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Theorem List for Intuitionistic Logic Explorer - 7601-7700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempncan3oi 7601 Subtraction and addition of equals. Almost but not exactly the same as pncan3i 7662 and pncan 7591, 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 7697. (Contributed by David A. Wheeler, 11-Oct-2018.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( ( A  +  B )  -  B )  =  A
 
Theoremmvrraddi 7602 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 7603 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 7604 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 7605 Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  CC  ->  ( A  -  0
 )  =  A )
 
Theoremnpncan 7606 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 7607 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 7608 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 7609 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 7610 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 7611 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 7612 Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  +  ( B  -  A ) )  =  0
 )
 
Theoremsubsub2 7613 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 7614 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 7615 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 7616 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 7617 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 7618 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 7619 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 7620 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 7621 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 7622 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 7623 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 7624 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 7625 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 7626 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 7627 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 7628 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 7629 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 7630 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 7631 Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
 |-  -u 0  =  0
 
Theoremnegid 7632 Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)
 |-  ( A  e.  CC  ->  ( A  +  -u A )  =  0 )
 
Theoremnegsub 7633 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 7634 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 7635 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 7636 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 7637 Negative contraposition law. (Contributed by NM, 9-May-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  =  B  <->  -u B  =  A ) )
 
Theoremnegcon2 7638 Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  =  -u B  <->  B  =  -u A ) )
 
Theoremnegeq0 7639 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 7640 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 7641 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 7642 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 7643 Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  -u ( A  +  B )  =  ( -u A  -  B ) )
 
Theoremnegsubdi2 7644 Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  -u ( A  -  B )  =  ( B  -  A ) )
 
Theoremneg2sub 7645 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 7646 Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.)
 |-  ( A  e.  RR  -> 
 -u A  e.  RR )
 
Theoremrenegcli 7647 Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 7646 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 7648 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 7649 Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B )  e.  RR )
 
Theoremnegreb 7650 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 7651 "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 7652 "Reverse" second Peano postulate analog for reals. (Contributed by NM, 6-Feb-2007.)
 |-  ( N  e.  RR  ->  ( N  -  1
 )  e.  RR )
 
Theoremnegcli 7653 Closure law for negative. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   =>    |-  -u A  e.  CC
 
Theoremnegidi 7654 Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   =>    |-  ( A  +  -u A )  =  0
 
Theoremnegnegi 7655 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 7656 Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   =>    |-  ( A  -  A )  =  0
 
Theoremsubid1i 7657 Identity law for subtraction. (Contributed by NM, 29-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  -  0 )  =  A
 
Theoremnegne0bi 7658 A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0 
 <->  -u A  =/=  0
 )
 
Theoremnegrebi 7659 The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( -u A  e.  RR  <->  A  e.  RR )
 
Theoremnegne0i 7660 The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  -u A  =/=  0
 
Theoremsubcli 7661 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 7662 Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  +  ( B  -  A ) )  =  B
 
Theoremnegsubi 7663 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 7664 Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  -  -u B )  =  ( A  +  B )
 
Theoremsubeq0i 7665 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 7666 Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( -u A  =  -u B 
 <->  A  =  B )
 
Theoremnegcon1i 7667 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( -u A  =  B  <->  -u B  =  A )
 
Theoremnegcon2i 7668 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  =  -u B 
 <->  B  =  -u A )
 
Theoremnegdii 7669 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 7670 Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  -u ( A  -  B )  =  ( -u A  +  B )
 
Theoremnegsubdi2i 7671 Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  -u ( A  -  B )  =  ( B  -  A )
 
Theoremsubaddi 7672 Relationship between subtraction and addition. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  -  B )  =  C  <->  ( B  +  C )  =  A )
 
Theoremsubadd2i 7673 Relationship between subtraction and addition. (Contributed by NM, 15-Dec-2006.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  -  B )  =  C  <->  ( C  +  B )  =  A )
 
Theoremsubaddrii 7674 Relationship between subtraction and addition. (Contributed by NM, 16-Dec-2006.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  ( B  +  C )  =  A   =>    |-  ( A  -  B )  =  C
 
Theoremsubsub23i 7675 Swap subtrahend and result of subtraction. (Contributed by NM, 7-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  -  B )  =  C  <->  ( A  -  C )  =  B )
 
Theoremaddsubassi 7676 Associative-type law for subtraction and addition. (Contributed by NM, 16-Sep-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  -  C )  =  ( A  +  ( B  -  C ) )
 
Theoremaddsubi 7677 Law for subtraction and addition. (Contributed by NM, 6-Aug-2003.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  -  C )  =  ( ( A  -  C )  +  B )
 
Theoremsubcani 7678 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  -  B )  =  ( A  -  C )  <->  B  =  C )
 
Theoremsubcan2i 7679 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 )
 
Theorempnncani 7680 Cancellation law for mixed addition and subtraction. (Contributed by NM, 14-Jan-2006.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  -  ( A  -  C ) )  =  ( B  +  C )
 
Theoremaddsub4i 7681 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 17-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   &    |-  D  e.  CC   =>    |-  ( ( A  +  B )  -  ( C  +  D )
 )  =  ( ( A  -  C )  +  ( B  -  D ) )
 
Theorem0reALT 7682 Alternate proof of 0re 7391. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  0  e.  RR
 
Theoremnegcld 7683 Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  -u A  e.  CC )
 
Theoremsubidd 7684 Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  -  A )  =  0 )
 
Theoremsubid1d 7685 Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  -  0 )  =  A )
 
Theoremnegidd 7686 Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  +  -u A )  =  0 )
 
Theoremnegnegd 7687 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  -u -u A  =  A )
 
Theoremnegeq0d 7688 A number is zero iff its negative is zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  =  0  <->  -u A  =  0 ) )
 
Theoremnegne0bd 7689 A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  =/=  0  <->  -u A  =/=  0
 ) )
 
Theoremnegcon1d 7690 Contraposition law for unary minus. Deduction form of negcon1 7637. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 -u A  =  B  <->  -u B  =  A )
 )
 
Theoremnegcon1ad 7691 Contraposition law for unary minus. One-way deduction form of negcon1 7637. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  -u A  =  B )   =>    |-  ( ph  ->  -u B  =  A )
 
Theoremneg11ad 7692 The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 7636. Generalization of neg11d 7708. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 -u A  =  -u B 
 <->  A  =  B ) )
 
Theoremnegned 7693 If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 7708. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  -u A  =/=  -u B )
 
Theoremnegne0d 7694 The negative of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  A  =/=  0 )   =>    |-  ( ph  ->  -u A  =/=  0 )
 
Theoremnegrebd 7695 The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  -u A  e.  RR )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremsubcld 7696 Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  -  B )  e.  CC )
 
Theorempncand 7697 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  +  B )  -  B )  =  A )
 
Theorempncan2d 7698 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  +  B )  -  A )  =  B )
 
Theorempncan3d 7699 Subtraction and addition of equals. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  +  ( B  -  A ) )  =  B )
 
Theoremnpcand 7700 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  -  B )  +  B )  =  A )
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