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Theorem List for Intuitionistic Logic Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaddsub12d 7101 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  -  C  + 
 -  C
 
Theoremnpncand 7102 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  +  -  C  -  C
 
Theoremnppcand 7103 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  +  C  +  +  C
 
Theoremnppcan2d 7104 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  +  C  +  C  -
 
Theoremnppcan3d 7105 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  +  C  +  +  C
 
Theoremsubsubd 7106 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  -  C  -  +  C
 
Theoremsubsub2d 7107 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  -  C  +  C  -
 
Theoremsubsub3d 7108 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  -  C  +  C  -
 
Theoremsubsub4d 7109 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  -  C  -  +  C
 
Theoremsub32d 7110 Swap the second and third terms in a double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  -  C  -  C  -
 
Theoremnnncand 7111 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  -  C 
 -  C  -
 
Theoremnnncan1d 7112 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  -  -  C  C  -
 
Theoremnnncan2d 7113 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  C  -  -  C  -
 
Theoremnpncan3d 7114 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  +  C  -  C  -
 
Theorempnpcand 7115 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  -  +  C  -  C
 
Theorempnpcan2d 7116 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  C  -  +  C  -
 
Theorempnncand 7117 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  -  -  C  +  C
 
Theoremppncand 7118 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     +  +  C  -  +  C
 
Theoremsubcand 7119 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   &     -  -  C   =>     C
 
Theoremsubcan2d 7120 Cancellation law for subtraction. (Contributed by Mario Carneiro, 22-Sep-2016.)
 CC   &     CC   &     C  CC   &     -  C  -  C   =>   
 
Theoremsubcanad 7121 Cancellation law for subtraction. Deduction form of subcan 7022. Generalization of subcand 7119. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   &     C  CC   =>     -  -  C  C
 
Theoremsubneintrd 7122 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcand 7119. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   &     C  CC   &     =/=  C   =>     -  =/=  -  C
 
Theoremsubcan2ad 7123 Cancellation law for subtraction. Deduction form of subcan2 6992. Generalization of subcan2d 7120. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   &     C  CC   =>     -  C  -  C
 
Theoremsubneintr2d 7124 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcan2d 7120. (Contributed by David Moews, 28-Feb-2017.)
 CC   &     CC   &     C  CC   &     =/=    =>     -  C  =/=  -  C
 
Theoremaddsub4d 7125 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 7126 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 7127 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 7128 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   &     D  CC   =>     +  +  C  -  D  +  C  -  D  +
 
Theoremaddsubeq4d 7129 Relation between sums and differences. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   &     D  CC   =>     +  C  +  D  C  -  -  D
 
Theoremsubeqrev 7130 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 7131 Cancellation law for addition and subtraction with 1. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
 CC  +  1  -  1
 
Theoremnpcan1 7132 Cancellation law for subtraction and addition with 1. (Contributed by Alexander van der Vekens, 5-Oct-2018.)
 CC  -  1  +  1
 
Theoremsubeq0bd 7133 If two complex numbers are equal, their difference is zero. Consequence of subeq0ad 7088. Converse of subeq0d 7086. Contrapositive of subne0ad 7089. (Contributed by David Moews, 28-Feb-2017.)
 CC   &       =>     -  0
 
Theoremrenegcld 7134 Closure law for negative of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   =>     -u  RR
 
Theoremresubcld 7135 Closure law for subtraction of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   =>     - 
 RR
 
3.3.3  Multiplication
 
Theoremkcnktkm1cn 7136 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 7137 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 7138 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 7139 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 7140 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 10-Aug-1999.)
 CC  0  x.  0
 
Theoremmul02lem2 7141 Zero times a real is zero. Although we prove it as a corollary of mul02 7140, the name is for consistency with the Metamath Proof Explorer which proves it before mul02 7140. (Contributed by Scott Fenton, 3-Jan-2013.)
 RR  0  x.  0
 
Theoremmul01 7142 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 7143 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.)
 CC   =>     0  x.  0
 
Theoremmul01i 7144 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 7145 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>    
 0  x.  0
 
Theoremmul01d 7146 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     x.  0  0
 
Theoremine0 7147 The imaginary unit  _i is not zero. (Contributed by NM, 6-May-1999.)
 _i  =/=  0
 
Theoremmulneg1 7148 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 7149 The product with a negative is the negative of the product. (Contributed by NM, 30-Jul-2004.)
 CC  CC  x.  -u  -u  x.
 
Theoremmulneg12 7150 Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.)
 CC  CC  -u  x.  x.  -u
 
Theoremmul2neg 7151 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 7152 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 7153 Product with minus one is negative. (Contributed by NM, 16-Nov-1999.)
 CC  -u 1  x.  -u
 
Theoremmulsub 7154 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 7155 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 7156 Product with minus one is negative. (Contributed by NM, 31-Jul-1999.)
 CC   =>     -u 1  x.  -u
 
Theoremmulneg1i 7157 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 7158 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 7159 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 7160 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
 
Theoremsubdiri 7161 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 8-May-1999.)
 CC   &     CC   &     C  CC   =>     -  x.  C  x.  C  -  x.  C
 
Theoremmuladdi 7162 Product of two sums. (Contributed by NM, 17-May-1999.)
 CC   &     CC   &     C  CC   &     D  CC   =>     +  x.  C  +  D  x.  C  +  D  x.  +  x.  D  +  C  x.
 
Theoremmulm1d 7163 Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   =>     -u 1  x.  -u
 
Theoremmulneg1d 7164 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -u  x.  -u  x.
 
Theoremmulneg2d 7165 Product with negative is negative of product. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     x.  -u  -u  x.
 
Theoremmul2negd 7166 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   =>     -u  x.  -u  x.
 
Theoremsubdid 7167 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     x.  -  C  x.  -  x.  C
 
Theoremsubdird 7168 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   =>     -  x.  C  x.  C  -  x.  C
 
Theoremmuladdd 7169 Product of two sums. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   &     D  CC   =>     +  x.  C  +  D  x.  C  +  D  x.  +  x.  D  +  C  x.
 
Theoremmulsubd 7170 Product of two differences. (Contributed by Mario Carneiro, 27-May-2016.)
 CC   &     CC   &     C  CC   &     D  CC   =>     -  x.  C  -  D  x.  C  +  D  x. 
 -  x.  D  +  C  x.
 
Theoremmulsubfacd 7171 Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018.)
 CC   &     CC   =>     x.  -  -  1  x.
 
3.3.4  Ordering on reals (cont.)
 
Theoremltadd2 7172 Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 RR  RR  C  RR  <  C  +  <  C  +
 
Theoremltadd2i 7173 Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.)
 RR   &     RR   &     C  RR   =>     <  C  +  <  C  +
 
Theoremltadd2d 7174 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 RR   &     RR   &     C  RR   =>     <  C  +  <  C  +
 
Theoremltadd2dd 7175 Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.)
 RR   &     RR   &     C  RR   &     <    =>     C  +  <  C  +
 
Theoremltletrd 7176 Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.)
 RR   &     RR   &     C  RR   &     <    &     <_  C   =>     <  C
 
Theoremgt0ne0 7177 Positive implies nonzero. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  0  <  =/=  0
 
Theoremlt0ne0 7178 A number which is less than zero is not zero. (Contributed by Stefan O'Rear, 13-Sep-2014.)
 RR  <  0  =/=  0
 
Theoremltadd1 7179 Addition to both sides of 'less than'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 12-Nov-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR  C  RR  <  +  C  <  +  C
 
Theoremleadd1 7180 Addition to both sides of 'less than or equal to'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 18-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR  C  RR  <_  +  C  <_  +  C
 
Theoremleadd2 7181 Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999.)
 RR  RR  C  RR  <_  C  +  <_  C  +
 
Theoremltsubadd 7182 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR  C  RR  -  <  C  <  C  +
 
Theoremltsubadd2 7183 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997.)
 RR  RR  C  RR  -  <  C  <  +  C
 
Theoremlesubadd 7184 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR  C  RR  -  <_  C  <_  C  +
 
Theoremlesubadd2 7185 'Less than or equal to' relationship between subtraction and addition. (Contributed by NM, 10-Aug-1999.)
 RR  RR  C  RR  -  <_  C  <_  +  C
 
Theoremltaddsub 7186 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
 RR  RR  C  RR  +  <  C  <  C  -
 
Theoremltaddsub2 7187 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
 RR  RR  C  RR  +  <  C  <  C  -
 
Theoremleaddsub 7188 'Less than or equal to' relationship between addition and subtraction. (Contributed by NM, 6-Apr-2005.)
 RR  RR  C  RR  +  <_  C  <_  C  -
 
Theoremleaddsub2 7189 'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005.)
 RR  RR  C  RR  +  <_  C  <_  C  -
 
Theoremsuble 7190 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.)
 RR  RR  C  RR  -  <_  C 
 -  C  <_
 
Theoremlesub 7191 Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR  RR  C  RR  <_  -  C  C  <_  -
 
Theoremltsub23 7192 'Less than' relationship between subtraction and addition. (Contributed by NM, 4-Oct-1999.)
 RR  RR  C  RR  -  <  C 
 -  C  <
 
Theoremltsub13 7193 'Less than' relationship between subtraction and addition. (Contributed by NM, 17-Nov-2004.)
 RR  RR  C  RR  <  -  C  C  <  -
 
Theoremle2add 7194 Adding both sides of two 'less than or equal to' relations. (Contributed by NM, 17-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR  C  RR  D  RR  <_  C  <_  D  +  <_  C  +  D
 
Theoremlt2add 7195 Adding both sides of two 'less than' relations. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 15-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 RR  RR  C  RR  D  RR  <  C  <  D  +  <  C  +  D
 
Theoremltleadd 7196 Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)
 RR  RR  C  RR  D  RR  <  C  <_  D  +  <  C  +  D
 
Theoremleltadd 7197 Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)
 RR  RR  C  RR  D  RR  <_  C  <  D  +  <  C  +  D
 
Theoremaddgt0 7198 The sum of 2 positive numbers is positive. (Contributed by NM, 1-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR  RR 
 0  <  0  <  0  <  +
 
Theoremaddgegt0 7199 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR  RR 
 0  <_  0  <  0  <  +
 
Theoremaddgtge0 7200 The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 RR  RR 
 0  <  0  <_  0  <  +
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