A  /\  0  <_  B )  ->  0  <_  ( A  x.  B ) )
 
Theoremmulge0d 8195 The product of two nonnegative numbers is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  0  <_  B )   =>    |-  ( ph  ->  0  <_  ( A  x.  B ) )
 
Theoremapti 8196 Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  =  B 
 <->  -.  A #  B ) )
 
Theoremapne 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.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A #  B  ->  A  =/=  B ) )
 
Theoremleltap 8198 '<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <_  B )  ->  ( A  <  B  <->  B #  A ) )
 
Theoremgt0ap0 8199 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  ( ( A  e.  RR  /\  0  <  A )  ->  A #  0 )
 
Theoremgt0ap0i 8200 Positive means apart from zero (useful for ordering theorems involving division). (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  RR   =>    |-  ( 0  <  A  ->  A #  0 )
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