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Theorem List for Intuitionistic Logic Explorer - 7701-7800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempnpcan2 7701 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 7702 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 7703 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 7704 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 7705 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 7706 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 7707 Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)
 |-  -u 0  =  0
 
Theoremnegid 7708 Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)
 |-  ( A  e.  CC  ->  ( A  +  -u A )  =  0 )
 
Theoremnegsub 7709 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 7710 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 7711 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 7712 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 7713 Negative contraposition law. (Contributed by NM, 9-May-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( -u A  =  B  <->  -u B  =  A ) )
 
Theoremnegcon2 7714 Negative contraposition law. (Contributed by NM, 14-Nov-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  =  -u B  <->  B  =  -u A ) )
 
Theoremnegeq0 7715 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 7716 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 7717 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 7718 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 7719 Distribution of negative over addition. (Contributed by NM, 1-Jan-2006.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  -u ( A  +  B )  =  ( -u A  -  B ) )
 
Theoremnegsubdi2 7720 Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  -u ( A  -  B )  =  ( B  -  A ) )
 
Theoremneg2sub 7721 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 7722 Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.)
 |-  ( A  e.  RR  -> 
 -u A  e.  RR )
 
Theoremrenegcli 7723 Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 7722 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 7724 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 7725 Closure law for subtraction of reals. (Contributed by NM, 20-Jan-1997.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  -  B )  e.  RR )
 
Theoremnegreb 7726 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 7727 "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 7728 "Reverse" second Peano postulate analog for reals. (Contributed by NM, 6-Feb-2007.)
 |-  ( N  e.  RR  ->  ( N  -  1
 )  e.  RR )
 
Theoremnegcli 7729 Closure law for negative. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   =>    |-  -u A  e.  CC
 
Theoremnegidi 7730 Addition of a number and its negative. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   =>    |-  ( A  +  -u A )  =  0
 
Theoremnegnegi 7731 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 7732 Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   =>    |-  ( A  -  A )  =  0
 
Theoremsubid1i 7733 Identity law for subtraction. (Contributed by NM, 29-May-1999.)
 |-  A  e.  CC   =>    |-  ( A  -  0 )  =  A
 
Theoremnegne0bi 7734 A number is nonzero iff its negative is nonzero. (Contributed by NM, 10-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( A  =/=  0 
 <->  -u A  =/=  0
 )
 
Theoremnegrebi 7735 The negative of a real is real. (Contributed by NM, 11-Aug-1999.)
 |-  A  e.  CC   =>    |-  ( -u A  e.  RR  <->  A  e.  RR )
 
Theoremnegne0i 7736 The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)
 |-  A  e.  CC   &    |-  A  =/=  0   =>    |-  -u A  =/=  0
 
Theoremsubcli 7737 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 7738 Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  +  ( B  -  A ) )  =  B
 
Theoremnegsubi 7739 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 7740 Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  -  -u B )  =  ( A  +  B )
 
Theoremsubeq0i 7741 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 7742 Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( -u A  =  -u B 
 <->  A  =  B )
 
Theoremnegcon1i 7743 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( -u A  =  B  <->  -u B  =  A )
 
Theoremnegcon2i 7744 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  =  -u B 
 <->  B  =  -u A )
 
Theoremnegdii 7745 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 7746 Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  -u ( A  -  B )  =  ( -u A  +  B )
 
Theoremnegsubdi2i 7747 Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  -u ( A  -  B )  =  ( B  -  A )
 
Theoremsubaddi 7748 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 7749 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 7750 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 7751 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 7752 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 7753 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 7754 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 7755 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 7756 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 7757 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 7758 Alternate proof of 0re 7467. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  0  e.  RR
 
Theoremnegcld 7759 Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  -u A  e.  CC )
 
Theoremsubidd 7760 Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  -  A )  =  0 )
 
Theoremsubid1d 7761 Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  -  0 )  =  A )
 
Theoremnegidd 7762 Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  +  -u A )  =  0 )
 
Theoremnegnegd 7763 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 7764 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 7765 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 7766 Contraposition law for unary minus. Deduction form of negcon1 7713. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 -u A  =  B  <->  -u B  =  A )
 )
 
Theoremnegcon1ad 7767 Contraposition law for unary minus. One-way deduction form of negcon1 7713. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  -u A  =  B )   =>    |-  ( ph  ->  -u B  =  A )
 
Theoremneg11ad 7768 The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 7712. Generalization of neg11d 7784. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  (
 -u A  =  -u B 
 <->  A  =  B ) )
 
Theoremnegned 7769 If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 7784. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  -u A  =/=  -u B )
 
Theoremnegne0d 7770 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 7771 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 7772 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 7773 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 7774 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 7775 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 7776 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  -  B )  +  B )  =  A )
 
Theoremnncand 7777 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  -  ( A  -  B ) )  =  B )
 
Theoremnegsubd 7778 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  +  -u B )  =  ( A  -  B ) )
 
Theoremsubnegd 7779 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  -  -u B )  =  ( A  +  B ) )
 
Theoremsubeq0d 7780 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A  -  B )  =  0
 )   =>    |-  ( ph  ->  A  =  B )
 
Theoremsubne0d 7781 Two unequal numbers have nonzero difference. See also subap0d 8093 which is the same thing for apartness rather than negated equality. (Contributed by Mario Carneiro, 1-Jan-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ( A  -  B )  =/=  0 )
 
Theoremsubeq0ad 7782 The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 7687. Generalization of subeq0d 7780. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( ( A  -  B )  =  0  <->  A  =  B ) )
 
Theoremsubne0ad 7783 If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d 7781. Contrapositive of subeq0bd 7836. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  ( A  -  B )  =/=  0
 )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremneg11d 7784 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  -u A  =  -u B )   =>    |-  ( ph  ->  A  =  B )
 
Theoremnegdid 7785 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  -u ( A  +  B )  =  ( -u A  +  -u B ) )
 
Theoremnegdi2d 7786 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  -u ( A  +  B )  =  ( -u A  -  B ) )
 
Theoremnegsubdid 7787 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  -u ( A  -  B )  =  ( -u A  +  B ) )
 
Theoremnegsubdi2d 7788 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 7789 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 7790 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 7791 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 7792 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 7793 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 7794 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 7795 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 7796 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 7797 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 7798 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 7799 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 7800 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 ) )
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