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Theorem List for Metamath Proof Explorer - 9001-9100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremadd12d 9001 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 9002 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 9003 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 9004 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 )
 ) )
 
5.3.2  Subtraction
 
Syntaxcmin 9005 Extend class notation to include subtraction.
 class  -
 
Syntaxcneg 9006 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 9005 ( -) to prevent syntax ambiguity. For example, looking at the syntax definition co 5792, 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 9007* Define subtraction. Theorem subval 9011 shows it value (and describes how this definition works), theorem subaddi 9101 relates it to addition, and theorems subcli 9090 and resubcli 9077 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 9008 Define the negative of a number (unary minus). We use different symbols for unary minus ( -u) and subtraction ( -) to prevent syntax ambiguity. See cneg 9006 for a discussion of this. (Contributed by NM, 10-Feb-1995.)
 |-  -u A  =  (
 0  -  A )
 
Theorem0cnALT 9009 0 is a complex number. (Proved without referencing ax-1cn 8763. Compare 0cn 8799.) (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 9010* 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 9011* 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 9012 Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)
 |-  ( A  =  B  -> 
 -u A  =  -u B )
 
Theoremnegeqi 9013 Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)
 |-  A  =  B   =>    |-  -u A  =  -u B
 
Theoremnegeqd 9014 Equality deduction for negatives. (Contributed by NM, 14-May-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  -u A  =  -u B )
 
Theoremnfnegd 9015 Deduction version of nfneg 9016. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/_ x -u A )
 
Theoremnfneg 9016 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 9017 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 )
 
Theoremnegex 9018 A negative is a set. (Contributed by NM, 4-Apr-2005.)
 |-  -u A  e.  _V
 
Theoremsubcl 9019 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 9020 Closure law for negative. (Contributed by NM, 6-Aug-2003.)
 |-  ( A  e.  CC  -> 
 -u A  e.  CC )
 
Theoremsubf 9021 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 9022 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 9023 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 9024 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 9025 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 9026 Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  -  A )  =  B )
 
Theorempncan3 9027 Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  ( B  -  A ) )  =  B )
 
Theoremnpcan 9028 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 9029 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 9030 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 9031 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 9032 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 9033 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 9034 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 ) ) )
 
Theoremsubid 9035 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 9036 Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  ( A  -  0
 )  =  A )
 
Theoremnpncan 9037 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 9038 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 ) )
 
Theoremnppcan3 9039 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 9040 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 9041 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 9042 Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  +  ( B  -  A ) )  =  0
 )
 
Theoremsubsub2 9043 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 9044 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 9045 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 9046 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 9047 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 9048 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 9049 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 9050 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 9051 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 9052 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 9053 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 9054 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 9055 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 9056 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 9057 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 9058 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 9059 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 9060 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 9061 Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
 |-  -u 0  =  0
 
Theoremnegid 9062 Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)
 |-  ( A  e.  CC  ->  ( A  +  -u A )  =  0 )
 
Theoremnegsub 9063 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 9064 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 9065 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 9066 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 9067 Negative contraposition law. (Contributed by NM, 9-May-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  =  B  <->  -u B  =  A ) )
 
Theoremnegcon2 9068 Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  =  -u B  <->  B  =  -u A ) )
 
Theoremnegeq0 9069 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 9070 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 9071 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 9072 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 9073 Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  -u ( A  +  B )  =  ( -u A  -  B ) )
 
Theoremnegsubdi2 9074 Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  -u ( A  -  B )  =  ( B  -  A ) )
 
Theoremneg2sub 9075 Relationship between subtraction and negative. (Contributed by Paul Chapman, 8-Oct-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  -  -u B )  =  ( B  -  A ) )
 
Theoremrenegcli 9076 Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 9078 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 9077 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
 
Theoremrenegcl 9078 Closure law for negative of reals. The weak deduction theorem dedth 3580 is used to convert hypothesis of the inference (deduction) form of this theorem, renegcli 9076, to an antecedent. (Contributed by NM, 20-Jan-1997.) (Proof modification is discouraged.)
 |-  ( A  e.  RR  -> 
 -u A  e.  RR )
 
Theoremresubcl 9079 Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B )  e.  RR )
 
Theoremnegreb 9080 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 ) )
 
Theorempeano2rem 9081 "Reverse" second Peano postulate analog for reals. (Contributed by NM, 6-Feb-2007.)
 |-  ( N  e.  RR  ->  ( N  -  1
 )  e.  RR )
 
Theoremnegcli 9082 Closure law for negative. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   =>    |-  -u A  e.  CC
 
Theoremnegidi 9083 Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   =>    |-  ( A  +  -u A )  =  0
 
Theoremnegnegi 9084 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
 
Theoremsubidi 9085 Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   =>    |-  ( A  -  A )  =  0
 
Theoremsubid1i 9086 Identity law for subtraction. (Contributed by NM, 29-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  -  0 )  =  A
 
Theoremnegne0bi 9087 A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0 
 <->  -u A  =/=  0
 )
 
Theoremnegrebi 9088 The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( -u A  e.  RR  <->  A  e.  RR )
 
Theoremnegne0i 9089 The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  -u A  =/=  0
 
Theoremsubcli 9090 Closure law for subtraction. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  -  B )  e.  CC
 
Theorempncan3i 9091 Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  +  ( B  -  A ) )  =  B
 
Theoremnegsubi 9092 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  +  -u B )  =  ( A  -  B )
 
Theoremsubnegi 9093 Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  -  -u B )  =  ( A  +  B )
 
Theoremsubeq0i 9094 If the difference between two numbers is zero, they are equal. (Contributed by NM, 8-May-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( A  -  B )  =  0  <->  A  =  B )
 
Theoremneg11i 9095 Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( -u A  =  -u B 
 <->  A  =  B )
 
Theoremnegcon1i 9096 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( -u A  =  B  <->  -u B  =  A )
 
Theoremnegcon2i 9097 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  =  -u B 
 <->  B  =  -u A )
 
Theoremnegdii 9098 Distribution of negative over addition. (Contributed by NM, 28-Jul-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  -u ( A  +  B )  =  ( -u A  +  -u B )
 
Theoremnegsubdii 9099 Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  -u ( A  -  B )  =  ( -u A  +  B )
 
Theoremnegsubdi2i 9100 Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  -u ( A  -  B )  =  ( B  -  A )
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