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Theorem List for Intuitionistic Logic Explorer - 8001-8100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsubled 8001 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 8002 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 8003 '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 8004 '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 8005 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 8006 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 8007 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 8008 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 8009 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 8010 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 8011 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 8012 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 8013 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 8014 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 8015 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 8016 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 8017 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 8018 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 8019 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 8020 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 8021 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 8022 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 8023 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 8024 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 8025  1  <_  1. Common special case. (Contributed by David A. Wheeler, 16-Jul-2016.)
 |-  1  <_  1
 
Theoremgt0add 8026 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 8027 Class of real apartness relation.
 class #
 
Definitiondf-reap 8028* Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although # is an apartness relation on the reals (see df-ap 8035 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 8040). (Contributed by Jim Kingdon, 26-Jan-2020.)
 |- #  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  ( x  <  y  \/  y  <  x ) ) }
 
Theoremreapval 8029 Real apartness in terms of classes. Beyond the development of # itself, proofs should use reaplt 8041 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 8030 Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 8058 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)
 |-  ( A  e.  RR  ->  -.  A #  A )
 
Theoremrecexre 8031* 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 8032 Real apartness is tight. Beyond the development of apartness itself, proofs should use apti 8075. (Contributed by Jim Kingdon, 30-Jan-2020.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  =  B 
 <->  -.  A #  B ) )
 
Theoremrecexgt0 8033* 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 8034 Class of complex apartness relation.
 class #
 
Definitiondf-ap 8035* 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 8120 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8058), symmetry (apsym 8059), and cotransitivity (apcotr 8060). Apartness implies negated equality, as seen at apne 8076, 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 8075).

(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 8036  _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 8037 The imaginary unit  _i is not a real number. (Contributed by NM, 6-May-1999.)
 |- 
 -.  _i  e.  RR
 
Theoremrimul 8038 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 8039 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 8040 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 8041 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 8042 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 8043 One is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
 |-  1 #  0
 
Theoremltmul1a 8044 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 8045 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 8046 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 8047 Lemma for reapmul1 8048. (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 8048 Multiplication of both sides of real apartness by a real number apart from zero. Special case of apmul1 8229. (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 8049 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 8050 Real negation respects apartness. (Contributed by Jim Kingdon, 13-Feb-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A #  B  <->  -u A #  -u B ) )
 
Theoremreapcotr 8051 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 8052 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 8053 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 8054 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 8055 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 8056 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 8057 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 8058 Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
 |-  ( A  e.  CC  ->  -.  A #  A )
 
Theoremapsym 8059 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 8060 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 8061 Addition respects apartness. Analogue of addcan 7641 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 8062 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 8063 Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5643. 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 8064 Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A #  B  <->  -u A #  -u B ) )
 
Theoremmulext1 8065 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 8066 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 8067 Strong extensionality for multiplication. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5643. 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 8068 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 8069 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 8070 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 8071 A square is nonnegative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  0  <_  ( A  x.  A ) )
 
Theoremmulge0 8072 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 8073 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 8074 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 8075 Complex apartness is tight. (Contributed by Jim Kingdon, 21-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  =  B 
 <->  -.  A #  B ) )
 
Theoremapne 8076 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 8077 '<_' 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 8078 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  ( ( A  e.  RR  /\  0  <  A )  ->  A #  0 )
 
Theoremgt0ap0i 8079 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 8080 Positive implies apart from zero. (Contributed by Jim Kingdon, 27-Feb-2020.)
 |-  A  e.  RR   &    |-  0  <  A   =>    |-  A #  0
 
Theoremgt0ap0d 8081 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 8082 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 8083 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 8084 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B ) 
 ->  B #  A )
 
Theoremgtapii 8085 'Greater than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  A  <  B   =>    |-  B #  A
 
Theoremltapii 8086 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  A  <  B   =>    |-  A #  B
 
Theoremltapi 8087 'Less than' implies apart. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  <  B  ->  B #  A )
 
Theoremgtapd 8088 '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 8089 '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 )
 
Theoremleltapd 8090 '<_' implies 'less than' is 'apart'. (Contributed by Jim Kingdon, 13-Aug-2021.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( A  <  B  <->  B #  A )
 )
 
Theoremap0gt0 8091 A nonnegative number is apart from zero if and only if it is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
 |-  ( ( A  e.  RR  /\  0  <_  A )  ->  ( A #  0  <->  0  <  A ) )
 
Theoremap0gt0d 8092 A nonzero nonnegative number is positive. (Contributed by Jim Kingdon, 11-Aug-2021.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0 
 <_  A )   &    |-  ( ph  ->  A #  0 )   =>    |-  ( ph  ->  0  <  A )
 
Theoremsubap0d 8093 Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A #  B )   =>    |-  ( ph  ->  ( A  -  B ) #  0 )
 
3.3.7  Reciprocals
 
Theoremrecextlem1 8094 Lemma for recexap 8096. (Contributed by Eric Schmidt, 23-May-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  ( _i  x.  B ) )  x.  ( A  -  ( _i  x.  B ) ) )  =  ( ( A  x.  A )  +  ( B  x.  B ) ) )
 
Theoremrecexaplem2 8095 Lemma for recexap 8096. (Contributed by Jim Kingdon, 20-Feb-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  (
 ( A  x.  A )  +  ( B  x.  B ) ) #  0 )
 
Theoremrecexap 8096* Existence of reciprocal of nonzero complex number. (Contributed by Jim Kingdon, 20-Feb-2020.)
 |-  ( ( A  e.  CC  /\  A #  0 ) 
 ->  E. x  e.  CC  ( A  x.  x )  =  1 )
 
Theoremmulap0 8097 The product of two numbers apart from zero is apart from zero. Lemma 2.15 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 22-Feb-2020.)
 |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( A  x.  B ) #  0 )
 
Theoremmulap0b 8098 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A #  0  /\  B #  0
 ) 
 <->  ( A  x.  B ) #  0 ) )
 
Theoremmulap0i 8099 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 23-Feb-2020.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  A #  0   &    |-  B #  0   =>    |-  ( A  x.  B ) #  0
 
Theoremmulap0bd 8100 The product of two numbers apart from zero is apart from zero. (Contributed by Jim Kingdon, 24-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A #  0  /\  B #  0 )  <->  ( A  x.  B ) #  0 )
 )
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