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Theorem List for Intuitionistic Logic Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlttri3d 8101 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  =  B  <->  ( -.  A  <  B  /\  -.  B  <  A ) ) )
 
Theoremletri3d 8102 Tightness of real apartness. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  =  B  <->  ( A  <_  B 
 /\  B  <_  A ) ) )
 
Theoremeqleltd 8103 Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  =  B  <->  ( A  <_  B 
 /\  -.  A  <  B ) ) )
 
Theoremlenltd 8104 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  <_  B  <->  -.  B  <  A ) )
 
Theoremltled 8105 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  A  <_  B )
 
Theoremltnsymd 8106 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  -.  B  <  A )
 
Theoremnltled 8107 'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  -.  B  <  A )   =>    |-  ( ph  ->  A  <_  B )
 
Theoremlensymd 8108 'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  -.  B  <  A )
 
Theoremmulgt0d 8109 The product of two positive numbers is positive. (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 ) )
 
Theoremletrd 8110 Transitive law deduction for 'less than or equal to'. (Contributed by NM, 20-May-2005.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  B 
 <_  C )   =>    |-  ( ph  ->  A  <_  C )
 
Theoremlelttrd 8111 Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <_  B )   &    |-  ( ph  ->  B  <  C )   =>    |-  ( ph  ->  A  <  C )
 
Theoremlttrd 8112 Transitive law deduction for 'less than'. (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 )
 
Theorem0lt1 8113 0 is less than 1. Theorem I.21 of [Apostol] p. 20. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 17-Jan-1997.)
 |-  0  <  1
 
Theoremltntri 8114 Negative trichotomy property for real numbers. It is well known that we cannot prove real number trichotomy,  A  <  B  \/  A  =  B  \/  B  <  A. Does that mean there is a pair of real numbers where none of those hold (that is, where we can refute each of those three relationships)? Actually, no, as shown here. This is another example of distinguishing between being unable to prove something, or being able to refute it. (Contributed by Jim Kingdon, 13-Aug-2023.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( -.  A  <  B  /\  -.  A  =  B  /\  -.  B  <  A ) )
 
4.2.5  Initial properties of the complex numbers
 
Theoremmul12 8115 Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C ) ) )
 
Theoremmul32 8116 Commutative/associative law. (Contributed by NM, 8-Oct-1999.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  B )  x.  C )  =  ( ( A  x.  C )  x.  B ) )
 
Theoremmul31 8117 Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  B )  x.  C )  =  ( ( C  x.  B )  x.  A ) )
 
Theoremmul4 8118 Rearrangement of 4 factors. (Contributed by NM, 8-Oct-1999.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  x.  B )  x.  ( C  x.  D ) )  =  ( ( A  x.  C )  x.  ( B  x.  D ) ) )
 
Theoremmuladd11 8119 A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( 1  +  A )  x.  ( 1  +  B ) )  =  (
 ( 1  +  A )  +  ( B  +  ( A  x.  B ) ) ) )
 
Theorem1p1times 8120 Two times a number. (Contributed by NM, 18-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  ( ( 1  +  1 )  x.  A )  =  ( A  +  A ) )
 
Theorempeano2cn 8121 A theorem for complex numbers analogous the second Peano postulate peano2 4612. (Contributed by NM, 17-Aug-2005.)
 |-  ( A  e.  CC  ->  ( A  +  1 )  e.  CC )
 
Theorempeano2re 8122 A theorem for reals analogous the second Peano postulate peano2 4612. (Contributed by NM, 5-Jul-2005.)
 |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
 
Theoremaddcom 8123 Addition commutes. (Contributed by Jim Kingdon, 17-Jan-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  =  ( B  +  A )
 )
 
Theoremaddrid 8124  0 is an additive identity. (Contributed by Jim Kingdon, 16-Jan-2020.)
 |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
 
Theoremaddlid 8125  0 is a left identity for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  CC  ->  ( 0  +  A )  =  A )
 
Theoremreaddcan 8126 Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( C  +  A )  =  ( C  +  B )  <->  A  =  B ) )
 
Theorem00id 8127  0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( 0  +  0 )  =  0
 
Theoremaddid1i 8128  0 is an additive identity. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  A  e.  CC   =>    |-  ( A  +  0 )  =  A
 
Theoremaddid2i 8129  0 is a left identity for addition. (Contributed by NM, 3-Jan-2013.)
 |-  A  e.  CC   =>    |-  ( 0  +  A )  =  A
 
Theoremaddcomi 8130 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  +  B )  =  ( B  +  A )
 
Theoremaddcomli 8131 Addition commutes. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  ( A  +  B )  =  C   =>    |-  ( B  +  A )  =  C
 
Theoremmul12i 8132 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C ) )
 
Theoremmul32i 8133 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  x.  B )  x.  C )  =  ( ( A  x.  C )  x.  B )
 
Theoremmul4i 8134 Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   =>    |-  ( ( A  x.  B )  x.  ( C  x.  D ) )  =  ( ( A  x.  C )  x.  ( B  x.  D ) )
 
Theoremaddridd 8135  0 is an additive identity. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  +  0 )  =  A )
 
Theoremaddlidd 8136  0 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 0  +  A )  =  A )
 
Theoremaddcomd 8137 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  +  B )  =  ( B  +  A ) )
 
Theoremmul12d 8138 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C ) ) )
 
Theoremmul32d 8139 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  x.  B )  x.  C )  =  ( ( A  x.  C )  x.  B ) )
 
Theoremmul31d 8140 Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  x.  B )  x.  C )  =  ( ( C  x.  B )  x.  A ) )
 
Theoremmul4d 8141 Rearrangement of 4 factors. (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  x.  B )  x.  ( C  x.  D ) )  =  ( ( A  x.  C )  x.  ( B  x.  D ) ) )
 
Theoremmuladd11r 8142 A simple product of sums expansion. (Contributed by AV, 30-Jul-2021.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  1 )  x.  ( B  +  1 ) )  =  ( ( ( A  x.  B )  +  ( A  +  B )
 )  +  1 ) )
 
Theoremcomraddd 8143 Commute RHS addition, in deduction form. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  A  =  ( B  +  C ) )   =>    |-  ( ph  ->  A  =  ( C  +  B ) )
 
4.3  Real and complex numbers - basic operations
 
4.3.1  Addition
 
Theoremadd12 8144 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  +  C )
 )  =  ( B  +  ( A  +  C ) ) )
 
Theoremadd32 8145 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  +  C )  =  ( ( A  +  C )  +  B ) )
 
Theoremadd32r 8146 Commutative/associative law that swaps the last two terms in a triple sum, rearranging the parentheses. (Contributed by Paul Chapman, 18-May-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  +  C )
 )  =  ( ( A  +  C )  +  B ) )
 
Theoremadd4 8147 Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  +  B )  +  ( C  +  D )
 )  =  ( ( A  +  C )  +  ( B  +  D ) ) )
 
Theoremadd42 8148 Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  +  B )  +  ( C  +  D )
 )  =  ( ( A  +  C )  +  ( D  +  B ) ) )
 
Theoremadd12i 8149 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( A  +  ( B  +  C ) )  =  ( B  +  ( A  +  C )
 )
 
Theoremadd32i 8150 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  +  C )  =  ( ( A  +  C )  +  B )
 
Theoremadd4i 8151 Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   =>    |-  ( ( A  +  B )  +  ( C  +  D )
 )  =  ( ( A  +  C )  +  ( B  +  D ) )
 
Theoremadd42i 8152 Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   =>    |-  ( ( A  +  B )  +  ( C  +  D )
 )  =  ( ( A  +  C )  +  ( D  +  B ) )
 
Theoremadd12d 8153 Commutative/associative law that swaps the first two terms in a triple sum. (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 )
 ) )
 
Theoremadd32d 8154 Commutative/associative law that swaps the last two terms in a triple sum. (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 ) )
 
Theoremadd4d 8155 Rearrangement of 4 terms in a sum. (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 )
 ) )
 
Theoremadd42d 8156 Rearrangement of 4 terms in a sum. (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 )
 ) )
 
4.3.2  Subtraction
 
Syntaxcmin 8157 Extend class notation to include subtraction.
 class  -
 
Syntaxcneg 8158 Extend class notation to include unary minus. The symbol  -u is not a class by itself but part of a compound class definition. We do this rather than making it a formal function since it is so commonly used. Note: We use different symbols for unary minus ( -u) and subtraction cmin 8157 ( -) to prevent syntax ambiguity. For example, looking at the syntax definition co 5895, if we used the same symbol then " (  -  A  -  B ) " could mean either " -  A " minus " B", or it could represent the (meaningless) operation of classes " - " and " -  B " connected with "operation" " A". On the other hand, " ( -u A  -  B ) " is unambiguous.
 class  -u A
 
Definitiondf-sub 8159* Define subtraction. Theorem subval 8178 shows its value (and describes how this definition works), Theorem subaddi 8273 relates it to addition, and Theorems subcli 8262 and resubcli 8249 prove its closure laws. (Contributed by NM, 26-Nov-1994.)
 |- 
 -  =  ( x  e.  CC ,  y  e.  CC  |->  ( iota_ z  e. 
 CC  ( y  +  z )  =  x ) )
 
Definitiondf-neg 8160 Define the negative of a number (unary minus). We use different symbols for unary minus ( -u) and subtraction ( -) to prevent syntax ambiguity. See cneg 8158 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
 |-  -u A  =  (
 0  -  A )
 
Theoremcnegexlem1 8161 Addition cancellation of a real number from two complex numbers. Lemma for cnegex 8164. (Contributed by Eric Schmidt, 22-May-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  =  ( A  +  C )  <->  B  =  C ) )
 
Theoremcnegexlem2 8162 Existence of a real number which produces a real number when multiplied by  _i. (Hint: zero is such a number, although we don't need to prove that yet). Lemma for cnegex 8164. (Contributed by Eric Schmidt, 22-May-2007.)
 |- 
 E. y  e.  RR  ( _i  x.  y
 )  e.  RR
 
Theoremcnegexlem3 8163* Existence of real number difference. Lemma for cnegex 8164. (Contributed by Eric Schmidt, 22-May-2007.)
 |-  ( ( b  e. 
 RR  /\  y  e.  RR )  ->  E. c  e.  RR  ( b  +  c )  =  y
 )
 
Theoremcnegex 8164* Existence of the negative of a complex number. (Contributed by Eric Schmidt, 21-May-2007.)
 |-  ( A  e.  CC  ->  E. x  e.  CC  ( A  +  x )  =  0 )
 
Theoremcnegex2 8165* Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  CC  ->  E. x  e.  CC  ( x  +  A )  =  0 )
 
Theoremaddcan 8166 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  =  ( A  +  C )  <->  B  =  C ) )
 
Theoremaddcan2 8167 Cancellation law for addition. (Contributed by NM, 30-Jul-2004.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  C )  =  ( B  +  C )  <->  A  =  B ) )
 
Theoremaddcani 8168 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  =  ( A  +  C )  <->  B  =  C )
 
Theoremaddcan2i 8169 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 14-May-2003.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  C )  =  ( B  +  C )  <->  A  =  B )
 
Theoremaddcand 8170 Cancellation law for addition. Theorem I.1 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 )  =  ( A  +  C )  <->  B  =  C ) )
 
Theoremaddcan2d 8171 Cancellation law for addition. Theorem I.1 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  +  C )  =  ( B  +  C )  <->  A  =  B ) )
 
Theoremaddcanad 8172 Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 8170. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  ( A  +  B )  =  ( A  +  C ) )   =>    |-  ( ph  ->  B  =  C )
 
Theoremaddcan2ad 8173 Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 8171. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  ( A  +  C )  =  ( B  +  C ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremaddneintrd 8174 Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 8172. Consequence of addcand 8170. (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 ) )
 
Theoremaddneintr2d 8175 Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 8173. Consequence of addcan2d 8171. (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 ) )
 
Theorem0cnALT 8176 Alternate proof of 0cn 7978. (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  0  e.  CC
 
Theoremnegeu 8177* Existential uniqueness of negatives. Theorem I.2 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  E! x  e. 
 CC  ( A  +  x )  =  B )
 
Theoremsubval 8178* Value of subtraction, which is the (unique) element  x such that  B  +  x  =  A. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 2-Nov-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B )  =  ( iota_ x  e.  CC  ( B  +  x )  =  A ) )
 
Theoremnegeq 8179 Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
 |-  ( A  =  B  -> 
 -u A  =  -u B )
 
Theoremnegeqi 8180 Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
 |-  A  =  B   =>    |-  -u A  =  -u B
 
Theoremnegeqd 8181 Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  -u A  =  -u B )
 
Theoremnfnegd 8182 Deduction version of nfneg 8183. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/_ x -u A )
 
Theoremnfneg 8183 Bound-variable hypothesis builder for the negative of a complex number. (Contributed by NM, 12-Jun-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x A   =>    |-  F/_ x -u A
 
Theoremcsbnegg 8184 Move class substitution in and out of the negative of a number. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ -u B  =  -u [_ A  /  x ]_ B )
 
Theoremsubcl 8185 Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B )  e.  CC )
 
Theoremnegcl 8186 Closure law for negative. (Contributed by NM, 6-Aug-2003.)
 |-  ( A  e.  CC  -> 
 -u A  e.  CC )
 
Theoremnegicn 8187  -u _i is a complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
 |-  -u _i  e.  CC
 
Theoremsubf 8188 Subtraction is an operation on the complex numbers. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)
 |- 
 -  : ( CC 
 X.  CC ) --> CC
 
Theoremsubadd 8189 Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  =  C  <->  ( B  +  C )  =  A ) )
 
Theoremsubadd2 8190 Relationship between subtraction and addition. (Contributed by Scott Fenton, 5-Jul-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  =  C  <->  ( C  +  B )  =  A ) )
 
Theoremsubsub23 8191 Swap subtrahend and result of subtraction. (Contributed by NM, 14-Dec-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  =  C  <->  ( A  -  C )  =  B ) )
 
Theorempncan 8192 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  -  B )  =  A )
 
Theorempncan2 8193 Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  -  A )  =  B )
 
Theorempncan3 8194 Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( B  -  A ) )  =  B )
 
Theoremnpcan 8195 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  +  B )  =  A )
 
Theoremaddsubass 8196 Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  -  C )  =  ( A  +  ( B  -  C ) ) )
 
Theoremaddsub 8197 Law for addition and subtraction. (Contributed by NM, 19-Aug-2001.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  -  C )  =  ( ( A  -  C )  +  B ) )
 
Theoremsubadd23 8198 Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  +  C )  =  ( A  +  ( C  -  B ) ) )
 
Theoremaddsub12 8199 Commutative/associative law for addition and subtraction. (Contributed by NM, 8-Feb-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  -  C ) )  =  ( B  +  ( A  -  C ) ) )
 
Theorem2addsub 8200 Law for subtraction and addition. (Contributed by NM, 20-Nov-2005.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( ( A  +  B )  +  C )  -  D )  =  ( (
 ( A  +  C )  -  D )  +  B ) )
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