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Theorem List for Intuitionistic Logic Explorer - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
4.3.2  Subtraction
 
Syntaxcmin 7901 Extend class notation to include subtraction.
 class  -
 
Syntaxcneg 7902 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 7901 ( -) to prevent syntax ambiguity. For example, looking at the syntax definition co 5742, 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 7903* Define subtraction. Theorem subval 7922 shows its value (and describes how this definition works), theorem subaddi 8017 relates it to addition, and theorems subcli 8006 and resubcli 7993 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 7904 Define the negative of a number (unary minus). We use different symbols for unary minus ( -u) and subtraction ( -) to prevent syntax ambiguity. See cneg 7902 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
 |-  -u A  =  (
 0  -  A )
 
Theoremcnegexlem1 7905 Addition cancellation of a real number from two complex numbers. Lemma for cnegex 7908. (Contributed by Eric Schmidt, 22-May-2007.)
 |-  ( ( A  e.  RR  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  =  ( A  +  C )  <->  B  =  C ) )
 
Theoremcnegexlem2 7906 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 7908. (Contributed by Eric Schmidt, 22-May-2007.)
 |- 
 E. y  e.  RR  ( _i  x.  y
 )  e.  RR
 
Theoremcnegexlem3 7907* Existence of real number difference. Lemma for cnegex 7908. (Contributed by Eric Schmidt, 22-May-2007.)
 |-  ( ( b  e. 
 RR  /\  y  e.  RR )  ->  E. c  e.  RR  ( b  +  c )  =  y
 )
 
Theoremcnegex 7908* 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 7909* 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 7910 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 7911 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 7912 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 7913 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 7914 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 7915 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 7916 Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 7914. (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 7917 Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 7915. (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 7918 Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 7916. Consequence of addcand 7914. (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 7919 Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 7917. Consequence of addcan2d 7915. (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 7920 Alternate proof of 0cn 7726. (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 7921* 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 7922* 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 7923 Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
 |-  ( A  =  B  -> 
 -u A  =  -u B )
 
Theoremnegeqi 7924 Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
 |-  A  =  B   =>    |-  -u A  =  -u B
 
Theoremnegeqd 7925 Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  -u A  =  -u B )
 
Theoremnfnegd 7926 Deduction version of nfneg 7927. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/_ x -u A )
 
Theoremnfneg 7927 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 7928 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 7929 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 7930 Closure law for negative. (Contributed by NM, 6-Aug-2003.)
 |-  ( A  e.  CC  -> 
 -u A  e.  CC )
 
Theoremnegicn 7931  -u _i is a complex number (common case). (Contributed by David A. Wheeler, 7-Dec-2018.)
 |-  -u _i  e.  CC
 
Theoremsubf 7932 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 7933 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 7934 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 7935 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 7936 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 7937 Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  -  A )  =  B )
 
Theorempncan3 7938 Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( B  -  A ) )  =  B )
 
Theoremnpcan 7939 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 7940 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 7941 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 7942 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 7943 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 7944 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 ) )
 
Theoremaddsubeq4 7945 Relation between sums and differences. (Contributed by Jeff Madsen, 17-Jun-2010.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  +  B )  =  ( C  +  D )  <->  ( C  -  A )  =  ( B  -  D ) ) )
 
Theorempncan3oi 7946 Subtraction and addition of equals. Almost but not exactly the same as pncan3i 8007 and pncan 7936, this order happens often when applying "operations to both sides" so create a theorem specifically for it. A deduction version of this is available as pncand 8042. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( A  +  B )  -  B )  =  A
 
Theoremmvrraddi 7947 Move RHS right addition to LHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  B  e.  CC   &    |-  C  e.  CC   &    |-  A  =  ( B  +  C )   =>    |-  ( A  -  C )  =  B
 
Theoremmvlladdi 7948 Move LHS left addition to RHS. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  ( A  +  B )  =  C   =>    |-  B  =  ( C  -  A )
 
Theoremsubid 7949 Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  ( A  -  A )  =  0 )
 
Theoremsubid1 7950 Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  ( A  -  0
 )  =  A )
 
Theoremnpncan 7951 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  +  ( B  -  C ) )  =  ( A  -  C ) )
 
Theoremnppcan 7952 Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( ( A  -  B )  +  C )  +  B )  =  ( A  +  C ) )
 
Theoremnnpcan 7953 Cancellation law for subtraction: ((a-b)-c)+b = a-c holds for complex numbers a,b,c. (Contributed by Alexander van der Vekens, 24-Mar-2018.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( ( A  -  B )  -  C )  +  B )  =  ( A  -  C ) )
 
Theoremnppcan3 7954 Cancellation law for subtraction. (Contributed by Mario Carneiro, 14-Sep-2015.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  +  ( C  +  B )
 )  =  ( A  +  C ) )
 
Theoremsubcan2 7955 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C )  =  ( B  -  C )  <->  A  =  B ) )
 
Theoremsubeq0 7956 If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  =  0  <->  A  =  B ) )
 
Theoremnpncan2 7957 Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  +  ( B  -  A ) )  =  0
 )
 
Theoremsubsub2 7958 Law for double subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( B  -  C ) )  =  ( A  +  ( C  -  B ) ) )
 
Theoremnncan 7959 Cancellation law for subtraction. (Contributed by NM, 21-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  ( A  -  B ) )  =  B )
 
Theoremsubsub 7960 Law for double subtraction. (Contributed by NM, 13-May-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( B  -  C ) )  =  ( ( A  -  B )  +  C ) )
 
Theoremnppcan2 7961 Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  ( B  +  C ) )  +  C )  =  ( A  -  B ) )
 
Theoremsubsub3 7962 Law for double subtraction. (Contributed by NM, 27-Jul-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( B  -  C ) )  =  ( ( A  +  C )  -  B ) )
 
Theoremsubsub4 7963 Law for double subtraction. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  -  C )  =  ( A  -  ( B  +  C ) ) )
 
Theoremsub32 7964 Swap the second and third terms in a double subtraction. (Contributed by NM, 19-Aug-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  -  C )  =  ( ( A  -  C )  -  B ) )
 
Theoremnnncan 7965 Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  ( B  -  C ) )  -  C )  =  ( A  -  B ) )
 
Theoremnnncan1 7966 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  -  ( A  -  C ) )  =  ( C  -  B ) )
 
Theoremnnncan2 7967 Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  C )  -  ( B  -  C ) )  =  ( A  -  B ) )
 
Theoremnpncan3 7968 Cancellation law for subtraction. (Contributed by Scott Fenton, 23-Jun-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  +  ( C  -  A ) )  =  ( C  -  B ) )
 
Theorempnpcan 7969 Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  -  ( A  +  C )
 )  =  ( B  -  C ) )
 
Theorempnpcan2 7970 Cancellation law for mixed addition and subtraction. (Contributed by Scott Fenton, 9-Jun-2006.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  C )  -  ( B  +  C )
 )  =  ( A  -  B ) )
 
Theorempnncan 7971 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  -  ( A  -  C ) )  =  ( B  +  C ) )
 
Theoremppncan 7972 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  +  ( C  -  B ) )  =  ( A  +  C ) )
 
Theoremaddsub4 7973 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  +  B )  -  ( C  +  D )
 )  =  ( ( A  -  C )  +  ( B  -  D ) ) )
 
Theoremsubadd4 7974 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 24-Aug-2006.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  -  B )  -  ( C  -  D ) )  =  ( ( A  +  D )  -  ( B  +  C ) ) )
 
Theoremsub4 7975 Rearrangement of 4 terms in a subtraction. (Contributed by NM, 23-Nov-2007.)
 |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( C  e.  CC  /\  D  e.  CC ) )  ->  ( ( A  -  B )  -  ( C  -  D ) )  =  ( ( A  -  C )  -  ( B  -  D ) ) )
 
Theoremneg0 7976 Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
 |-  -u 0  =  0
 
Theoremnegid 7977 Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)
 |-  ( A  e.  CC  ->  ( A  +  -u A )  =  0 )
 
Theoremnegsub 7978 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.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
 
Theoremsubneg 7979 Relationship between subtraction and negative. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  -u B )  =  ( A  +  B ) )
 
Theoremnegneg 7980 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.)
 |-  ( A  e.  CC  -> 
 -u -u A  =  A )
 
Theoremneg11 7981 Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  =  -u B  <->  A  =  B ) )
 
Theoremnegcon1 7982 Negative contraposition law. (Contributed by NM, 9-May-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  =  B  <->  -u B  =  A ) )
 
Theoremnegcon2 7983 Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  =  -u B  <->  B  =  -u A ) )
 
Theoremnegeq0 7984 A number is zero iff its negative is zero. (Contributed by NM, 12-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  ( A  =  0  <->  -u A  =  0 ) )
 
Theoremsubcan 7985 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  =  ( A  -  C )  <->  B  =  C ) )
 
Theoremnegsubdi 7986 Distribution of negative over subtraction. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  -u ( A  -  B )  =  ( -u A  +  B ) )
 
Theoremnegdi 7987 Distribution of negative over addition. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  -u ( A  +  B )  =  ( -u A  +  -u B ) )
 
Theoremnegdi2 7988 Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  -u ( A  +  B )  =  ( -u A  -  B ) )
 
Theoremnegsubdi2 7989 Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  -u ( A  -  B )  =  ( B  -  A ) )
 
Theoremneg2sub 7990 Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  -  -u B )  =  ( B  -  A ) )
 
Theoremrenegcl 7991 Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.)
 |-  ( A  e.  RR  -> 
 -u A  e.  RR )
 
Theoremrenegcli 7992 Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 7991 is deprecated, but is retained for its demonstration value.) (Contributed by NM, 17-Jan-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  A  e.  RR   =>    |-  -u A  e.  RR
 
Theoremresubcli 7993 Closure law for subtraction of reals. (Contributed by NM, 17-Jan-1997.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  -  B )  e.  RR
 
Theoremresubcl 7994 Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B )  e.  RR )
 
Theoremnegreb 7995 The negative of a real is real. (Contributed by NM, 11-Aug-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)
 |-  ( A  e.  CC  ->  ( -u A  e.  RR  <->  A  e.  RR ) )
 
Theorempeano2cnm 7996 "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  e.  CC  ->  ( N  -  1
 )  e.  CC )
 
Theorempeano2rem 7997 "Reverse" second Peano postulate analog for reals. (Contributed by NM, 6-Feb-2007.)
 |-  ( N  e.  RR  ->  ( N  -  1
 )  e.  RR )
 
Theoremnegcli 7998 Closure law for negative. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   =>    |-  -u A  e.  CC
 
Theoremnegidi 7999 Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   =>    |-  ( A  +  -u A )  =  0
 
Theoremnegnegi 8000 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.)
 |-  A  e.  CC   =>    |-  -u -u A  =  A
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