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Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsubcan2i 7101 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
 CC   &     CC   &     C  CC   =>     -  C  -  C
 
Theorempnncani 7102 Cancellation law for mixed addition and subtraction. (Contributed by NM, 14-Jan-2006.)
 CC   &     CC   &     C  CC   =>     +  -  -  C  +  C
 
Theoremaddsub4i 7103 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 7104 Alternate proof of 0re 6825. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 0  RR
 
Theoremnegcld 7105 Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     -u  CC
 
Theoremsubidd 7106 Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     -  0
 
Theoremsubid1d 7107 Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     -  0
 
Theoremnegidd 7108 Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     +  -u  0
 
Theoremnegnegd 7109 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 7110 A number is zero iff its negative is zero. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     0  -u  0
 
Theoremnegne0bd 7111 A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     =/=  0  -u  =/=  0
 
Theoremnegcon1d 7112 Contraposition law for unary minus. Deduction form of negcon1 7059. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   =>     -u  -u
 
Theoremnegcon1ad 7113 Contraposition law for unary minus. One-way deduction form of negcon1 7059. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     -u    =>     -u
 
Theoremneg11ad 7114 The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 7058. Generalization of neg11d 7130. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   =>     -u  -u
 
Theoremnegned 7115 If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 7130. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   &     =/=    =>     -u  =/=  -u
 
Theoremnegne0d 7116 The negative of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     =/=  0   =>     -u  =/=  0
 
Theoremnegrebd 7117 The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016.)
 CC   &     -u  RR   =>     RR
 
Theoremsubcld 7118 Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     - 
 CC
 
Theorempncand 7119 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     +  -
 
Theorempncan2d 7120 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     +  -
 
Theorempncan3d 7121 Subtraction and addition of equals. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     +  -
 
Theoremnpcand 7122 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -  +
 
Theoremnncand 7123 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -  -
 
Theoremnegsubd 7124 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     +  -u  -
 
Theoremsubnegd 7125 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -  -u  +
 
Theoremsubeq0d 7126 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     -  0   =>   
 
Theoremsubne0d 7127 Two unequal numbers have nonzero difference. (Contributed by Mario Carneiro, 1-Jan-2017.)
 CC   &     CC   &     =/=    =>     -  =/=  0
 
Theoremsubeq0ad 7128 The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 7033. Generalization of subeq0d 7126. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   =>     -  0
 
Theoremsubne0ad 7129 If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d 7127. Contrapositive of subeq0bd 7173. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   &     -  =/=  0   =>     =/=
 
Theoremneg11d 7130 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     -u  -u   =>   
 
Theoremnegdid 7131 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -u  +  -u  +  -u
 
Theoremnegdi2d 7132 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -u  +  -u  -
 
Theoremnegsubdid 7133 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -u  -  -u  +
 
Theoremnegsubdi2d 7134 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -u  -  -
 
Theoremneg2subd 7135 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -u  -  -u  -
 
Theoremsubaddd 7136 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  C  +  C
 
Theoremsubadd2d 7137 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  C  C  +
 
Theoremaddsubassd 7138 Associative-type law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  -  C  +  -  C
 
Theoremaddsubd 7139 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  -  C  -  C  +
 
Theoremsubadd23d 7140 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  +  C  +  C  -
 
Theoremaddsub12d 7141 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  -  C  + 
 -  C
 
Theoremnpncand 7142 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  +  -  C  -  C
 
Theoremnppcand 7143 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  +  C  +  +  C
 
Theoremnppcan2d 7144 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  +  C  +  C  -
 
Theoremnppcan3d 7145 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  +  C  +  +  C
 
Theoremsubsubd 7146 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  -  C  -  +  C
 
Theoremsubsub2d 7147 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  -  C  +  C  -
 
Theoremsubsub3d 7148 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  -  C  +  C  -
 
Theoremsubsub4d 7149 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  -  C  -  +  C
 
Theoremsub32d 7150 Swap the second and third terms in a double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  -  C  -  C  -
 
Theoremnnncand 7151 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  -  C 
 -  C  -
 
Theoremnnncan1d 7152 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  -  -  C  C  -
 
Theoremnnncan2d 7153 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  C  -  -  C  -
 
Theoremnpncan3d 7154 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  +  C  -  C  -
 
Theorempnpcand 7155 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  -  +  C  -  C
 
Theorempnpcan2d 7156 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  C  -  +  C  -
 
Theorempnncand 7157 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  -  -  C  +  C
 
Theoremppncand 7158 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  +  C  -  +  C
 
Theoremsubcand 7159 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   &     -  -  C   =>     C
 
Theoremsubcan2d 7160 Cancellation law for subtraction. (Contributed by Mario Carneiro, 22-Sep-2016.)
 CC   &     CC   &     C  CC   &     -  C  -  C   =>   
 
Theoremsubcanad 7161 Cancellation law for subtraction. Deduction form of subcan 7062. Generalization of subcand 7159. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   &     C  CC   =>     -  -  C  C
 
Theoremsubneintrd 7162 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcand 7159. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   &     C  CC   &     =/=  C   =>     -  =/=  -  C
 
Theoremsubcan2ad 7163 Cancellation law for subtraction. Deduction form of subcan2 7032. Generalization of subcan2d 7160. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   &     C  CC   =>     -  C  -  C
 
Theoremsubneintr2d 7164 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcan2d 7160. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   &     C  CC   &     =/=    =>     -  C  =/=  -  C
 
Theoremaddsub4d 7165 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   &     D  CC   =>     +  -  C  +  D  -  C  +  -  D
 
Theoremsubadd4d 7166 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   &     D  CC   =>     -  -  C  -  D  +  D  -  +  C
 
Theoremsub4d 7167 Rearrangement of 4 terms in a subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   &     D  CC   =>     -  -  C  -  D  -  C  -  -  D
 
Theorem2addsubd 7168 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   &     D  CC   =>     +  +  C  -  D  +  C  -  D  +
 
Theoremaddsubeq4d 7169 Relation between sums and differences. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   &     D  CC   =>     +  C  +  D  C  -  -  D
 
Theoremsubeqrev 7170 Reverse the order of subtraction in an equality. (Contributed by Scott Fenton, 8-Jul-2013.)
 CC  CC  C  CC  D  CC  -  C  -  D 
 -  D  -  C
 
Theorempncan1 7171 Cancellation law for addition and subtraction with 1. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
 CC  +  1  -  1
 
Theoremnpcan1 7172 Cancellation law for subtraction and addition with 1. (Contributed by Alexander van der Vekens, 5-Oct-2018.)
 CC  -  1  +  1
 
Theoremsubeq0bd 7173 If two complex numbers are equal, their difference is zero. Consequence of subeq0ad 7128. Converse of subeq0d 7126. Contrapositive of subne0ad 7129. (Contributed by David Moews, 28-Feb-2017.)
 CC   &       =>     -  0
 
Theoremrenegcld 7174 Closure law for negative of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   =>     -u  RR
 
Theoremresubcld 7175 Closure law for subtraction of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>     - 
 RR
 
3.3.3  Multiplication
 
Theoremkcnktkm1cn 7176 k times k minus 1 is a complex number if k is a complex number. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
 K  CC  K  x.  K  -  1  CC
 
Theoremmuladd 7177 Product of two sums. (Contributed by NM, 14-Jan-2006.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 CC  CC  C  CC  D  CC  +  x.  C  +  D  x.  C  +  D  x.  +  x.  D  +  C  x.
 
Theoremsubdi 7178 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 18-Nov-2004.)
 CC  CC  C  CC  x.  -  C  x.  -  x.  C
 
Theoremsubdir 7179 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 30-Dec-2005.)
 CC  CC  C  CC  -  x.  C  x.  C  -  x.  C
 
Theoremmul02 7180 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 10-Aug-1999.)
 CC  0  x.  0
 
Theoremmul02lem2 7181 Zero times a real is zero. Although we prove it as a corollary of mul02 7180, the name is for consistency with the Metamath Proof Explorer which proves it before mul02 7180. (Contributed by Scott Fenton, 3-Jan-2013.)
 RR  0  x.  0
 
Theoremmul01 7182 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
 CC  x.  0  0
 
Theoremmul02i 7183 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.)
 CC   =>     0  x.  0
 
Theoremmul01i 7184 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
 CC   =>     x.  0  0
 
Theoremmul02d 7185 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>    
 0  x.  0
 
Theoremmul01d 7186 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     x.  0  0
 
Theoremine0 7187 The imaginary unit  _i is not zero. (Contributed by NM, 6-May-1999.)
 _i  =/=  0
 
Theoremmulneg1 7188 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 CC  CC  -u  x.  -u  x.
 
Theoremmulneg2 7189 The product with a negative is the negative of the product. (Contributed by NM, 30-Jul-2004.)
 CC  CC  x.  -u  -u  x.
 
Theoremmulneg12 7190 Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.)
 CC  CC  -u  x.  x.  -u
 
Theoremmul2neg 7191 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 30-Jul-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 CC  CC  -u  x.  -u  x.
 
Theoremsubmul2 7192 Convert a subtraction to addition using multiplication by a negative. (Contributed by NM, 2-Feb-2007.)
 CC  CC  C  CC  -  x.  C  +  x.  -u C
 
Theoremmulm1 7193 Product with minus one is negative. (Contributed by NM, 16-Nov-1999.)
 CC  -u 1  x.  -u
 
Theoremmulsub 7194 Product of two differences. (Contributed by NM, 14-Jan-2006.)
 CC  CC  C  CC  D  CC  -  x.  C  -  D  x.  C  +  D  x. 
 -  x.  D  +  C  x.
 
Theoremmulsub2 7195 Swap the order of subtraction in a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)
 CC  CC  C  CC  D  CC  -  x.  C  -  D  -  x.  D  -  C
 
Theoremmulm1i 7196 Product with minus one is negative. (Contributed by NM, 31-Jul-1999.)
 CC   =>     -u 1  x.  -u
 
Theoremmulneg1i 7197 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 10-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -u  x.  -u  x.
 
Theoremmulneg2i 7198 Product with negative is negative of product. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     x.  -u  -u  x.
 
Theoremmul2negi 7199 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -u  x.  -u  x.
 
Theoremsubdii 7200 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.)
 CC   &     CC   &     C  CC   =>     x.  -  C  x.  -  x.  C
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