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Theorem List for Metamath Proof Explorer - 9001-9100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnegne0i 9001 The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.)

Theoremsubcli 9002 Closure law for subtraction. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theorempncan3i 9003 Subtraction and addition of equals. (Contributed by NM, 26-Nov-1994.)

Theoremnegsubi 9004 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.)

Theoremsubnegi 9005 Relationship between subtraction and negative. (Contributed by NM, 1-Dec-2005.)

Theoremsubeq0i 9006 If the difference between two numbers is zero, they are equal. (Contributed by NM, 8-May-1999.)

Theoremneg11i 9007 Negative is one-to-one. (Contributed by NM, 1-Aug-1999.)

Theoremnegcon1i 9008 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)

Theoremnegcon2i 9009 Negative contraposition law. (Contributed by NM, 25-Aug-1999.)

Theoremnegdii 9010 Distribution of negative over addition. (Contributed by NM, 28-Jul-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremnegsubdii 9011 Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.)

Theoremnegsubdi2i 9012 Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.)

Theoremsubaddi 9013 Relationship between subtraction and addition. (Contributed by NM, 26-Nov-1994.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremsubadd2i 9014 Relationship between subtraction and addition. (Contributed by NM, 15-Dec-2006.)

Theoremsubaddrii 9015 Relationship between subtraction and addition. (Contributed by NM, 16-Dec-2006.)

Theoremsubsub23i 9016 Swap subtrahend and result of subtraction. (Contributed by NM, 7-Oct-1999.)

Theoremaddsubassi 9017 Associative-type law for subtraction and addition. (Contributed by NM, 16-Sep-1999.)

Theoremaddsubi 9018 Law for subtraction and addition. (Contributed by NM, 6-Aug-2003.)

Theoremsubcani 9019 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)

Theoremsubcan2i 9020 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)

Theorempnncani 9021 Cancellation law for mixed addition and subtraction. (Contributed by NM, 14-Jan-2006.)

Theoremaddsub4i 9022 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 17-Oct-1999.)

Theorem0reALT 9023 0 is a real number. (Contributed by NM, 19-Feb-2005.) (Proof modification is discouraged.)

Theoremnegcld 9024 Closure law for negative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubidd 9025 Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubid1d 9026 Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegidd 9027 Addition of a number and its negative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegnegd 9028 A number is equal to the negative of its negative. Theorem I.4 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegeq0d 9029 A number is zero iff its negative is zero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegne0bd 9030 A number is nonzero iff its negative is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegcon1d 9031 Contraposition law for unary minus. Deduction form of negcon1 8979. (Contributed by David Moews, 28-Feb-2017.)

Theoremnegcon1ad 9032 Contraposition law for unary minus. One-way deduction form of negcon1 8979. (Contributed by David Moews, 28-Feb-2017.)

Theoremneg11ad 9033 The negatives of two complex numbers are equal iff they are equal. Deduction form of neg11 8978. Generalization of neg11d 9049. (Contributed by David Moews, 28-Feb-2017.)

Theoremnegned 9034 If two complex numbers are unequal, so are their negatives. Contrapositive of neg11d 9049. (Contributed by David Moews, 28-Feb-2017.)

Theoremnegne0d 9035 The negative of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegrebd 9036 The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremsubcld 9037 Closure law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempncand 9038 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempncan2d 9039 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempncan3d 9040 Subtraction and addition of equals. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnpcand 9041 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnncand 9042 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegsubd 9043 Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubnegd 9044 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubeq0d 9045 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubne0d 9046 Two unequal numbers have nonzero difference. (Contributed by Mario Carneiro, 1-Jan-2017.)

Theoremsubeq0ad 9047 The difference of two complex numbers is zero iff they are equal. Deduction form of subeq0 8953. Generalization of subeq0d 9045. (Contributed by David Moews, 28-Feb-2017.)

Theoremsubne0ad 9048 If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d 9046. Contrapositive of subeq0bd 9089. (Contributed by David Moews, 28-Feb-2017.)

Theoremneg11d 9049 If the difference between two numbers is zero, they are equal. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegdid 9050 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegdi2d 9051 Distribution of negative over addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegsubdid 9052 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnegsubdi2d 9053 Distribution of negative over subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremneg2subd 9054 Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubaddd 9055 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubadd2d 9056 Relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddsubassd 9057 Associative-type law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddsubd 9058 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubadd23d 9059 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddsub12d 9060 Commutative/associative law for addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnpncand 9061 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnppcand 9062 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnppcan2d 9063 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnppcan3d 9064 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubsubd 9065 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubsub2d 9066 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubsub3d 9067 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubsub4d 9068 Law for double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsub32d 9069 Swap the second and third terms in a double subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnncand 9070 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnncan1d 9071 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnnncan2d 9072 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremnpncan3d 9073 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempnpcand 9074 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempnpcan2d 9075 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorempnncand 9076 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremppncand 9077 Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubcand 9078 Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubcan2d 9079 Cancellation law for subtraction. (Contributed by Mario Carneiro, 22-Sep-2016.)

Theoremsubcanad 9080 Cancellation law for subtraction. Deduction form of subcan 8982. Generalization of subcand 9078. (Contributed by David Moews, 28-Feb-2017.)

Theoremsubneintrd 9081 Introducing subtraction on both sides of a statement of nonequality. Contrapositive of subcand 9078. (Contributed by David Moews, 28-Feb-2017.)

Theoremsubcan2ad 9082 Cancellation law for subtraction. Deduction form of subcan2 8952. Generalization of subcan2d 9079. (Contributed by David Moews, 28-Feb-2017.)

Theoremsubneintr2d 9083 Introducing subtraction on both sides of a statement of nonequality. Contrapositive of subcan2d 9079. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddsub4d 9084 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubadd4d 9085 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsub4d 9086 Rearrangement of 4 terms in a subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

Theorem2addsubd 9087 Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddsubeq4d 9088 Relation between sums and differences. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremsubeq0bd 9089 If two complex numbers are equal, their difference is zero. Consequence of subeq0ad 9047. Converse of subeq0d 9045. Contrapositive of subne0ad 9048. (Contributed by David Moews, 28-Feb-2017.)

Theoremrenegcld 9090 Closure law for negative of reals. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremresubcld 9091 Closure law for subtraction of reals. (Contributed by Mario Carneiro, 27-May-2016.)

5.3.3  Multiplication

Theoremmuladd 9092 Product of two sums. (Contributed by NM, 14-Jan-2006.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremsubdi 9093 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 18-Nov-2004.)

Theoremsubdir 9094 Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 30-Dec-2005.)

Theoremine0 9095 The imaginary unit is not zero. (Contributed by NM, 6-May-1999.)

Theoremmulneg1 9096 Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremmulneg2 9097 The product with a negative is the negative of the product. (Contributed by NM, 30-Jul-2004.)

Theoremmulneg12 9098 Swap the negative sign in a product. (Contributed by NM, 30-Jul-2004.)

Theoremmul2neg 9099 Product of two negatives. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 30-Jul-2004.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremsubmul2 9100 Convert a subtraction to addition using multiplication by a negative. (Contributed by NM, 2-Feb-2007.)

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