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Theorem List for Intuitionistic Logic Explorer - 7601-7700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsubge02d 7601 Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( 0  <_  B  <->  ( A  -  B )  <_  A ) )
 
Theoremltadd1d 7602 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 7603 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 7604 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 7605 '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 7606 '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 7607 '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 7608 '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 7609 '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 7610 '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 7611 '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 7612 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 7613 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 7614 '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 7615 '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 7616 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 7617 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 7618 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 7619 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 7620 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 7621 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 7622 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 7623 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 7624 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 7625 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 7626 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 7627 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 7628 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 7629 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 7630 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 7631 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 7632 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 7633 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 7634 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 7635 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 7636  1  <_  1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
 |-  1  <_  1
 
Theoremgt0add 7637 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 7638 Class of real apartness relation.
 class #
 
Definitiondf-reap 7639* Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although # is an apartness relation on the reals (see df-ap 7646 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 7651). (Contributed by Jim Kingdon, 26-Jan-2020.)
 |- #  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  ( x  <  y  \/  y  <  x ) ) }
 
Theoremreapval 7640 Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 7652 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 7641 Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 7669 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
 |-  ( A  e.  RR  ->  -.  A #  A )
 
Theoremrecexre 7642* 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 7643 Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 7686. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B 
 <->  -.  A #  B ) )
 
Theoremrecexgt0 7644* 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 7645 Class of complex apartness relation.
 class #
 
Definitiondf-ap 7646* 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 7731 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 7669), symmetry (apsym 7670), and cotransitivity (apcotr 7671). Apartness implies negated equality, as seen at apne 7687, 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 7686).

(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 7647  _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 7648 The imaginary unit  _i is not a real number. (Contributed by NM, 6-May-1999.)
 |- 
 -.  _i  e.  RR
 
Theoremrimul 7649 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 7650 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 7651 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 7652 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 7653 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 7654 One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
 |-  1 #  0
 
Theoremltmul1a 7655 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 7656 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 7657 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 7658 Lemma for reapmul1 7659. (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 7659 Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 7838. (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 7660 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 7661 Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  -u A #  -u B ) )
 
Theoremreapcotr 7662 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 7663 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 7664 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 7665 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 7666 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 7667 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 7668 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 7669 Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
 |-  ( A  e.  CC  ->  -.  A #  A )
 
Theoremapsym 7670 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 7671 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 7672 Addition respects apartness. Analogue of addcan 7253 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 7673 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 7674 Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5548. 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 7675 Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A #  B  <->  -u A #  -u B ) )
 
Theoremmulext1 7676 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 7677 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 7678 Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5548. 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 7679 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 7680 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 7681 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 7682 A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  0  <_  ( A  x.  A ) )
 
Theoremmulge0 7683 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 7684 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 7685 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 7686 Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  =  B 
 <->  -.  A #  B ) )
 
Theoremapne 7687 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 7688 '<_' 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 7689 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  ( ( A  e.  RR  /\  0  <  A )  ->  A #  0 )
 
Theoremgt0ap0i 7690 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 7691 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  RR   &    |-  0  <  A   =>    |-  A #  0
 
Theoremgt0ap0d 7692 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 7693 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 7694 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 7695 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B ) 
 ->  B #  A )
 
Theoremgtapii 7696 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  A  <  B   =>    |-  B #  A
 
Theoremltapii 7697 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  A  <  B   =>    |-  A #  B
 
Theoremltapi 7698 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  ->  B #  A )
 
Theoremgtapd 7699 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  B #  A )
 
Theoremltapd 7700 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  A #  B )
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