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Theorem List for Intuitionistic Logic Explorer - 7001-7100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsub32 7001 Swap the second and third terms in a double subtraction. (Contributed by NM, 19-Aug-2005.)
 CC  CC  C  CC  -  -  C  -  C  -
 
Theoremnnncan 7002 Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)
 CC  CC  C  CC  -  -  C  -  C  -
 
Theoremnnncan1 7003 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 CC  CC  C  CC  -  -  -  C  C  -
 
Theoremnnncan2 7004 Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)
 CC  CC  C  CC  -  C  -  -  C  -
 
Theoremnpncan3 7005 Cancellation law for subtraction. (Contributed by Scott Fenton, 23-Jun-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 CC  CC  C  CC  -  +  C  -  C  -
 
Theorempnpcan 7006 Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 CC  CC  C  CC  +  -  +  C  -  C
 
Theorempnpcan2 7007 Cancellation law for mixed addition and subtraction. (Contributed by Scott Fenton, 9-Jun-2006.)
 CC  CC  C  CC  +  C  -  +  C  -
 
Theorempnncan 7008 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 CC  CC  C  CC  +  -  -  C  +  C
 
Theoremppncan 7009 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.)
 CC  CC  C  CC  +  +  C  -  +  C
 
Theoremaddsub4 7010 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.)
 CC  CC  C  CC  D  CC  +  -  C  +  D  -  C  +  -  D
 
Theoremsubadd4 7011 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 24-Aug-2006.)
 CC  CC  C  CC  D  CC  -  -  C  -  D  +  D  -  +  C
 
Theoremsub4 7012 Rearrangement of 4 terms in a subtraction. (Contributed by NM, 23-Nov-2007.)
 CC  CC  C  CC  D  CC  -  -  C  -  D  -  C  -  -  D
 
Theoremneg0 7013 Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
 -u 0  0
 
Theoremnegid 7014 Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)
 CC  +  -u  0
 
Theoremnegsub 7015 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 CC  CC  +  -u  -
 
Theoremsubneg 7016 Relationship between subtraction and negative. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 CC  CC  -  -u  +
 
Theoremnegneg 7017 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by NM, 12-Jan-2002.) (Revised by Mario Carneiro, 27-May-2016.)
 CC  -u -u
 
Theoremneg11 7018 Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 CC  CC  -u  -u
 
Theoremnegcon1 7019 Negative contraposition law. (Contributed by NM, 9-May-2004.)
 CC  CC  -u  -u
 
Theoremnegcon2 7020 Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
 CC  CC  -u  -u
 
Theoremnegeq0 7021 A number is zero iff its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 CC  0 
 -u  0
 
Theoremsubcan 7022 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 CC  CC  C  CC  -  -  C  C
 
Theoremnegsubdi 7023 Distribution of negative over subtraction. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 CC  CC  -u  -  -u  +
 
Theoremnegdi 7024 Distribution of negative over addition. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 CC  CC  -u  +  -u  +  -u
 
Theoremnegdi2 7025 Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)
 CC  CC  -u  +  -u  -
 
Theoremnegsubdi2 7026 Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)
 CC  CC  -u  -  -
 
Theoremneg2sub 7027 Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
 CC  CC  -u  -  -u  -
 
Theoremrenegcl 7028 Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.)
 RR  -u  RR
 
Theoremrenegcli 7029 Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 7028 is deprecated, but is retained for its demonstration value.) (Contributed by NM, 17-Jan-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 RR   =>     -u  RR
 
Theoremresubcli 7030 Closure law for subtraction of reals. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>     -  RR
 
Theoremresubcl 7031 Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)
 RR  RR  -  RR
 
Theoremnegreb 7032 The negative of a real is real. (Contributed by NM, 11-Aug-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
 CC  -u  RR  RR
 
Theorempeano2cnm 7033 "Reverse" second Peano postulate analog for complex numbers: A complex number minus 1 is a complex number. (Contributed by Alexander van der Vekens, 18-Mar-2018.)
 N  CC  N  -  1  CC
 
Theorempeano2rem 7034 "Reverse" second Peano postulate analog for reals. (Contributed by NM, 6-Feb-2007.)
 N  RR  N  -  1  RR
 
Theoremnegcli 7035 Closure law for negative. (Contributed by NM, 26-Nov-1994.)
 CC   =>     -u  CC
 
Theoremnegidi 7036 Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)
 CC   =>     +  -u  0
 
Theoremnegnegi 7037 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by NM, 8-Feb-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 CC   =>     -u -u
 
Theoremsubidi 7038 Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)
 CC   =>     -  0
 
Theoremsubid1i 7039 Identity law for subtraction. (Contributed by NM, 29-May-1999.)
 CC   =>     -  0
 
Theoremnegne0bi 7040 A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)
 CC   =>     =/=  0  -u  =/=  0
 
Theoremnegrebi 7041 The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
 CC   =>     -u  RR  RR
 
Theoremnegne0i 7042 The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)
 CC   &     =/=  0   =>     -u  =/=  0
 
Theoremsubcli 7043 Closure law for subtraction. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
 CC   &     CC   =>     -  CC
 
Theorempncan3i 7044 Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.)
 CC   &     CC   =>     +  -
 
Theoremnegsubi 7045 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 CC   &     CC   =>     +  -u  -
 
Theoremsubnegi 7046 Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.)
 CC   &     CC   =>     -  -u  +
 
Theoremsubeq0i 7047 If the difference between two numbers is zero, they are equal. (Contributed by NM, 8-May-1999.)
 CC   &     CC   =>     -  0
 
Theoremneg11i 7048 Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
 CC   &     CC   =>     -u  -u
 
Theoremnegcon1i 7049 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
 CC   &     CC   =>     -u  -u
 
Theoremnegcon2i 7050 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
 CC   &     CC   =>     -u  -u
 
Theoremnegdii 7051 Distribution of negative over addition. (Contributed by NM, 28-Jul-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 CC   &     CC   =>     -u  +  -u  +  -u
 
Theoremnegsubdii 7052 Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)
 CC   &     CC   =>     -u  -  -u  +
 
Theoremnegsubdi2i 7053 Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)
 CC   &     CC   =>     -u  -  -
 
Theoremsubaddi 7054 Relationship between subtraction and addition. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
 CC   &     CC   &     C  CC   =>     -  C  +  C
 
Theoremsubadd2i 7055 Relationship between subtraction and addition. (Contributed by NM, 15-Dec-2006.)
 CC   &     CC   &     C  CC   =>     -  C  C  +
 
Theoremsubaddrii 7056 Relationship between subtraction and addition. (Contributed by NM, 16-Dec-2006.)
 CC   &     CC   &     C  CC   &     +  C    =>     -  C
 
Theoremsubsub23i 7057 Swap subtrahend and result of subtraction. (Contributed by NM, 7-Oct-1999.)
 CC   &     CC   &     C  CC   =>     -  C  -  C
 
Theoremaddsubassi 7058 Associative-type law for subtraction and addition. (Contributed by NM, 16-Sep-1999.)
 CC   &     CC   &     C  CC   =>     +  -  C  +  -  C
 
Theoremaddsubi 7059 Law for subtraction and addition. (Contributed by NM, 6-Aug-2003.)
 CC   &     CC   &     C  CC   =>     +  -  C  -  C  +
 
Theoremsubcani 7060 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
 CC   &     CC   &     C  CC   =>     -  -  C  C
 
Theoremsubcan2i 7061 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
 CC   &     CC   &     C  CC   =>     -  C  -  C
 
Theorempnncani 7062 Cancellation law for mixed addition and subtraction. (Contributed by NM, 14-Jan-2006.)
 CC   &     CC   &     C  CC   =>     +  -  -  C  +  C
 
Theoremaddsub4i 7063 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 17-Oct-1999.)
 CC   &     CC   &     C  CC   &     D  CC   =>     +  -  C  +  D  -  C  +  -  D
 
Theorem0reALT 7064 Alternate proof of 0re 6785. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 0  RR
 
Theoremnegcld 7065 Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     -u  CC
 
Theoremsubidd 7066 Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     -  0
 
Theoremsubid1d 7067 Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     -  0
 
Theoremnegidd 7068 Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     +  -u  0
 
Theoremnegnegd 7069 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     -u -u
 
Theoremnegeq0d 7070 A number is zero iff its negative is zero. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     0  -u  0
 
Theoremnegne0bd 7071 A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     =/=  0  -u  =/=  0
 
Theoremnegcon1d 7072 Contraposition law for unary minus. Deduction form of negcon1 7019. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   =>     -u  -u
 
Theoremnegcon1ad 7073 Contraposition law for unary minus. One-way deduction form of negcon1 7019. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     -u    =>     -u
 
Theoremneg11ad 7074 The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 7018. Generalization of neg11d 7090. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   =>     -u  -u
 
Theoremnegned 7075 If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 7090. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   &     =/=    =>     -u  =/=  -u
 
Theoremnegne0d 7076 The negative of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     =/=  0   =>     -u  =/=  0
 
Theoremnegrebd 7077 The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016.)
 CC   &     -u  RR   =>     RR
 
Theoremsubcld 7078 Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     - 
 CC
 
Theorempncand 7079 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     +  -
 
Theorempncan2d 7080 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     +  -
 
Theorempncan3d 7081 Subtraction and addition of equals. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     +  -
 
Theoremnpcand 7082 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -  +
 
Theoremnncand 7083 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -  -
 
Theoremnegsubd 7084 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     +  -u  -
 
Theoremsubnegd 7085 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -  -u  +
 
Theoremsubeq0d 7086 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     -  0   =>   
 
Theoremsubne0d 7087 Two unequal numbers have nonzero difference. (Contributed by Mario Carneiro, 1-Jan-2017.)
 CC   &     CC   &     =/=    =>     -  =/=  0
 
Theoremsubeq0ad 7088 The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 6993. Generalization of subeq0d 7086. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   =>     -  0
 
Theoremsubne0ad 7089 If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d 7087. Contrapositive of subeq0bd 7133. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   &     -  =/=  0   =>     =/=
 
Theoremneg11d 7090 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     -u  -u   =>   
 
Theoremnegdid 7091 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -u  +  -u  +  -u
 
Theoremnegdi2d 7092 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -u  +  -u  -
 
Theoremnegsubdid 7093 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -u  -  -u  +
 
Theoremnegsubdi2d 7094 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -u  -  -
 
Theoremneg2subd 7095 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -u  -  -u  -
 
Theoremsubaddd 7096 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  C  +  C
 
Theoremsubadd2d 7097 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  C  C  +
 
Theoremaddsubassd 7098 Associative-type law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  -  C  +  -  C
 
Theoremaddsubd 7099 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  -  C  -  C  +
 
Theoremsubadd23d 7100 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  +  C  +  C  -
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