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

Theoremmul02i 9001 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.)

Theoremmul01i 9002 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddcomi 9003 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Theoremaddcani 9005 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 27-Oct-1999.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremaddcan2i 9006 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 14-May-2003.) (Revised by Scott Fenton, 3-Jan-2013.)

Theoremmul12i 9007 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremmul32i 9008 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)

Theoremmul4i 9009 Rearrangement of 4 factors. (Contributed by NM, 16-Feb-1995.)

Theoremmul02d 9010 Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul01d 9011 Multiplication by . Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddid2d 9013 is a left identity for addition. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcomd 9014 Addition commutes. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremaddcand 9015 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcan2d 9016 Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremaddcanad 9017 Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand 9015. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddcan2ad 9018 Cancelling a term on the right-hand side of a sum in an equality. Consequence of addcan2d 9016. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddneintrd 9019 Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad 9017. Consequence of addcand 9015. (Contributed by David Moews, 28-Feb-2017.)

Theoremaddneintr2d 9020 Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 9018. Consequence of addcan2d 9016. (Contributed by David Moews, 28-Feb-2017.)

Theoremmul12d 9021 Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul32d 9022 Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul31d 9023 Commutative/associative law. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremmul4d 9024 Rearrangement of 4 factors. (Contributed by Mario Carneiro, 27-May-2016.)

5.3  Real and complex numbers - basic operations

Theoremadd12 9025 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.)

Theoremadd32 9026 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.)

Theoremadd4 9027 Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremadd42 9028 Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005.)

Theoremadd12i 9029 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)

Theoremadd32i 9030 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)

Theoremadd4i 9031 Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.)

Theoremadd42i 9032 Rearrangement of 4 terms in a sum. (Contributed by NM, 22-Aug-1999.)

Theoremadd12d 9033 Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadd32d 9034 Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadd4d 9035 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)

Theoremadd42d 9036 Rearrangement of 4 terms in a sum. (Contributed by Mario Carneiro, 27-May-2016.)

5.3.2  Subtraction

Syntaxcmin 9037 Extend class notation to include subtraction.

Syntaxcneg 9038 Extend class notation to include unary minus. The symbol 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 () and subtraction cmin 9037 () to prevent syntax ambiguity. For example, looking at the syntax definition co 5858, if we used the same symbol then " " could mean either " " minus "", or it could represent the (meaningless) operation of classes " " and " " connected with "operation" "". On the other hand, " " is unambiguous.

Definitiondf-sub 9039* Define subtraction. Theorem subval 9043 shows it value (and describes how this definition works), theorem subaddi 9133 relates it to addition, and theorems subcli 9122 and resubcli 9109 prove its closure laws. (Contributed by NM, 26-Nov-1994.)

Definitiondf-neg 9040 Define the negative of a number (unary minus). We use different symbols for unary minus () and subtraction () to prevent syntax ambiguity. See cneg 9038 for a discussion of this. (Contributed by NM, 10-Feb-1995.)

Theorem0cnALT 9041 0 is a complex number. (Proved without referencing ax-1cn 8795. Compare 0cn 8831.) (Contributed by NM, 19-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnegeu 9042* 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.)

Theoremsubval 9043* Value of subtraction, which is the (unique) element such that . (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 2-Nov-2013.)

Theoremnegeq 9044 Equality theorem for negatives. (Contributed by NM, 10-Feb-1995.)

Theoremnegeqi 9045 Equality inference for negatives. (Contributed by NM, 14-Feb-1995.)

Theoremnegeqd 9046 Equality deduction for negatives. (Contributed by NM, 14-May-1999.)

Theoremnfnegd 9047 Deduction version of nfneg 9048. (Contributed by NM, 29-Feb-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremnfneg 9048 Bound-variable hypothesis builder for the negative of a complex number. (Contributed by NM, 12-Jun-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)

Theoremcsbnegg 9049 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.)

Theoremnegex 9050 A negative is a set. (Contributed by NM, 4-Apr-2005.)

Theoremsubcl 9051 Closure law for subtraction. (Contributed by NM, 10-May-1999.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremnegcl 9052 Closure law for negative. (Contributed by NM, 6-Aug-2003.)

Theoremsubf 9053 Subtraction is an operation on the complex numbers. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)

Theoremsubadd 9054 Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.)

Theoremsubadd2 9055 Relationship between subtraction and addition. (Contributed by Scott Fenton, 5-Jul-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theoremsubsub23 9056 Swap subtrahend and result of subtraction. (Contributed by NM, 14-Dec-2007.)

Theorempncan 9057 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theorempncan2 9058 Cancellation law for subtraction. (Contributed by NM, 17-Apr-2005.)

Theorempncan3 9059 Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.)

Theoremnpcan 9060 Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremaddsubass 9061 Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremaddsub 9062 Law for addition and subtraction. (Contributed by NM, 19-Aug-2001.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Theoremsubadd23 9063 Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007.)

Theoremaddsub12 9064 Commutative/associative law for addition and subtraction. (Contributed by NM, 8-Feb-2005.)

Theoremaddsubeq4 9066 Relation between sums and differences. (Contributed by Jeff Madsen, 17-Jun-2010.)

Theoremsubid 9067 Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremsubid1 9068 Identity law for subtraction. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnpncan 9069 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)

Theoremnppcan 9070 Cancellation law for subtraction. (Contributed by NM, 1-Sep-2005.)

Theoremnppcan3 9071 Cancellation law for subtraction. (Contributed by Mario Carneiro, 14-Sep-2015.)

Theoremsubcan2 9072 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.)

Theoremsubeq0 9073 If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999.)

Theoremnpncan2 9074 Cancellation law for subtraction. (Contributed by Scott Fenton, 21-Jun-2013.)

Theoremsubsub2 9075 Law for double subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnncan 9076 Cancellation law for subtraction. (Contributed by NM, 21-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremsubsub 9077 Law for double subtraction. (Contributed by NM, 13-May-2004.)

Theoremnppcan2 9078 Cancellation law for subtraction. (Contributed by NM, 29-Sep-2005.)

Theoremsubsub3 9079 Law for double subtraction. (Contributed by NM, 27-Jul-2005.)

Theoremsubsub4 9080 Law for double subtraction. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremsub32 9081 Swap the second and third terms in a double subtraction. (Contributed by NM, 19-Aug-2005.)

Theoremnnncan 9082 Cancellation law for subtraction. (Contributed by NM, 4-Sep-2005.)

Theoremnnncan1 9083 Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Theoremnnncan2 9084 Cancellation law for subtraction. (Contributed by NM, 1-Oct-2005.)

Theoremnpncan3 9085 Cancellation law for subtraction. (Contributed by Scott Fenton, 23-Jun-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Theorempnpcan 9086 Cancellation law for mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theorempnpcan2 9087 Cancellation law for mixed addition and subtraction. (Contributed by Scott Fenton, 9-Jun-2006.)

Theorempnncan 9088 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremppncan 9089 Cancellation law for mixed addition and subtraction. (Contributed by NM, 30-Jun-2005.)

Theoremaddsub4 9090 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.)

Theoremsubadd4 9091 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 24-Aug-2006.)

Theoremsub4 9092 Rearrangement of 4 terms in a subtraction. (Contributed by NM, 23-Nov-2007.)

Theoremneg0 9093 Minus 0 equals 0. (Contributed by NM, 17-Jan-1997.)

Theoremnegid 9094 Addition of a number and its negative. (Contributed by NM, 14-Mar-2005.)

Theoremnegsub 9095 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.)

Theoremsubneg 9096 Relationship between subtraction and negative. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnegneg 9097 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.)

Theoremneg11 9098 Negative is one-to-one. (Contributed by NM, 8-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)

Theoremnegcon1 9099 Negative contraposition law. (Contributed by NM, 9-May-2004.)

Theoremnegcon2 9100 Negative contraposition law. (Contributed by NM, 14-Nov-2004.)

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