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Theorem List for Intuitionistic Logic Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnegsubdi2d 8101 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  -u ( A  -  B )  =  ( B  -  A ) )
 
Theoremneg2subd 8102 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 -u A  -  -u B )  =  ( B  -  A ) )
 
Theoremsubaddd 8103 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  =  C  <->  ( B  +  C )  =  A ) )
 
Theoremsubadd2d 8104 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  =  C  <->  ( C  +  B )  =  A ) )
 
Theoremaddsubassd 8105 Associative-type law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  -  C )  =  ( A  +  ( B  -  C ) ) )
 
Theoremaddsubd 8106 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  -  C )  =  ( ( A  -  C )  +  B ) )
 
Theoremsubadd23d 8107 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  +  C )  =  ( A  +  ( C  -  B ) ) )
 
Theoremaddsub12d 8108 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  +  ( B  -  C ) )  =  ( B  +  ( A  -  C ) ) )
 
Theoremnpncand 8109 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  +  ( B  -  C ) )  =  ( A  -  C ) )
 
Theoremnppcand 8110 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( ( A  -  B )  +  C )  +  B )  =  ( A  +  C ) )
 
Theoremnppcan2d 8111 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  ( B  +  C )
 )  +  C )  =  ( A  -  B ) )
 
Theoremnppcan3d 8112 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  +  ( C  +  B ) )  =  ( A  +  C ) )
 
Theoremsubsubd 8113 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  -  ( B  -  C ) )  =  ( ( A  -  B )  +  C ) )
 
Theoremsubsub2d 8114 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  -  ( B  -  C ) )  =  ( A  +  ( C  -  B ) ) )
 
Theoremsubsub3d 8115 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  -  ( B  -  C ) )  =  ( ( A  +  C )  -  B ) )
 
Theoremsubsub4d 8116 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  -  C )  =  ( A  -  ( B  +  C )
 ) )
 
Theoremsub32d 8117 Swap the second and third terms in a double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  -  C )  =  ( ( A  -  C )  -  B ) )
 
Theoremnnncand 8118 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  ( B  -  C ) )  -  C )  =  ( A  -  B ) )
 
Theoremnnncan1d 8119 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  -  ( A  -  C ) )  =  ( C  -  B ) )
 
Theoremnnncan2d 8120 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  C )  -  ( B  -  C ) )  =  ( A  -  B ) )
 
Theoremnpncan3d 8121 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  +  ( C  -  A ) )  =  ( C  -  B ) )
 
Theorempnpcand 8122 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  -  ( A  +  C ) )  =  ( B  -  C ) )
 
Theorempnpcan2d 8123 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  C )  -  ( B  +  C ) )  =  ( A  -  B ) )
 
Theorempnncand 8124 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  -  ( A  -  C ) )  =  ( B  +  C ) )
 
Theoremppncand 8125 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  +  ( C  -  B ) )  =  ( A  +  C ) )
 
Theoremsubcand 8126 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  ( A  -  B )  =  ( A  -  C ) )   =>    |-  ( ph  ->  B  =  C )
 
Theoremsubcan2d 8127 Cancellation law for subtraction. (Contributed by Mario Carneiro, 22-Sep-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  ( A  -  C )  =  ( B  -  C ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremsubcanad 8128 Cancellation law for subtraction. Deduction form of subcan 8029. Generalization of subcand 8126. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  =  ( A  -  C )  <->  B  =  C ) )
 
Theoremsubneintrd 8129 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcand 8126. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  B  =/=  C )   =>    |-  ( ph  ->  ( A  -  B )  =/=  ( A  -  C ) )
 
Theoremsubcan2ad 8130 Cancellation law for subtraction. Deduction form of subcan2 7999. Generalization of subcan2d 8127. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  C )  =  ( B  -  C )  <->  A  =  B ) )
 
Theoremsubneintr2d 8131 Introducing subtraction on both sides of a statement of inequality. Contrapositive of subcan2d 8127. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ( A  -  C )  =/=  ( B  -  C ) )
 
Theoremaddsub4d 8132 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  -  ( C  +  D ) )  =  ( ( A  -  C )  +  ( B  -  D ) ) )
 
Theoremsubadd4d 8133 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  -  ( C  -  D ) )  =  ( ( A  +  D )  -  ( B  +  C )
 ) )
 
Theoremsub4d 8134 Rearrangement of 4 terms in a subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  -  ( C  -  D ) )  =  ( ( A  -  C )  -  ( B  -  D ) ) )
 
Theorem2addsubd 8135 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( ( A  +  B )  +  C )  -  D )  =  ( ( ( A  +  C )  -  D )  +  B ) )
 
Theoremaddsubeq4d 8136 Relation between sums and differences. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  =  ( C  +  D )  <->  ( C  -  A )  =  ( B  -  D ) ) )
 
Theoremsubeqxfrd 8137 Transfer two terms of a subtraction in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  ( A  -  B )  =  ( C  -  D ) )   =>    |-  ( ph  ->  ( A  -  C )  =  ( B  -  D ) )
 
Theoremmvlraddd 8138 Move LHS right addition to RHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A  +  B )  =  C )   =>    |-  ( ph  ->  A  =  ( C  -  B ) )
 
Theoremmvlladdd 8139 Move LHS left addition to RHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A  +  B )  =  C )   =>    |-  ( ph  ->  B  =  ( C  -  A ) )
 
Theoremmvrraddd 8140 Move RHS right addition to LHS. (Contributed by David A. Wheeler, 15-Oct-2018.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =  ( B  +  C ) )   =>    |-  ( ph  ->  ( A  -  C )  =  B )
 
Theoremmvrladdd 8141 Move RHS left addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =  ( B  +  C ) )   =>    |-  ( ph  ->  ( A  -  B )  =  C )
 
Theoremassraddsubd 8142 Associate RHS addition-subtraction. (Contributed by David A. Wheeler, 15-Oct-2018.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  A  =  ( ( B  +  C )  -  D ) )   =>    |-  ( ph  ->  A  =  ( B  +  ( C  -  D ) ) )
 
Theoremsubaddeqd 8143 Transfer two terms of a subtraction to an addition in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  ( A  +  B )  =  ( C  +  D ) )   =>    |-  ( ph  ->  ( A  -  D )  =  ( C  -  B ) )
 
Theoremaddlsub 8144 Left-subtraction: Subtraction of the left summand from the result of an addition. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  =  C  <->  A  =  ( C  -  B ) ) )
 
Theoremaddrsub 8145 Right-subtraction: Subtraction of the right summand from the result of an addition. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  =  C  <->  B  =  ( C  -  A ) ) )
 
Theoremsubexsub 8146 A subtraction law: Exchanging the subtrahend and the result of the subtraction. (Contributed by BJ, 6-Jun-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  =  ( C  -  B )  <->  B  =  ( C  -  A ) ) )
 
Theoremaddid0 8147 If adding a number to a another number yields the other number, the added number must be  0. This shows that  0 is the unique (right) identity of the complex numbers. (Contributed by AV, 17-Jan-2021.)
 |-  ( ( X  e.  CC  /\  Y  e.  CC )  ->  ( ( X  +  Y )  =  X  <->  Y  =  0
 ) )
 
Theoremaddn0nid 8148 Adding a nonzero number to a complex number does not yield the complex number. (Contributed by AV, 17-Jan-2021.)
 |-  ( ( X  e.  CC  /\  Y  e.  CC  /\  Y  =/=  0 ) 
 ->  ( X  +  Y )  =/=  X )
 
Theorempnpncand 8149 Addition/subtraction cancellation law. (Contributed by Scott Fenton, 14-Dec-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  ( B  -  C ) )  +  ( C  -  B ) )  =  A )
 
Theoremsubeqrev 8150 Reverse the order of subtraction in an equality. (Contributed by Scott Fenton, 8-Jul-2013.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  -  B )  =  ( C  -  D )  <->  ( B  -  A )  =  ( D  -  C ) ) )
 
Theorempncan1 8151 Cancellation law for addition and subtraction with 1. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
 |-  ( A  e.  CC  ->  ( ( A  +  1 )  -  1
 )  =  A )
 
Theoremnpcan1 8152 Cancellation law for subtraction and addition with 1. (Contributed by Alexander van der Vekens, 5-Oct-2018.)
 |-  ( A  e.  CC  ->  ( ( A  -  1 )  +  1
 )  =  A )
 
Theoremsubeq0bd 8153 If two complex numbers are equal, their difference is zero. Consequence of subeq0ad 8095. Converse of subeq0d 8093. Contrapositive of subne0ad 8096. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( A  -  B )  =  0 )
 
Theoremrenegcld 8154 Closure law for negative of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  -u A  e.  RR )
 
Theoremresubcld 8155 Closure law for subtraction of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  -  B )  e.  RR )
 
Theoremnegf1o 8156* Negation is an isomorphism of a subset of the real numbers to the negated elements of the subset. (Contributed by AV, 9-Aug-2020.)
 |-  F  =  ( x  e.  A  |->  -u x )   =>    |-  ( A  C_  RR  ->  F : A -1-1-onto-> { n  e.  RR  |  -u n  e.  A } )
 
4.3.3  Multiplication
 
Theoremkcnktkm1cn 8157 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  e.  CC  ->  ( K  x.  ( K  -  1 ) )  e.  CC )
 
Theoremmuladd 8158 Product of two sums. (Contributed by NM, 14-Jan-2006.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  +  B )  x.  ( C  +  D )
 )  =  ( ( ( A  x.  C )  +  ( D  x.  B ) )  +  ( ( A  x.  D )  +  ( C  x.  B ) ) ) )
 
Theoremsubdi 8159 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 18-Nov-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  -  C ) )  =  ( ( A  x.  B )  -  ( A  x.  C ) ) )
 
Theoremsubdir 8160 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 30-Dec-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  x.  C )  =  ( ( A  x.  C )  -  ( B  x.  C ) ) )
 
Theoremmul02 8161 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 10-Aug-1999.)
 |-  ( A  e.  CC  ->  ( 0  x.  A )  =  0 )
 
Theoremmul02lem2 8162 Zero times a real is zero. Although we prove it as a corollary of mul02 8161, the name is for consistency with the Metamath Proof Explorer which proves it before mul02 8161. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  RR  ->  ( 0  x.  A )  =  0 )
 
Theoremmul01 8163 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 15-May-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  CC  ->  ( A  x.  0
 )  =  0 )
 
Theoremmul02i 8164 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   =>    |-  ( 0  x.  A )  =  0
 
Theoremmul01i 8165 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  A  e.  CC   =>    |-  ( A  x.  0 )  =  0
 
Theoremmul02d 8166 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 0  x.  A )  =  0 )
 
Theoremmul01d 8167 Multiplication by  0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  x.  0 )  =  0 )
 
Theoremine0 8168 The imaginary unit  _i is not zero. (Contributed by NM, 6-May-1999.)
 |-  _i  =/=  0
 
Theoremmulneg1 8169 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.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  x.  B )  =  -u ( A  x.  B ) )
 
Theoremmulneg2 8170 The product with a negative is the negative of the product. (Contributed by NM, 30-Jul-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  -u B )  =  -u ( A  x.  B ) )
 
Theoremmulneg12 8171 Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  x.  B )  =  ( A  x.  -u B ) )
 
Theoremmul2neg 8172 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.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  x.  -u B )  =  ( A  x.  B ) )
 
Theoremsubmul2 8173 Convert a subtraction to addition using multiplication by a negative. (Contributed by NM, 2-Feb-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( B  x.  C ) )  =  ( A  +  ( B  x.  -u C ) ) )
 
Theoremmulm1 8174 Product with minus one is negative. (Contributed by NM, 16-Nov-1999.)
 |-  ( A  e.  CC  ->  ( -u 1  x.  A )  =  -u A )
 
Theoremmulsub 8175 Product of two differences. (Contributed by NM, 14-Jan-2006.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  -  B )  x.  ( C  -  D ) )  =  ( ( ( A  x.  C )  +  ( D  x.  B ) )  -  ( ( A  x.  D )  +  ( C  x.  B ) ) ) )
 
Theoremmulsub2 8176 Swap the order of subtraction in a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  -  B )  x.  ( C  -  D ) )  =  ( ( B  -  A )  x.  ( D  -  C ) ) )
 
Theoremmulm1i 8177 Product with minus one is negative. (Contributed by NM, 31-Jul-1999.)
 |-  A  e.  CC   =>    |-  ( -u 1  x.  A )  =  -u A
 
Theoremmulneg1i 8178 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.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( -u A  x.  B )  =  -u ( A  x.  B )
 
Theoremmulneg2i 8179 Product with negative is negative of product. (Contributed by NM, 31-Jul-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  x.  -u B )  =  -u ( A  x.  B )
 
Theoremmul2negi 8180 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-Feb-1995.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( -u A  x.  -u B )  =  ( A  x.  B )
 
Theoremsubdii 8181 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( A  x.  ( B  -  C ) )  =  ( ( A  x.  B )  -  ( A  x.  C ) )
 
Theoremsubdiri 8182 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 8-May-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  -  B )  x.  C )  =  ( ( A  x.  C )  -  ( B  x.  C ) )
 
Theoremmuladdi 8183 Product of two sums. (Contributed by NM, 17-May-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   =>    |-  ( ( A  +  B )  x.  ( C  +  D )
 )  =  ( ( ( A  x.  C )  +  ( D  x.  B ) )  +  ( ( A  x.  D )  +  ( C  x.  B ) ) )
 
Theoremmulm1d 8184 Product with minus one is negative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( -u 1  x.  A )  =  -u A )
 
Theoremmulneg1d 8185 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 -u A  x.  B )  =  -u ( A  x.  B ) )
 
Theoremmulneg2d 8186 Product with negative is negative of product. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  x.  -u B )  =  -u ( A  x.  B ) )
 
Theoremmul2negd 8187 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 -u A  x.  -u B )  =  ( A  x.  B ) )
 
Theoremsubdid 8188 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  x.  ( B  -  C ) )  =  ( ( A  x.  B )  -  ( A  x.  C ) ) )
 
Theoremsubdird 8189 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  x.  C )  =  ( ( A  x.  C )  -  ( B  x.  C ) ) )
 
Theoremmuladdd 8190 Product of two sums. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  x.  ( C  +  D ) )  =  ( ( ( A  x.  C )  +  ( D  x.  B ) )  +  (
 ( A  x.  D )  +  ( C  x.  B ) ) ) )
 
Theoremmulsubd 8191 Product of two differences. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  x.  ( C  -  D ) )  =  ( ( ( A  x.  C )  +  ( D  x.  B ) )  -  (
 ( A  x.  D )  +  ( C  x.  B ) ) ) )
 
Theoremmulsubfacd 8192 Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  x.  B )  -  B )  =  ( ( A  -  1 )  x.  B ) )
 
4.3.4  Ordering on reals (cont.)
 
Theoremltadd2 8193 Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B ) ) )
 
Theoremltadd2i 8194 Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  C  e.  RR   =>    |-  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B ) )
 
Theoremltadd2d 8195 Addition to 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  +  A )  <  ( C  +  B ) ) )
 
Theoremltadd2dd 8196 Addition to 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  +  A )  <  ( C  +  B ) )
 
Theoremltletrd 8197 Transitive law deduction for 'less than', 'less than or equal to'. (Contributed by NM, 9-Jan-2006.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  B )   &    |-  ( ph  ->  B 
 <_  C )   =>    |-  ( ph  ->  A  <  C )
 
Theoremltaddneg 8198 Adding a negative number to another number decreases it. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  0  <-> 
 ( B  +  A )  <  B ) )
 
Theoremltaddnegr 8199 Adding a negative number to another number decreases it. (Contributed by AV, 19-Mar-2021.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  0  <-> 
 ( A  +  B )  <  B ) )
 
Theoremlelttrdi 8200 If a number is less than another number, and the other number is less than or equal to a third number, the first number is less than the third number. (Contributed by Alexander van der Vekens, 24-Mar-2018.)
 |-  ( ph  ->  ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR ) )   &    |-  ( ph  ->  B 
 <_  C )   =>    |-  ( ph  ->  ( A  <  B  ->  A  <  C ) )
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