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Theorem List for Intuitionistic Logic Explorer - 7701-7800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaddge02d 7701 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 7702 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <_  ( A  -  B )  <->  B  <_  A ) )
 
Theoremsuble0d 7703 Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( ( A  -  B )  <_  0  <->  A  <_  B ) )
 
Theoremsubge02d 7704 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <_  B  <->  ( A  -  B )  <_  A ) )
 
Theoremltadd1d 7705 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 7706 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 7707 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 7708 '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 7709 '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 7710 '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 7711 '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 7712 '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 7713 '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 7714 '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 7715 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 7716 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 7717 '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 7718 '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 7719 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 7720 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 7721 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 7722 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 7723 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 7724 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 7725 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 7726 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 7727 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 7728 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 7729 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 7730 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 7731 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 7732 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 7733 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 7734 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 7735 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 7736 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 7737 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 7738 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 7739  1  <_  1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
 |-  1  <_  1
 
Theoremgt0add 7740 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 7741 Class of real apartness relation.
 class #
 
Definitiondf-reap 7742* Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although # is an apartness relation on the reals (see df-ap 7749 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 7754). (Contributed by Jim Kingdon, 26-Jan-2020.)
 |- #  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  ( x  <  y  \/  y  <  x ) ) }
 
Theoremreapval 7743 Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 7755 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 7744 Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 7772 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
 |-  ( A  e.  RR  ->  -.  A #  A )
 
Theoremrecexre 7745* 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 7746 Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 7789. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B 
 <->  -.  A #  B ) )
 
Theoremrecexgt0 7747* 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 7748 Class of complex apartness relation.
 class #
 
Definitiondf-ap 7749* 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 7834 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 7772), symmetry (apsym 7773), and cotransitivity (apcotr 7774). Apartness implies negated equality, as seen at apne 7790, 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 7789).

(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 7750  _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 7751 The imaginary unit  _i is not a real number. (Contributed by NM, 6-May-1999.)
 |- 
 -.  _i  e.  RR
 
Theoremrimul 7752 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 7753 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 7754 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 7755 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 7756 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 7757 One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
 |-  1 #  0
 
Theoremltmul1a 7758 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 7759 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 7760 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 7761 Lemma for reapmul1 7762. (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 7762 Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 7943. (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 7763 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 7764 Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  -u A #  -u B ) )
 
Theoremreapcotr 7765 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 7766 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 7767 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 7768 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 7769 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 7770 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 7771 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 7772 Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
 |-  ( A  e.  CC  ->  -.  A #  A )
 
Theoremapsym 7773 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 7774 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 7775 Addition respects apartness. Analogue of addcan 7355 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 7776 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 7777 Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5552. 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 7778 Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A #  B  <->  -u A #  -u B ) )
 
Theoremmulext1 7779 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 7780 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 7781 Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5552. 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 7782 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 7783 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 7784 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 7785 A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  0  <_  ( A  x.  A ) )
 
Theoremmulge0 7786 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 7787 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 7788 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 7789 Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  =  B 
 <->  -.  A #  B ) )
 
Theoremapne 7790 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 7791 '<_' 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 7792 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  ( ( A  e.  RR  /\  0  <  A )  ->  A #  0 )
 
Theoremgt0ap0i 7793 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 )
 
Theoremgt0ap0ii 7794 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  RR   &    |-  0  <  A   =>    |-  A #  0
 
Theoremgt0ap0d 7795 Positive implies apart from zero. Because of the way we define #,  A must be an element of  RR, not just  RR*. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <  A )   =>    |-  ( ph  ->  A #  0 )
 
Theoremnegap0 7796 A number is apart from zero iff its negative is apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  ( A  e.  CC  ->  ( A #  0  <->  -u A #  0 ) )
 
Theoremltleap 7797 Less than in terms of non-strict order and apartness. (Contributed by Jim Kingdon, 28-Feb-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <-> 
 ( A  <_  B  /\  A #  B ) ) )
 
Theoremltap 7798 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B ) 
 ->  B #  A )
 
Theoremgtapii 7799 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  A  <  B   =>    |-  B #  A
 
Theoremltapii 7800 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  A  <  B   =>    |-  A #  B
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